A gamesemantic model of computation
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Abstract
The present paper introduces a novel notion of ‘(effective) computability,’ called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church–Turing computability), noninductive, nonaxiomatic manner and shows, as a main technical achievement, that viable strategies are Turing complete. Consequently, we have given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higherorder computation, in a more abstract fashion. As immediate corollaries, some of the wellknown theorems in computability theory such as the smn theorem and the first recursion theorem are generalized. Notably, our gamesemantic framework distinguishes highlevel computational processes that operate directly on mathematical objects such as natural numbers (not on their symbolic representations) and their symbolic implementations that define their ‘computability,’ which sheds new light on the very concept of computation. This work is intended to be a stepping stone toward a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.
Mathematics Subject Classification
Primary 03D10 Secondary 68Q051 Introduction
The present work introduces an intrinsic, noninductive, nonaxiomatic formulation of ‘(effectively) computable’ strategies in game semantics and proves as a main theorem that they are Turing complete. This result leads to a novel mathematical foundation of computation beyond classical computation, e.g., higherorder computation, that distinguishes highlevel and lowlevel computational processes, where the latter defines ‘effective computability’ of the former.
Convention
We shall informally use computational processes and algorithms almost as synonyms of computation, but they put more emphasis on ‘processes.’
1.1 Search for Turing machines beyond classical computation
Turing machines (TMs) introduced in the classic work [67] by Alan Turing have been widely accepted as giving a reasonable, highly convincing definition of ‘effectivity’ or ‘(effective) computability’ of (partial) functions on (finite sequences of) natural numbers, which let us call in this paper recursiveness, classical computability or Church–Turing computability, in a mathematically rigorous manner. This is because ‘computability’ of a function intuitively means the very existence of an algorithm that implements the function’s input/output behavior, and TMs are none other than a mathematical formulation of this informal concept.
In mathematics, however, there are various kinds of nonclassical computation, where by classical computation we mean what merely implements a function on natural numbers, since there are a variety of mathematical objects other than natural numbers, for which TMs have certain limitations.
As an example of nonclassical computation, consider higherorder computation [50], i.e., computation that takes (as an input) or produces (as an output) another computation, which abounds in mathematics, e.g., quantification in mathematical logic, differentiation in analysis or simply an application \((f, a) \mapsto f(a)\) of a function \(f : A \rightarrow B\) to an argument \(a \in A\). However, TMs cannot capture higherorder computation in a natural or systematic fashion. In fact, although TMs may compute on symbols that encode other TMs, e.g., consider universal TMs [35, 48, 64], they cannot compute on ‘external behavior’ of an input computation, which implies that the input is limited to a recursive one (to be encoded); however, it makes perfect sense to consider computation on nonrecursive objects such as nonrecursive real numbers. For this point, one may argue that oracle TMs [47, 64] may treat an input computation as an oracle, a blackboxlike computation that does not have to be recursive; however, it is like a function (rather than a computational process) that computes just in a single step, which appears conceptually mysterious and technically ad hoc. (Another approach is to give an input computation as a potentially infinite sequence of symbols on the input tape [69], but it may be criticized in a similar manner.)
On the other hand, most of the other models of higherorder computation are, unlike TMs, either syntactic (such as \(\lambda \)calculi and programming languages [9, 50]), inductive and/or axiomatic (such as Kleene’s schemata S1S9 [40, 41]) or extrinsic (i.e., reducing to classical computation by encoding whose ‘effectivity’ is usually left imprecise [16, 50]), thus lacking the semantic, direct, intrinsic nature of TMs. Also, unlike classical computability, a confluence between different notions of higherorder computability has been rarely established [50]. For this problem, it would be a key step to establish a TMslike model of higherorder computation since it may tell us which notion of higherorder computability is a ‘correct’ one.
1.2 Search for mathematics of highlevel computational processes
Perhaps more crucially than the limitation for nonclassical computation mentioned above, one may argue that TMs are not appropriate as mathematics of computational processes since computational steps of TMs are often too lowlevel to see what they are supposed to compute. In other words, we need mathematics of highlevel computational processes that gives a ‘birdseyeview’ of lowlevel computational processes.^{1} Also, what TMs formulate is essentially symbol manipulations; however, the content of computation on mathematical, semantic, nonsymbolic objects seems completely independent of its symbolic representation, e.g., consider a process (not a function) to add numbers or to take the union of sets.
Therefore, it would be rather appropriate, at least from the conceptual and the mathematical points of view, to formulate such highlevel computational processes in a more abstract, in particular syntaxindependent, manner, in order to explain lowlevel computational processes, and then regard the latter as executable symbolic implementations of the former.
1.3 Our research problem: mathematics of computational processes
To summarize, it would be reasonable and meaningful from both of the conceptual and the mathematical viewpoints to develop mathematics of abstract (in particular syntaxindependent), highlevel computational processes as well as executable, lowlevel ones beyond classical computation such that the former is defined to be ‘effectively computable’ if it is implementable or representable by the latter.
We address this problem in the present paper. However, since there are so many kinds of computation, e.g., parallel, concurrent, probabilistic, nondeterministic and quantum, as the first step, this paper focuses on a certain kind of higherorder, sequential (i.e., at most one computational step may be performed at a time) computation, which is based on (sequential) game semantics^{2} introduced below.... we should have achieved a mathematical model of computation, perhaps highly abstract in contrast with the concrete nature of paper and register machines, but such that programming languages are merely executable fragment of the theory ...[52]
1.4 Game semantics
Game semantics (of computation) [3, 6, 37] is a particular kind of denotational semantics of programming languages [8, 32, 70], in which types and terms are modeled as games and strategies (whose definitions are given in Sect. 2), respectively. Historically, having its roots in ‘gamesbased’ approaches in mathematical logic to capture validity [19, 59], higherorder computability [21, 42, 43, 44, 45, 46] and proofs in linear logic [5, 11, 36], combined with ideas from sequential algorithms [10], process calculi [34, 53] and geometry of interaction [24, 25, 26, 28, 29, 30] in theoretical computer science, several variants of game semantics in its modern form were developed in the early 1990s to give the first syntaxindependent characterization of the higherorder programming language PCF [7, 38, 55]; since then, a variety of games and strategies have been proposed to model various programming features [6].
An advantage of game semantics is this flexibility: It models a wide range of programming languages by simply varying constraints on strategies [6], which enables one to systematically compare and relate different languages ignoring syntactic details. Also, as full completeness and full abstraction results [15] in the literature have demonstrated, game semantics in general has an appropriate degree of abstraction (and thus it has a good potential to be mathematics of highlevel computational processes). Finally, yet another strong point of game semantics is its conceptual naturality: It interprets syntax as ‘dynamic interactions’ between the participants of games, providing a computational, intensional explanation of syntax in a natural, intuitive (yet mathematically precise) manner. Informally, one can imagine that games provide a highlevel description of interactive computation between a TM and an oracle, and therefore, they seem appropriate as an approach to the research problem defined in Sect. 1.3. Note that such an intensional nature stands in sharp contrast to the traditional domaintheoretic denotational semantics [8] which, e.g., cannot capture sequentiality of PCF (but the game models [7, 38, 55] can).
In the following, let us give a brief, informal introduction to games and strategies (as defined in [6]) in order to sketch the main idea of the present paper.
A game, roughly, is a certain kind of a rooted forest whose branches represent possible ‘developments’ or (valid) positions of a ‘game in the usual sense’ (such as chess, poker, etc.). Moves of a game are nodes of the game, where some moves are distinguished and called initial; only initial moves can be the first element (or occurrence) of a position of the game. Plays of a game are increasing sequences \(\varvec{\epsilon }, m_1, m_1 m_2, \ldots \) of positions of the game, where \(\varvec{\epsilon }\) is the empty sequence. For our purpose, it suffices to focus on rather standard sequential (as opposed to concurrent [2]) and unpolarized (as opposed to polarized [49]) games played by two participants, Player (P), who represents a ‘computational agent,’ and Opponent (O), who represents a ‘computational environment,’ in each of which O always starts a play (i.e., unpolarized), and then they alternately and separately (i.e., sequential) perform moves allowed by the rules of the game. Strictly speaking, a position of each game is not just a sequence of moves: Each occurrence m of O’s or O (resp. P’s or P) noninitial move in a position points to a previous occurrence \(m'\) of P (resp. O) move in the position, representing that m is performed specifically as a response to \(m'\). A strategy on a game, on the other hand, is what tells P which move (together with a pointer) she should make at each of her turns in the game. Hence, a game semantics \(\llbracket \_ \rrbracket _{\mathcal {G}}\) of a programming language \(\mathcal {L}\) interprets a type \(\mathsf {A}\) of \(\mathcal {L}\) as a game \(\llbracket \mathsf {A} \rrbracket _{\mathcal {G}}\) that specifies possible plays between P and O, and a term \(\mathsf {M : A}\)^{3} of \(\mathcal {L}\) as a strategy \(\llbracket \mathsf {M} \rrbracket _{\mathcal {G}}\) that describes for P how to play on \(\llbracket \mathsf {A} \rrbracket _{\mathcal {G}}\); an execution of the term \(\mathsf {M}\) is then modeled as a play of \(\llbracket \mathsf {A} \rrbracket _{\mathcal {G}}\) in which P follows \(\llbracket \mathsf {M} \rrbracket _{\mathcal {G}}\).
in which a play starts with O’s question q (‘What is your number?’) and ends with P’s answer \(n \in \mathbb {N}\) (‘My number is n!’), where \(\mathbb {N}\) is the set of all natural numbers, and n points to q (though this pointer is omitted in the diagram). A strategy \(\underline{10}\) on N, for instance, that corresponds to \(10 \in \mathbb {N}\) can be represented by the map \(q \mapsto 10\) equipped with a pointer from 10 to q (though it is the only choice). In the following, the pointers of most strategies are obvious, and thus, we often omit them.
Next, a fundamental construction ! on games, called exponential, is basically the countably infinite iteration of \(\otimes \), i.e., \(!A {\mathop {=}\limits ^{\mathrm {df. }}} A \otimes A \otimes \dots \) for each game A, where the ‘tag’ for each copy of A is typically given as \((\_, i)\), where \(i \in \mathbb {N}\).
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The first occurrence of the position must be a move of B;
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A change of ABparity in the position must be made by P;
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Each occurrence of an initial move (called an initial occurrence) of A points to an initial occurrence of B.
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O’s question \(q^{[1]}\) for an output (‘What is your output?’);
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P’s question \(q^{[0]}\) for an input (‘Wait, what is your input?’);
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O’s answer, say, \(n^{[0]}\), to \(q^{[0]}\) (‘OK, here is an input n.’);
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P’s answer, say, \(m^{[1]}\), to \(q^{[1]}\) (‘Alright, the output is then m.’).
Let us remark here that the following play, which corresponds to a constant linear function that maps \(x \mapsto m\) for all \(x \in \mathbb {N}\), is also possible: \(\varvec{\epsilon }, q^{[1]}, q^{[1]}m^{[1]}\). Thus, strictly speaking, \(A \multimap B\) is the game of affine functions from A to B, but we follow the standard convention to call \(\multimap \) linear implication.
Another construction & on games, called product, is similar to yet simpler than tensor: A position \(\varvec{s}\) of the product \( A \& B\) of A and B is either a position \(\varvec{t}^{[0]}\) of \(A^{[0]}\) or a position \(\varvec{u}^{[1]}\) of \(B^{[1]}\). It is the product in the category \(\mathcal {G}\) of games and strategies, e.g., there is the pairing\( \langle \sigma , \tau \rangle : \ !C \multimap A \& B\) of given strategies \(\sigma : \ !C \multimap A\) and \(\tau : \ !C \multimap B\) that plays as \(\sigma \) (resp. \(\tau \)) if O initiates a play by a move of A (resp. B). Clearly, we may generalize product and pairing to nary ones for any \(n \in \mathbb {N}\).
These four constructions \(\otimes \), !, \(\multimap \) and & come from the corresponding ones in linear logic [5, 27]. Thus, in particular, the usual implication (or the function space) \(\Rightarrow \) is recovered by Girard translation [27]: \(A \Rightarrow B {\mathop {=}\limits ^{\mathrm {df. }}} \ !A \multimap B\).
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O’s question q for an output (‘What is your output?’);
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P’s question (q, j) for an input function (‘Wait, your first output please!’);
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O’s question ((q, i), j) for an input (‘What is your first input then?’);
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P’s answer, say, ((n, i), j), to ((q, i), j) (‘Here is my first input n.’);
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O’s answer, say, (m, j), to (q, j) (‘OK, then here is my first output m.’);
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P’s question \((q, j')\) for an input function (‘Your second output please!’);
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O’s question \(((q, i'), j')\) for an input (‘What is your second input then?’);
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P’s answer, say, \(((n', i'), j')\), to \(((q, i'), j')\) (‘Here is my second input \(n'\).’);
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O’s answer, say, \((m', j')\), to \((q, j')\) (‘OK, then here is my second output \(m'\).’);
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P’s answer, say, l, to q (‘Alright, my output is then l.’).
1.5 Toward a gamesemantic model of computation
As seen in the examples given above, games and strategies capture higherorder computation in an abstract, conceptually natural fashion, where O plays the role of an oracle as part of the formalization. Note also that P computes on ‘external behavior’ of O, and thus O’s computation does not have to be recursive at all. Thus, one may expect that games and strategies would be appropriate as mathematics of highlevel computational processes, solving the research problem of Sect. 1.3.
However, conventional games and strategies have never been formulated as a mathematical model of computation (in the sense of TMs); rather, the primary focus of the field has been full abstraction [8, 15], i.e., to characterize observational equivalences in syntax. In other words, game semantics has not been concerned that much with stepbystep processes in computation or their ‘effective computability,’ and it has been identifying programs with the same value [32, 70].
For instance, strategies on the game \(N \Rightarrow N\) typically play by q. (q, i) . (n, i) .m, where \(n, m, i \in \mathbb {N}\), as described above, and so they are essentially functions that map \(n \mapsto m\); in particular, it is not formulated at all how they calculate the fourth move m from the third one (n, i).^{5} As a consequence, ‘effective computability’ in game semantics has been extrinsic: A strategy has been defined to be ‘effective’ or recursive if it is representable by a partial recursive function [7, 20, 38].
This situation is in a sense frustrating since games and strategies seem to have a good potential to give a semantic, intrinsic (i.e., without recourse to an established model of computation), nonaxiomatic, noninductive formulation of higherorder computation, but they have not taken advantage of this potential.
For the potential, we have decided to employ games and strategies as our basic mathematical framework and extend them to give mathematics of computational processes in the sense described in Sect. 1.3. For this aim, we shall first refine the category \(\mathcal {G}\) of games and strategies in such a way that accommodates stepbystep processes in computation, and then define their ‘effectivity’ in terms of their atomic computational steps. Fortunately, there is already the bicategory \(\mathcal {DG}\) of dynamic games and strategies [71], which addresses the first point.
1.6 Dynamic games and strategies
In the literature, there are several game models [13, 18, 31, 56] that exhibit stepbystep processes in computation to serve as a tool for program verification and analysis (the work [17, 23] may be called ‘intensional game semantics,’ but they rather keep track of costs in computation, not computational steps themselves). However, these variants of games and strategies are just conventional ones, and consequently, such stepbystep processes have no official status in their categories.
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O’s question \(q^{[3]}\) for an output in \(!N^{[0]} \multimap N^{[3]}\) (‘What is your output?’);
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P’s question Open image in new window by \( double \) for an input in \(!N^{[2]} \multimap N^{[3]}\) (‘Wait, what is your input?’);
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Open image in new window in turn triggers the question Open image in new window for an output in \(!N^{[0]} \multimap \ !N^{[1]}\) (‘What is your output?’);
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P’s question \((q, \langle 0, 0 \rangle )^{[0]}\) by \( succ ^{\dagger }\) for an input in \(!N^{[0]} \multimap \ !N^{[1]}\) (‘Wait, what is your input?’);
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O’s answer, say, \((n, \langle 0, 0 \rangle )^{[0]}\), to the question \((q, \langle 0, 0 \rangle )^{[0]}\) in \(!N^{[0]} \multimap \ !N^{[3]}\) (‘Here is an input n.’);
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P’s answer Open image in new window to the question Open image in new window by \( succ ^{\dagger }\) in \(!N^{[0]} \multimap \ !N^{[1]}\) (‘The output is then \(n+1\).’);
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Open image in new window in turn triggers the answer Open image in new window to the question Open image in new window in \(!N^{[2]} \multimap N^{[3]}\) (‘Here is the input \(n+1\).’);
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P’s answer \(2 \cdot (n+1)^{[3]}\) to the initial question \(q^{[3]}\) by \( double \) in \(!N^{[0]} \multimap N^{[3]}\) (‘The output is then \(2 \cdot (n+1)\)!’).
Nevertheless, the present author and Samson Abramsky have introduced a novel, dynamic variant of games and strategies that systematically model dynamics and intensionality of computation, and also studied their algebraic structures [71]. In contrast to the previous work mentioned above, dynamic strategies themselves embody stepbystep processes in computation by retaining intermediate occurrences of moves, and composition of them is parallel composition without hiding. In addition, the categorical structure of existing game semantics is not lost but rather refined by the cartesian closed bicategory [57] \(\mathcal {DG}\) of dynamic games and strategies, forming a categorical ‘universe’ of highlevel computational processes.
1.7 Viable strategies
Not surprisingly, perhaps, this problem has turned out to be challenging, and the main technical achievement of the present paper is to give a positive answer to it.Is there any intrinsic, noninductive, nonaxiomatic notion of ‘effectivity’ of dynamic strategies that is Turing complete?
As already mentioned, our solution is to give lowlevel computational processes (which are clearly ‘executable’) in order to define ‘effectivity’ of dynamic strategies (or highlevel computational processes). This is achieved roughly as follows.
Remark
The concepts introduced below make sense for conventional (i.e., nondynamic) games and strategies too, but they do not give rise to a Turing complete model of computation for composition of conventional strategies does not preserve our notion of ‘effectivity’ or viability as we shall see.
Next, as we shall see, dynamic strategies modeling PCF only need to refer to at most three moves in the history of previous moves which may be ‘effectively’ identified by pointers (specifically the last three moves in the Pview [6, 38]; see Definition 16). Thus, it may seem at first glance that finitary dynamic strategies in the following sense suffice: A strategy is finitary if its representation by a partial function [38, 51] that assigns the next Pmove to previous moves, called its table, is finite. However, it is not the case: Finitary strategies cannot handle unboundedly many manipulations of ‘tags’ for exponential (more precisely, manipulations such that the length of input or output ‘tags’ is unbounded), but such manipulations seem to be necessary for Turing completeness, e.g., a strategy that models primitive recursion or minimization has to interact with input strategies unboundedly many times, and thus, it must handle unboundedly many ‘tags.’
Then, the main idea of our solution is to define a strategy to be viable if its table is ‘describable’ by a finitary strategy. To state it more precisely, let us note that there is the terminal gameT which has only the empty sequence \(\varvec{\epsilon }\) as a position, and each game G is identical up to ‘tags’ to the implication \(T \Rightarrow G\). Hence, we may regard strategies \(\sigma : G\) as the one on the implication \(T \Rightarrow G\) up to ‘tags,’ and vice versa; we shall not take the trouble of distinguishing the two. Also, we define for each move \(m = [m']_{\varvec{e}}\) of a game G, where \(\varvec{e} = e_1 . e_2 \dots e_{k}\) is a unary representation of the ‘tag’ for exponential on m, the strategy \(\underline{m}\) on a suitable game \(\mathcal {G}(M_G)\) that plays as \(\hat{q} . \mathsf {m'} . q . \mathsf {e_1} . q . \mathsf {e_2} \dots q . \mathsf {e_{k}} . q . \checkmark \), where each noninitial element points to the last element, and the font difference between the moves is just for clarity. In this manner, the strategy \(\underline{m} : \mathcal {G}(M_G)\) encodes the move m. Then, viability of strategies is given more precisely as follows: A strategy \(\sigma : G\) is defined to be viable if its partial function representation \((m_3, m_2, m_1) \mapsto m\), where \(m_1\), \(m_2\) and \(m_3\) are the last, the second last and the third last moves of the current Pview, respectively, is ‘implementable’ by a finitary strategy \( \mathcal {A}(\sigma )^\circledS : \mathcal {G}(M_{G}) \& \mathcal {G}(M_{G}) \& \mathcal {G}(M_{G}) \Rightarrow \mathcal {G}(M_{G})\), called an instruction strategy for \(\sigma \), in the sense that the composition \(\mathcal {A}(\sigma )^\circledS \circ \langle \underline{m_3}, \underline{m_2}, \underline{m_1} \rangle ^{\dagger } : \mathcal {G}(M_G)\) coincides with \(\underline{m} : \mathcal {G}(M_{G})\) for all quadruples \((m_3, m_2, m_1) \mapsto m\) in the table of \(\sigma \), where \( \langle \underline{m_3}, \underline{m_2}, \underline{m_1} \rangle : T \Rightarrow \mathcal {G}(M_G) \& \mathcal {G}(M_G) \& \mathcal {G}(M_G)\) is the ternary pairing of the strategies \(\underline{m_i} : T \Rightarrow \mathcal {G}(M_G)\) (i = 1, 2, 3).
This example illustrates why we need viable (not only finitary) dynamic strategies for Turing completeness, where recall that minimization (in the general form) or an equivalent construction is vital to construct all partial recursive functions [16]. Also, it should be intuitively clear now why we have to employ composition of strategies without hiding: An instruction strategy for the composition of strategies \(\phi : A \Rightarrow B\) and \(\psi : B \Rightarrow C\) without hiding can be obtained simply as the disjoint union of instruction strategies for \(\phi ^{\dagger }\) and \(\psi \) (see the proof of Theorem 76 for the details), but it is not possible for composition with hiding. (In fact, there is no obvious way to construct an instruction strategy for the composition of \(\phi \) and \(\psi \) with hiding.)
We advocate that viability of strategies gives a reasonable notion of ‘effective computability’ as finitary strategies are clearly ‘effective,’ and so their descriptions or instruction strategies can be ‘effectively read off’ by P. Note also that viability is defined solely in terms of games and strategies without any axiom or induction. Moreover, viability is at least as strong as Church–Turing computability: As the main results of the present work, we show that dynamic strategies definable by PCF are all viable (Theorem 81), and therefore, they are Turing complete in particular (Corollary 82).
Also, viable dynamic strategies solve the problem defined in Sect. 1.3 in the following sense. First, as we have seen via examples, games and strategies give an abstract, syntaxindependent formulation of highlevel computational processes, e.g., the lazy natural number game \(\mathcal {N}\) defines natural numbers (not their symbolic representation) as ‘counting processes’ in an abstract, syntaxindependent fashion, beyond classical computation, e.g., higherorder computation. Moreover, an instruction strategy for a viable dynamic strategy describes a lowlevel computational process that implements the dynamic strategy. In this manner, we have obtained a single mathematical framework for both highlevel and lowlevel computational processes as well as ‘effective computability’ of the former in terms of the latter.
1.8 Our contribution and related work
Our main technical achievement is to define an intrinsic, noninductive, nonaxiomatic notion of ‘effectivity’ of strategies in game semantics, namely viable dynamic strategies, and show that they are Turing complete (Corollary 82). We have also shown the converse (though it is not that surprising): The input/output behavior of each viable dynamic strategy computing on natural numbers coincides with a partial recursive function (Theorem 85). This result immediately implies a universality result [7, 15, 58] as well: Every viable dynamic strategy on a dynamic game interpreting a type of PCF is (up to intrinsic equivalence) the denotation of a term of PCF (Corollary 89). In addition, some of the wellknown theorems in computability theory [16, 60] such as the smn theorem and the first recursion theorem are generalized to nonclassical computation (Corollaries 83 and 84). We hope that these technical results would convince the reader that viability of dynamic strategies is a natural, reasonable generalization of Church–Turing computability.
Another, more conceptual contribution of the present work is to establish a single mathematical framework for both highlevel and lowlevel computational processes, where the former defines what computation does, while the latter describes how to execute the former. In comparison with existing mathematical models of computation, our gamesemantic approach has some novel features. First, in comparison with computation by TMs or programming languages, plays of games are a more abstract concept; in particular they are not necessarily symbol manipulations, which is why they are suitable for abstract, highlevel computational processes. Next, computation in a game proceeds as an interaction between P and O, which may be seen as a generalization of computation by TMs in which just one interaction occurs (i.e., O gives an input on the infinite tape, and then P returns an output on the tape); this in particular means that O’s computation does not have to be recursive, and it is part of the formalization, which is why game semantics in general captures higherorder computation in a natural, systematic manner. The present work inherits this interactive nature of game semantics. Last but not least, games are a semantic counterpart of types, where note that types do not a priori exist in TMs, and types in programming languages are syntactic entities. Hence, our approach provides a deeper clarification of types in the context of theory of computation.
Moreover, by exploiting the flexibility of game semantics, our approach would be applicable to a wide range of computation though it is left as future work. Also, game semantics has interpreted various logics as well [1, 5, 37, 72], and so it would be possible to employ our framework for a realizability interpretation of constructive logic [65, 68], for which viable dynamic strategies would be more suitable as realizers than existing strategies such as [12] since the former contains more ‘computational contents’ and makes more sense as a model of computation than the latter. Furthermore, the game models [1, 72] interpret MartinLöf type theory, one of the most prominent foundations of constructive mathematics, and thus our framework would provide a mathematical, syntaxindependent formalization of constructive mathematics too.^{8} Of course, we need to work out details for these developments, which is out of the scope of the present paper, but it is in principle clear how to apply our framework to existing game semantics. In this sense, the present work would serve as a stepping stone toward these extensions.
In the literature, there have been several attempts to provide a mathematical foundation of computation beyond classical or symbolic ones. We do not claim at all our gamesemantic approach is best or canonical in comparison with the previous work; however, our approach certainly has some advantages. For instance, Robin Gandy proposed in the famous paper [22] a notion of ‘mechanical devices,’ now known as Gandy machines (GMs), which appear more general than TMs, but showed that TMs are actually as powerful as GMs. However, since GMs are an axiomatic approach to define a general class of ‘mechanical devices’ that are ‘effectively executable,’ they do not give a distinction between highlevel and lowlevel computational processes, where GMs formulate the latter. More recent abstract state machines (ASMs) [33] introduced by Yuri Gurevich employ a similar idea to that of GMs for ‘effectivity,’ namely to require an upper bound of elements that may change in a single step of computation, utilizing structures in the sense of mathematical logic [63]. Notably, ASMs define a very general notion of computation, namely computation as structure transition. However, it seems that this framework is in some sense too general; for instance, it is possible that an ASM computes a real number in a single step, but then its ‘effectivity’ is questionable. In general, an appropriate notion of ‘effective computability’ of ASMs has been missing. Also, the way of computing a function by an ASM is to update input/output pairs of the function in the elementwise fashion, but it does not seem to be a common or natural processes in practice. In addition, Yiannis Moschovakis considered a mathematical foundation of algorithms [54] in which, similarly to us, he proposed that algorithms and their ‘implementations’ should be distinguished, where by algorithms he refers to what we call highlevel computational processes. However, his framework, called recursors, is also based on structures, and his notion of algorithms is relative to atomic operations given in each structure; thus, it does not give a foundational analysis on the notion of ‘effective computability.’ Therefore, although the previous work captures broader notions of computation than the present work, our approach has the advantage of achieving both of the distinction between highlevel and lowlevel computational processes, and the primitive, intrinsic notion of ‘effective computability.’ Also, the interactive, typed nature of game semantics stands in sharp contrast to the previous work as well.
At this point, we need to mention computability logic [39] developed by Giorgi Japaridze since his idea is similar to ours; he defines ‘effective computability’ via computing machines playing in games. Nevertheless, there are notable differences between computability logic and the present work. First, computing machines in computability logic are a variant of TMs, and thus they are less novel as a model of computation than our approach; in fact, the definition of ‘effective computability’ in computability logic can be seen more or less as a consequence of just spelling out the standard notion of recursive strategies [7, 20, 38]. Next, our framework inherits the categorical structure of existing game semantics (see [71] for this point), providing a compositional formulation of logic and computation, i.e., a compound proof or program is constructed from its components, while there has been no known categorical structure of computability logic. Nevertheless, it would be interesting to adopt his TMsbased approach in our framework and compare the resulting computational power with that of the present work.
Finally, let us mention some of the precursors of game semantics. To clarify the notion of higherorder computability, Stephen Cole Kleene considered a model of higherorder computation based on dialogues between computational oracles in a series of papers [42, 43, 44], which can be seen as the first attempt to define a mathematical notion of algorithms in a higherorder setting [50]. Moreover, Gandy and his student Giovanni Pani refined these works by Kleene to obtain a model of PCF that satisfies universality though this work was not published. These previous papers are direct ancestors of game semantics (in particular the socalled HOgames [38] by Martin Hyland and Luke Ong). As another line of research (motivated by the full abstraction problem for PCF [58]), PierreLouis Curien and Gerard Berry conceived of sequential algorithms [10] which was the first attempt to go beyond (extensional) functions to capture sequentiality of PCF. Sequential algorithms preceded and became highly influential to the development of game semantics; in fact, sequential algorithms are presented in the style of game semantics in [50], and it is shown in [14] that the oracle computation developed by Kleene can be represented by sequential algorithms (though the converse does not hold). Nevertheless, a point we would like to emphasize here is that neither of the previous attempts defines ‘effective computability’ in a similar manner to the present work; our approach has an advantage in its intrinsic, noninductive, nonaxiomatic nature.
1.9 Structure of the paper
The rest of the paper proceeds roughly as follows. This introduction ends with fixing some notation. Then, recalling dynamic games and strategies in Sect. 2, we define viability of strategies and establish, as the main theorem, the fact that viable dynamic strategies may interpret all terms of PCF in Sect. 3, proving their Turing completeness as a corollary. Finally, we draw a conclusion and propose future work in Sect. 4.
Notation

We use bold letters \(\varvec{s}, \varvec{t}, \varvec{u}, \varvec{v}\), etc. for sequences, in particular \(\varvec{\epsilon }\) for the empty sequence, and letters a, b, c, d, m, n, x, y, z, etc. for elements of sequences;

We often abbreviate a finite sequence \(\varvec{s} = (x_1, x_2, \dots , x_{\varvec{s}})\) as \(x_1 x_2 \dots x_{\varvec{s}}\), where \(\varvec{s}\) denotes the length (i.e., the number of elements) of \(\varvec{s}\), and write \(\varvec{s}(i)\), where \(i \in \{ 1, 2, \dots , \varvec{s} \}\), as another notation for \(x_i\);

A concatenation of sequences is represented by the juxtaposition of them, but we often write \(a \varvec{s}\), \(\varvec{t} b\), \(\varvec{u} c \varvec{v}\) for \((a) \varvec{s}\), \(\varvec{t} (b)\), \(\varvec{u} (c) \varvec{v}\), etc., and also write \(\varvec{s} . \varvec{t}\) for \(\varvec{s t}\);

We define \(\varvec{s}^n {\mathop {=}\limits ^{\mathrm {df. }}} \underbrace{\varvec{s} \varvec{s} \cdots \varvec{s}}_n\) for a sequence \(\varvec{s}\) and a natural number \(n \in \mathbb {N}\);

We write \(\mathsf {Even}(\varvec{s})\) (resp. \(\mathsf {Odd}(\varvec{s})\)) iff \(\varvec{s}\) is of evenlength (resp. oddlength);

We define \(S^\mathsf {P} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in S \mid \mathsf {P}(\varvec{s}) \}\) for a set S of sequences and \(\mathsf {P} \in \{ \mathsf {Even}, \mathsf {Odd} \}\);

\(\varvec{s} \preceq \varvec{t}\) means \(\varvec{s}\) is a prefix of \(\varvec{t}\), i.e., \(\varvec{t} = \varvec{s} . \varvec{u}\) for some sequence \(\varvec{u}\), and given a set S of sequences, we define \(\mathsf {Pref}(S) {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \mid \exists \varvec{t} \in S . \ \! \varvec{s} \preceq \varvec{t} \ \! \}\);

For a poset P and a subset \(S \subseteq P\), \(\mathsf {Sup}(S)\) denotes the supremum of S;

\(X^* {\mathop {=}\limits ^{\mathrm {df. }}} \{ x_1 x_2 \ldots x_n \mid n \in \mathbb {N}, \forall i \in \{ 1, 2, \ldots , n \} . \ \! x_i \in X \ \! \}\) for each set X;

For a function \(f : A \rightarrow B\) and a subset \(S \subseteq A\), we define \(f \upharpoonright S : S \rightarrow B\) to be the restriction of f to S, and \(f^*: A^*\rightarrow B^*\) by \(f^*(a_1 a_2 \dots a_n) {\mathop {=}\limits ^{\mathrm {df. }}} f(a_1) f(a_2) \dots f(a_n) \in B^*\) for all \(a_1 a_2 \dots a_n \in A^*\);

Given sets \(X_1, X_2, \dots , X_n\), and an index \(i \in \{ 1, 2, \dots , n \}\), we write \(\pi _i\) (or \(\pi _i^{(n)}\)) for the ith projection function\(X_1 \times X_2 \times \dots \times X_n \rightarrow X_i\) that maps \((x_1, x_2, \dots , x_n) \mapsto x_i\) for all \(x_j \in X_j\) (\(j = 1, 2, \ldots , n\));

\(\simeq \) denote the Kleene equality, i.e., \(x \simeq y {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (x \downarrow \wedge \ y \downarrow \wedge \ x = y) \vee (x \uparrow \wedge \ y \uparrow )\), where we write \(x \downarrow \) if an element x is defined, and \(x \uparrow \) otherwise.
2 Preliminary: games and strategies
Our games and strategies are essentially the ‘dynamic refinement’ of McCusker’s variant [6, 51],^{9} which has been proposed under the name of dynamic games and strategies by the present author and Abramsky in [71] to capture dynamics (or rewriting) and intensionality (or algorithms) of computation by mathematical, particularly syntaxindependent, concepts. As already explained, we have chosen this variant since, in contrast to conventional games and strategies, dynamic games and strategies capture stepbystep processes in computation, which is essential for a TMslike model of computation.
However, we need some modifications of dynamic games and strategies. First, although disjoint union of sets of moves (for constructions on games) is usually treated informally for brevity, we need to adopt a particular formalization of ‘tags’ for the disjoint union because we are concerned with ‘effective computability’ of strategies, and thus, we must show that manipulations of ‘tags’ are all ‘effectively executable’ by strategies. In particular, we have to employ exponential ! in which different ‘rounds’ or threads are distinguished by such ‘effective tags.’
In addition, we slightly refine the original definition of dynamic games by requiring that an intermediate occurrence of an Omove in a position of a dynamic game must be a mere copy of the last occurrence of a Pmove, which reflects the example of composition without hiding in the introduction. This modification is due to our computabilitytheoretic motivation: Intermediate occurrences of moves are ‘invisible’ to O (as in the example of composition without hiding), and therefore, P has to ‘effectively’ compute intermediate occurrences of Omoves too (though this point does not matter in [71]); note that it is clearly ‘effective’ to just copy and repeat a move. Also, it conceptually makes sense as well: Intermediate occurrences of Omoves are just copies or dummies of those of Pmoves, and thus what happens in the intermediate part of each play is essentially P’s calculation only. Technically, this is achieved by introducing dummy internal Omoves (Definition 8) and strengthening the axiom DP2 (Definition 18). Let us remark, however, that this refinement is technically trivial, and it is not our main contribution.
This section presents the resulting variant of games and strategies. Fixing an implementation of ‘tags’ in Sect. 2.1 as a preparation, we recall (the slightly modified) dynamic games and strategies in Sects. 2.2 and 2.3, respectively. To make this paper essentially selfcontained, we shall explain motivations and intuitions behind the definitions.
2.1 On ‘tags’ for disjoint union of sets
Let us begin with fixing ‘tags’ for disjoint union of sets that can be ‘effectively’ manipulated. We first define outer tags (Definition 3) for exponential (Definition 33), and then inner tags (Definition 5) for other constructions on games.
Definition 1
(Effective tags) An effective tag is a finite sequence over the twoelement set \(\Sigma = \{ \ell , \hbar \}\), where \(\ell \) and \(\hbar \) are arbitrarily fixed elements such that \(\ell \ne \hbar \).
Definition 2
Clearly, the functions \( de : \Sigma ^*\leftrightarrows \mathbb {N}^*: en \) are mutually inverses (n.b., they both map the empty sequence \(\varvec{\epsilon }\) to itself). In fact, each effective tag \(\varvec{\gamma } \in \Sigma ^*\) is intended to be a binary representation of the finite sequence \( de (\varvec{\gamma }) \in \mathbb {N}^*\) of natural numbers.
However, effective tags are not sufficient for our purpose: For nested exponentials occurring in promotion (Definition 57) and fixedpoint strategies (Example 75), we need to ‘effectively’ associate a natural number to each pair of natural numbers in an ‘effectively’ invertible manner. Of course it is possible as there is a recursive bijection \(\mathbb {N} \times \mathbb {N} {\mathop {\rightarrow }\limits ^{\sim }} \mathbb {N}\) whose inverse is recursive too, which is an elementary fact in computability theory [16, 60], but we cannot rely on it for we are aiming at developing an autonomous foundation of ‘effective computability.’
On the other hand, such a bijection is necessary only for manipulating effective tags, and so we would like to avoid an involved mechanism to achieve it. Then, our solution for this problem is to simply introduce elements to denote the bijection:
Definition 3
(Outer tags) An outer tag or an extended effective tags is an expression Open image in new window , where Open image in new window and Open image in new window are arbitrarily fixed elements such that Open image in new window and Open image in new window , generated by the grammar \(\varvec{e} {\mathop {\equiv }\limits ^{\mathrm {df. }}} \varvec{\gamma } \mid \varvec{e}_1 \hbar \ \! \varvec{e}_2 \mid \) Open image in new window , where \(\varvec{\gamma }\) ranges over effective tags.
Notation
Let \(\mathcal {T}\) denote the set of all outer tags.
Definition 4
Of course, we lose the bijectivity between \(\Sigma ^*\) and \(\mathbb {N}^*\) for outer tags (e.g., if Open image in new window , then \( ede (\ell ^{i}) = (i)\), but Open image in new window ), but in return, we may ‘effectively execute’ the bijection \(\wp : \mathbb {N}^*{\mathop {\rightarrow }\limits ^{\sim }} \mathbb {N}\) by just inserting the elements Open image in new window and Open image in new window .^{10} We shall utilize outer tags for exponential !; see Definition 33.
On the other hand, for ‘tags’ on moves for other constructions on games, i.e., \((\_)^{[i]}\) in the introduction, let us employ just four distinguished elements:
Definition 5
(Inner tags) Let \(\mathscr {W}\), \(\mathscr {E}\), \(\mathscr {N}\) and \(\mathscr {S}\) be arbitrarily fixed, pairwise distinct elements. A finite sequence \(\varvec{s} \in \{ \mathscr {W}, \mathscr {E}, \mathscr {N}, \mathscr {S} \}^*\) is called an inner tag.
We shall focus on games whose moves are all tagged elements:
Definition 6
(Inner elements) An inner element is a finitely nested pair \(( \dots ((m, t_1), t_2), \dots , t_k)\), usually written \(m_{t_1 t_2 \dots t_k}\), such that m is a distinguished element, called the substance of \(m_{t_1 t_2 \dots t_k}\), and \(t_1 t_2 \dots t_k\) is an inner tag.
Definition 7
(Tagged elements) A tagged element is any pair \((m_{t_1 t_2 \dots t_k}, \varvec{e})\), usually written \([m_{t_1 t_2 \dots t_k}]_{\varvec{e}}\), of an inner element \(m_{t_1 t_2 \dots t_k}\) and an outer tag \(\varvec{e} \in \mathcal {T}\).
Convention
We often abbreviate an inner element \(m_{t_1 t_2 \dots t_k}\) as m if the inner tag \(t_1 t_2 \dots t_k\) is not very important.
2.2 Games
As already stated, our games are (slightly modified) dynamic games introduced in [71]. The main idea of dynamic games is to introduce, in McCusker’s games [6, 51], a distinction between internal and external moves, where internal moves constitute internal communication between strategies (i.e., moves with square boxes in the introduction), and they are to be a posteriori hidden by the hiding operation, in order to capture intensionality and dynamics of computation by internal moves and the hiding operation, respectively. Conceptually, internal moves are ‘invisible’ to O as they represent how P ‘internally’ calculates the next external Pmove (i.e., stepbystep processes in computation). In addition, unlike [71], we restrict internal Omoves to dummies of internal Pmoves (Definition 8) for the computabilitytheoretic motivation already mentioned at the beginning of Sect. 2.
We first review (the slightly modified) dynamic games in the present section; see [71] for the details, and [3, 6, 37] for a general introduction to game semantics.
Convention
To distinguish our ‘dynamic concepts’ from conventional ones [6, 51], we add the word static in front of the latter, e.g., static arenas, static games, etc.
2.2.1 Arenas and legal positions
Similarly to McCusker’s games, dynamic games are based on two preliminary concepts: (dynamic) arenas and legal positions. An arena defines the basic components of a game, which in turn induces its legal positions that specify the basic rules of the game. Let us begin with recalling these two concepts.
Definition 8

\(M_G\) is a set of tagged elements, called moves, such that: (M) the set \(\pi _1(M_G)\) of all inner elements of G is finite;

\(\lambda _G\) is a function \(M_G \rightarrow \{ \mathsf {O}, \mathsf {P} \} \times \{ \mathsf {Q}, \mathsf {A} \} \times \mathbb {N}\), called the labeling function, where \(\mathsf {O}\), \(\mathsf {P}\), \(\mathsf {Q}\) and \(\mathsf {A}\) are arbitrarily fixed, pairwise distinct symbols, called the labels, that satisfies: (L)\(\mu (G) {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Sup}(\{ \lambda _G^{\mathbb {N}}(m) \mid m \in M_G \ \! \}) \in \mathbb {N}\);
 \(\vdash _G\) is a subset of \((\{ \star \} \cup M_G) \times M_G\), where \(\star \) is an arbitrarily fixed symbol such that \(\star \not \in M_G\), called the enabling relation, that satisfies:

(E1) If \(\star \vdash _G m\), then \(\lambda _G(m) = (\mathsf {O}, \mathsf {Q}, 0)\), and \(n = \star \) whenever \(n \vdash _G m\);

(E2) If \(m \vdash _G n\) and \(\lambda _G^{\mathsf {QA}}(n) = \mathsf {A}\), then \(\lambda _G^{\mathsf {QA}}(m) = \mathsf {Q}\) and \(\lambda _G^{\mathbb {N}}(m) = \lambda _G^{\mathbb {N}}(n)\);

(E3) If \(m \vdash _G n\) and \(m \ne \star \), then \(\lambda _G^{\mathsf {OP}}(m) \ne \lambda _G^{\mathsf {OP}}(n)\);

(E4) If \(m \vdash _G n\), \(m \ne \star \) and \(\lambda _G^{\mathbb {N}}(m) \ne \lambda _G^{\mathbb {N}}(n)\), then \(\lambda _G^{\mathsf {OP}}(m) = \mathsf {O}\);


\(\Delta _G\) is a bijection \(M_G^{\mathsf {PInt}} {\mathop {\rightarrow }\limits ^{\sim }} M_G^{\mathsf {OInt}}\), called the dummy function, that satisfies: (D) there exists some finite partial function \(\delta _G\) on inner tags such that if \([m_{\varvec{t}}]_{\varvec{e}} \in M_G^{\mathsf {PInt}}\), \([n_{\varvec{u}}]_{\varvec{f}} \in M_G^{\mathsf {OInt}}\) and \(\Delta _G([m_{\varvec{t}}]_{\varvec{e}}) = [n_{\varvec{u}}]_{\varvec{f}}\), then \(m = n\), \(\varvec{e} = \varvec{f}\), \(\lambda _G^{\mathsf {QA}}([m_{\varvec{t}}]_{\varvec{e}}) = \lambda _G^{\mathsf {QA}}([n_{\varvec{u}}]_{\varvec{f}})\), \(\lambda _G^{\mathbb {N}}([m_{\varvec{t}}]_{\varvec{e}}) = \lambda _G^{\mathbb {N}}([n_{\varvec{u}}]_{\varvec{f}})\) and \(\varvec{u} = \delta _G(\varvec{t})\)
Notation
We write \(M_G^{\mathsf {Init}}\) (resp. \(M_G^{\mathsf {Int}}\), \(M_G^{\mathsf {Ext}}\)) for the set of all initial (resp. internal, external) moves of a dynamic arena G.
A dynamic arena is a static arena defined in [6], equipped with another labeling \(\lambda _G^{\mathbb {N}}\) on moves and dummies of internal Pmoves, satisfying additional axioms about them. From the opposite angle, dynamic arenas are a generalization of static arenas: A static arena is equivalent to a dynamic arena whose moves are all external.

E1 sets the convention that an initial move must be O’s question, and an initial move cannot be performed for a previous move;

E2 states that an answer must be performed for a question;

E3 mentions that an Omove must be performed for a Pmove, and vice versa.
In addition, unlike [71], we have introduced the additional structure of dummy functions for the computabilitytheoretic motivation mentioned at the beginning of Sect. 2. The idea is that each internal Omove \(m \in M_G^{\mathsf {OInt}}\) of a dynamic game G must be the dummy of a unique internal Pmove \(m' \in M_G^{\mathsf {PInt}}\), i.e., \(m = \Delta _G(m')\), and m may occur in a position only right after an occurrence of \(m'\), which axiomatizes the phenomenon of intermediate occurrences of moves in the composition of \( succ \) and \( double \) without hiding described in the introduction. We shall formalize this restriction on occurrences of internal Omoves by the axiom DP2 in Definition 18.

M requires the set \(\pi _1 (M_G)\) to be finite so that each move is distinguishable, which is not required in [71] yet necessary to define ‘effectivity’ in the present work;

L requires the least upper bound \(\mu (G)\) to be finite as it is conceptually natural and technically necessary for concatenation \(\ddagger \) of games (Definition 36);

E1 adds \(\lambda _G^{\mathbb {N}}(m) = 0\) for all \(m \in M_G^{\mathsf {Init}}\) as O cannot ‘see’ internal moves, and thus, he cannot initiate a play with an internal move;

E2 additionally requires the priority orders between a ‘QA pair’ to be the same since otherwise an output of the hiding operation may not be well defined;

E4 states that only P can perform a move for a previous move if they have different priority orders because internal moves are ‘invisible’ to O (as we shall see, if \(\lambda _G^{\mathbb {N}}(m_1) = k_1 < k_2 = \lambda _G^{\mathbb {N}}(m_2)\), then after the \(k_1\)many iteration of the hiding operation, \(m_1\) and \(m_2\) become external and internal, respectively, i.e., the I/Eparity of moves is relative, which is why E4 is not only concerned with I/Eparity but more finegrained priority orders);

D requires that each internal Pmove \(p \in M_G^{\mathsf {PInt}}\) and its dummy \(\Delta _G(p) \in M_G^{\mathsf {OInt}}\) may differ only in their inner tags since the latter is the dummy of the former (n.b., it reflects the informal example in the introduction), and \(\Delta _G(p)\) is ‘effectively’ obtainable from p by a finitary calculation \(\delta _G\) on inner tags.
Convention
From now on, arenas refer to dynamic arenas by default.
As explained previously, an interaction between P and O in a game is represented by a finite sequence of moves that satisfies certain axioms (under the name of (valid) positions; see Definition 18). Strictly speaking, however, we equip such sequences with an additional structure, called justifiers or pointers, to distinguish similar yet different computational processes (see, e.g., [6] for this point):
Definition 9
(Occurrences of moves) Given a finite sequence \(\varvec{s} \in M_G^*\) of moves of an arena G, an occurrence (of a move) in \(\varvec{s}\) is a pair \((\varvec{s}(i), i)\) such that \(i \in \{ 1, 2, \dots , \varvec{s} \}\). More specifically, we call the pair \((\varvec{s}(i), i)\) an initial occurrence (resp. a noninitial occurrence) in \(\varvec{s}\) if \(\star \vdash _G \varvec{s}(i)\) (resp. otherwise).
Remark
We have been so far casual about the distinction between moves and occurrences, but we shall be more precise from now on.
Definition 10
(Jsequences [6, 38]) A justified (j) sequence of an arena G is a pair \(\varvec{s} = (\varvec{s}, \mathcal {J}_{\varvec{s}})\) of a finite sequence \(\varvec{s} \in M_G^*\) and a map \(\mathcal {J}_{\varvec{s}} : \{ 1, 2, \dots , \varvec{s} \} \rightarrow \{ 0, 1, 2, \dots , \varvec{s} \}\) such that for all \(i \in \{ 1, 2, \dots , \varvec{s} \}\)\(\mathcal {J}_{\varvec{s}}(i) = 0\) if \(\star \vdash _G \varvec{s}(i)\), and \(0< \mathcal {J}_{\varvec{s}}(i) < i \wedge \varvec{s}({\mathcal {J}_{\varvec{s}}(i)}) \vdash _G \varvec{s}(i)\) otherwise. The justifier of each noninitial occurrence \((\varvec{s}(i), i)\) in \(\varvec{s}\) is the occurrence \((\varvec{s}({\mathcal {J}_{\varvec{s}}(i)}), \mathcal {J}_{\varvec{s}}(i))\) in \(\varvec{s}\). We say that \((\varvec{s}(i), i)\) is justified by \((\varvec{s}({\mathcal {J}_{\varvec{s}}(i)}), \mathcal {J}_{\varvec{s}}(i))\), or there is a (necessarily unique) pointer from the former to the latter.
Notation
We write \(\mathscr {J}_G\) for the set of all jsequences of an arena G.
Convention
By abuse of notation, we usually keep the pointer structure \(\mathcal {J}_{\varvec{s}}\) of each jsequence \(\varvec{s} = (\varvec{s}, \mathcal {J}_{\varvec{s}})\) implicit and often abbreviate occurrences \((\varvec{s}(i), i)\) in \(\varvec{s}\) as \(\varvec{s}(i)\). Thus, \(\varvec{s} = \varvec{t} \in \mathscr {J}_G\) means \(\varvec{s} = \varvec{t}\)and\(\mathcal {J}_{\varvec{s}} = \mathcal {J}_{\varvec{t}}\). Moreover, we usually write \(\mathcal {J}_{\varvec{s}}(\varvec{s}(i)) = \varvec{s}(j)\) for \(\mathcal {J}_{\varvec{s}}(i) = j\). This convention is mathematically imprecise, but it is very convenient in practice, and it does not bring any serious confusion (in fact, it has been standard in the literature of game semantics).
The idea is that each noninitial occurrence in a jsequence must be performed for a specific previous occurrence, viz. its justifier. Since the present paper is not concerned with a faithful interpretation of programs, one may wonder if justifiers would play any important role in the rest of the paper; however, they do in a novel manner: They allow P to ‘effectively’ collect, from the history of previous occurrences, a bounded number of necessary ones, as we shall see in Sect. 3.1.
Note that the first element m of each nonempty jsequence \(m \varvec{s} \in \mathscr {J}_G\) must be initial; we particularly call m the opening occurrence of \(m \varvec{s}\). Clearly, an opening occurrence must be an initial occurrence, but not necessarily vice versa.
Let us now consider justifiers, jsequences and arenas from the ‘external viewpoint’ (Definitions 12, 13 and 14):
Definition 11
(Jsubsequences [71]) Given an arena G and a jsequence \(\varvec{s} \in \mathscr {J}_G\), a jsubsequence of \(\varvec{s}\) is a jsequence \(\varvec{t} \in \mathscr {J}_G\) such that \(\varvec{t}\) is a subsequence of \(\varvec{s}\), and \(\mathcal {J}_{\varvec{t}}(n) = m\) iff there are occurrences \(m_1, m_2, \dots , m_k\) (\(k \in \mathbb {N}\)) in \(\varvec{s}\) eliminated in \(\varvec{t}\) such that \(\mathcal {J}_{\varvec{s}}(n) = m_1 \wedge \mathcal {J}_{\varvec{s}}(m_1) = m_2 \dots \wedge \mathcal {J}_{\varvec{s}}(m_{k1}) = m_k \wedge \mathcal {J}_{\varvec{s}}(m_{k}) = m\).
Definition 12
(External justifiers [71]) Let G be an arena, \(\varvec{s} \in \mathscr {J}_G\) and \(d \in \mathbb {N} \cup \{ \omega \}\). Each noninitial occurrence n in \(\varvec{s}\) has a unique sequence of justifiers \(m m_k \dots m_2 m_1 n\)\((k \in \mathbb {N})\) such that \(\mathcal {J}_{\varvec{s}}(n) = m_1\), \(\mathcal {J}_{\varvec{s}}(m_1) = m_{2}\), ..., \(\mathcal {J}_{\varvec{s}}(m_{k1}) = m_k\), \(\mathcal {J}_{\varvec{s}}(m_k) = m\), \(\lambda _G^{\mathbb {N}}(m) = 0 \vee \lambda _G^{\mathbb {N}}(m) > d\) and \(0 < \lambda _G^{\mathbb {N}}(m_i) \leqslant d\) for \(i = 1, 2, \dots , k\). The occurrence m is called the \({{{\varvec{d}}}}\)external justifier of n in \(\varvec{s}\), and written \(\mathcal {J}_{\varvec{s}}^{\circleddash d}(n)\).
Note that dexternal justifiers are a simple generalization of justifiers: 0external justifiers coincide with justifiers. dexternal justifiers are intended to be justifiers after the dtimes iteration of the hiding operation \(\mathcal {H}\), as we shall see shortly.
Definition 13
(External jsubsequences [71]) Given an arena G, \(\varvec{s} \in \mathscr {J}_G\) and \(d \in \mathbb {N} \cup \{ \omega \}\), the \({{{\varvec{d}}}}\)external justified (j) subsequence\(\mathcal {H}^d_G(\varvec{s})\) of \(\varvec{s}\) is obtained from \(\varvec{s}\) by deleting occurrences of internal moves m such that \(0 < \lambda _G^{\mathbb {N}}(m) \leqslant d\) and equipping the resulting subsequence of \(\varvec{s}\) with pointers \(\mathscr {J}_{\mathcal {H}_G^d(\varvec{s})} : n \mapsto \mathscr {J}_{\varvec{s}}^{\circleddash d}(n)\).
Remark
It should be clear how to reformulate Definitions 11, 12 and 13 more formally, following Definitions 9 and 10.
Definition 14

\(M_{\mathcal {H}^d(G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ m \in M_G \mid \lambda _G^{\mathbb {N}}(m) = 0 \vee \lambda _G^{\mathbb {N}}(m) > d \ \! \}\);

\(\lambda _{\mathcal {H}^d(G)} {\mathop {=}\limits ^{\mathrm {df. }}} \lambda _G^{\circleddash d} \upharpoonright M_{\mathcal {H}^d(G)}\), where \(\lambda _G^{\circleddash d} {\mathop {=}\limits ^{\mathrm {df. }}} \langle \lambda _G^{\mathsf {OP}}, \lambda _G^{\mathsf {QA}}, m \mapsto \lambda _G^{\mathbb {N}} (m) \circleddash d \rangle \) and \(n \circleddash d {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} n  d &{}\text {if }n \geqslant d; \\ 0 &{}\text {otherwise} \end{array}\right. }\) for all \(n \in \mathbb {N}\);

\(m \vdash _{\mathcal {H}^d(G)} \! n {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \exists k \in \mathbb {N}, m_1, m_2, \dots , m_{2k1}, m_{2k} \in M_G {\setminus } M_{\mathcal {H}^d(G)} . \ \! m \vdash _G m_1 \wedge m_1 \vdash _G m_2 \dots \wedge m_{2k1} \vdash _G m_{2k} \wedge m_{2k} \vdash _G n\) (\(\Leftrightarrow m \vdash _G n\) if \(k = 0\));

\(\Delta _{\mathcal {H}^d(G)} {\mathop {=}\limits ^{\mathrm {df. }}} \Delta _G \upharpoonright M_{\mathcal {H}^d(G)}\).
That is, \(\mathcal {H}^d(G)\) is obtained from G by deleting internal moves m such that \(0 < \lambda _G^{\mathbb {N}}(m) \leqslant d\), decreasing by d the priority orders of the remaining internal moves, ‘concatenating’ the enabling relation to form the ‘dexternal’ one and taking the obvious restrictions of the labeling and the dummy functions.
Convention
Given \(d \in \mathbb {N} \cup \{ \omega \}\), we regard \(\mathcal {H}^d\) as an operation on arenas G, called the \({{{\varvec{d}}}}\)hiding operation (on arenas), and \(\mathcal {H}_G^d\) as an operation on jsequences of G, called the \({{{\varvec{d}}}}\)hiding operation (on jsequences).
Lemma 15
(Closure of arenas and jsequences under hiding [71]) If G is an arena, then, for all \(d \in \mathbb {N} \cup \{ \omega \}\), so is \(\mathcal {H}^d(G)\), and \(\mathcal {H}_G^d(\varvec{s}) \in \mathscr {J}_{\mathcal {H}^d(G)}\) for all \(\varvec{s} \in \mathscr {J}_G\). Also, \(\underbrace{\mathcal {H}^1 \circ \mathcal {H}^1 \cdots \circ \mathcal {H}^1}_i(G) = \mathcal {H}^i (G)\) and \(\mathcal {H}^1_{\mathcal {H}^{i1}(G)} \circ \mathcal {H}^1_{\mathcal {H}^{i2}(G)} \cdots \circ \mathcal {H}^1_{\mathcal {H}^{1}(G)} \circ \mathcal {H}^1_{G}(\varvec{s}) = \mathcal {H}_G^i(\varvec{s})\) for any \(i \in \mathbb {N}\) (it means \(G = G\) and \(\varvec{s} = \varvec{s}\) if \(i = 0\)), arena G and \(\varvec{s} \in \mathscr {J}_G\).
Proof
We need to consider the additional structure of dummy functions; everything else has been proved in [71]. Let G be an arena, and \(d \in \mathbb {N} \cup \{ \omega \}\). For each \(p \in M_G^{\mathsf {PInt}}\), we clearly have \(p \in M_{\mathcal {H}^d(G)} \Leftrightarrow \Delta _G(p) \in M_{\mathcal {H}^d(G)}\) by the axiom D on \(\Delta _G\); thus, \(\Delta _{\mathcal {H}^d(G)}\) is a welldefined bijection \(M_{\mathcal {H}^d(G)}^{\mathsf {PInt}} {\mathop {\rightarrow }\limits ^{\sim }} M_{\mathcal {H}^d(G)}^{\mathsf {OInt}}\). Finally, the axiom D on \(\Delta _{\mathcal {H}^d(G)}\) clearly follows from that on \(\Delta _G\), completing the proof. \(\square \)
Convention
Thanks to Lemma 15, we henceforth regard the ihiding operations \(\mathcal {H}^i\) and \(\mathcal {H}^i_G\) as the itimes iteration of the 1hiding operations \(\mathcal {H}^1\) and \(\mathcal {H}^1_G\), respectively, for all \(i \in \mathbb {N}\). For this reason, we write \(\mathcal {H}\) and \(\mathcal {H}_G\) for \(\mathcal {H}^1\) and \(\mathcal {H}^1_G\), respectively, and call them the hiding operations (on arenas and jsequences, respectively).
Next, let us recall the notion of ‘relevant part’ of previous moves, called views:
Definition 16

\(\lceil \varvec{\epsilon } \rceil _G {\mathop {=}\limits ^{\mathrm {df. }}} \varvec{\epsilon }\);

\(\lceil \varvec{s} m \rceil _G {\mathop {=}\limits ^{\mathrm {df. }}} \lceil \varvec{s} \rceil _G . m\) if m is a Pmove;

\(\lceil \varvec{s} m \rceil _G {\mathop {=}\limits ^{\mathrm {df. }}} m\) if m is initial;

\(\lceil \varvec{s} m \varvec{t} n \rceil _G {\mathop {=}\limits ^{\mathrm {df. }}} \lceil \varvec{s} \rceil _G . m n\) if n is an Omove with \(\mathcal {J}_{\varvec{s} m \varvec{t} n}(n) = m\);

\(\lfloor \varvec{\epsilon } \rfloor _G {\mathop {=}\limits ^{\mathrm {df. }}} \varvec{\epsilon }\);

\(\lfloor \varvec{s} m \rfloor _G {\mathop {=}\limits ^{\mathrm {df. }}} \lfloor \varvec{s} \rfloor _G . m\) if m is an Omove;

\(\lfloor \varvec{s} m \varvec{t} n \rfloor _G {\mathop {=}\limits ^{\mathrm {df. }}} \lfloor \varvec{s} \rfloor _G . m n\) if n is a Pmove with \(\mathcal {J}_{\varvec{s} m \varvec{t} n}(n) = m\)
The idea behind Definition 16 is as follows. For a jsequence \(\varvec{t}m\) of an arena G such that m is a Pmove (resp. an Omove), the Pview \(\lceil \varvec{t} \rceil \) (resp. the Oview \(\lfloor \varvec{t} \rfloor \)) is intended to be the currently ‘relevant part’ of \(\varvec{t}\) for P (resp. O). That is, P (resp. O) is concerned only with the last Omove (resp. Pmove), its justifier and that justifier’s Pview (resp. Oview), which then recursively proceeds.
As explained in [6], strategies (Definition 42) that model computation without state refer only to Pviews, not entire histories of previous occurrences, as inputs; they are called innocent strategies (Definition 46). In this sense, innocence captures statefreeness of strategies. In this paper, however, Pviews play a different yet fundamental role for our notion of ‘effective computability’ in Sect. 3.1.
We are now ready to define:
Definition 17

(Alternation) If \(\varvec{s} = \varvec{t} m n \varvec{u}\), then \(\lambda _G^\mathsf {OP} (m) \ne \lambda _G^\mathsf {OP} (n)\);

(Generalized visibility) If \(\varvec{s} = \varvec{v} m \varvec{w}\) with m noninitial, and \(d \in \mathbb {N} \cup \{ \omega \}\) satisfy \(\lambda _G^{\mathbb {N}}(m) = 0 \vee \lambda _G^{\mathbb {N}}(m) > d\), then \(\mathcal {J}^{\circleddash d}_{\varvec{s}}(m)\) occurs in \(\lceil \mathcal {H}_G^d(\varvec{v}) \rceil _{\mathcal {H}^d(G)}\) if m is a Pmove, and it occurs in \(\lfloor \mathcal {H}_G^d(\varvec{v}) \rfloor _{\mathcal {H}^d(G)}\) if m is an Omove;

(IEswitch) If \(\varvec{s} = \varvec{t} m n \varvec{u}\) with \(\lambda _G^{\mathbb {N}}(m) \ne \lambda _G^{\mathbb {N}}(n)\), then m is an Omove.
Notation
We write \(\mathscr {L}_G\) for the set of all dynamic legal positions of an arena G.
Recall that a static legal position defined in [6] is a jsequence that satisfies alternation and visibility, i.e., generalized visibility only for \(d = 0\) [6, 38, 51], which is technically to guarantee that the P and the Oviews of a jsequence are again jsequences and conceptually to ensure that the justifier of each noninitial occurrence belongs to the ‘relevant part’ of the history of previous occurrences.

In a position of the game, O always performs the first move by a question, and then P and O alternately play (by alternation), in which every noninitial occurrence is performed for a specific previous occurrence (by justification);

The justifier of each noninitial occurrence in a position belongs to the ‘relevant part’ of the previous occurrences in the position (by visibility).

Generalized visibility is a generalization of visibility; it requires that visibility holds after any iteration of the hiding operations on arenas and jsequences;

IEswitch states that only P can change a priority order during a play because internal moves are ‘invisible’ to O, where the same remark as in E4 is applied for its finer distinction of priority orders than the I/Eparity.
Convention
Legal positions henceforth mean dynamic legal positions.
2.2.2 Games
We are now ready to recall dynamic games:
Definition 18

(P1)\(P_G\) is nonempty and prefixclosed (i.e., \(\varvec{s}m \in P_G \Rightarrow \varvec{s} \in P_G\));

(DP2) If \(\varvec{s} = \varvec{t} . p . o' . \varvec{u} . p' . o\) (resp. \(\varvec{s} = \varvec{t} . o' . \varvec{u} . p' . o\)), where o is an internal Omove, and \(o'\) is an internal (resp. external) Omove such that \(o' = \mathcal {J}_{\varvec{s}}(p')\), then \(o = \Delta _G(p')\) and \(\mathcal {J}_{\varvec{s}}(o) = p\) (resp. \(\mathcal {J}_{\varvec{s}}(o) = p'\)).
Remark
In [71], each dynamic game G is equipped with an equivalence relation \(\simeq _G\) on its positions in order to ignore permutations of ‘tags’ for exponential ! as in [7] and Section 3.6 of [51]. Naturally, dynamic strategies \(\sigma : G\) are identified up to \(\simeq _G\), i.e., the equivalence class \([\sigma ]\) of \(\sigma \) with respect to \(\simeq _G\) is a morphism in the bicategory of dynamic games and strategies [71], which matches the syntactic equality on terms. However, our notion of ‘effective computability’ or viability (Definition 70) is defined on dynamic strategies, not their equivalence classes, and our focus is not a fully complete interpretation of a programming language; thus, we do not have to take such equivalence classes at all. Hence, for simplicity, we exclude such equivalence relations on positions from the structure of dynamic games in the present paper.
Of course, we may easily adopt the full definition of dynamic games G (i.e., with \(\simeq _G\)) and equivalence classes \([\sigma ]\) of dynamic strategies \(\sigma : G\) as in [71]: We may simply define \([\sigma ]\) to be viable if there is some viable representative \(\tau \in [\sigma ]\).
Thus, dynamic games are static games defined in [6, 51] except that their arenas are dynamic ones and they additionally satisfy the axiom DP2. The axiom P1 talks about the natural phenomenon that each nonempty position or ‘moment’ of a play must have the previous ‘moment.’ In addition, by the axiom DP2 for dynamic games, internal Omoves must be performed as dummies of the last internal Pmoves, where the pointers specified by the axiom would make sense if one considers the example of composition of \( succ \) and \( double \) without hiding in the introduction. Conceptually, we impose the axiom for O cannot ‘see’ internal moves, and thus the internal part of each play must be essentially P’s calculation only; technically, it is to ensure external consistency of dynamic strategies: Dynamic strategies act always in the same way from the viewpoint of O, i.e., the external part of each play by a dynamic strategy does not depend on the internal part (see [71] for the details).
Remark
The axiom DP2 defined in [71] just requires determinacy of internal Omoves in each play (it is similar to determinacy of Pmoves for strategies), which works for the purpose of the work. However, in the present paper, we are concerned with ‘effective computability’ of strategies, and thus in particular computation of internal Omoves by P must be ‘effective’ (since O cannot compute them). For this point, we have strengthened the axiom DP2 as above so that computation of internal Omoves becomes trivial.
Convention
Henceforth, games refer to dynamic games by default.
Definition 19
(Subgames [71]) A subgame of a game G is a game H that satisfies \(M_H \subseteq M_G\), \(\lambda _H = \lambda _G \upharpoonright M_H\), \(\vdash _H \ \subseteq \ \vdash _G \cap \ (\{ \star \} \cup M_H) \times M_H\), \(\Delta _H = \Delta _G \upharpoonright M_H\), \(P_H \subseteq P_G\) and \(\mu (H) = \mu (G)\). In this case, we write Open image in new window .
Example 20
The terminal gameT is defined by \(T {\mathop {=}\limits ^{\mathrm {df. }}} (\emptyset , \emptyset , \emptyset , \emptyset , \{ \varvec{\epsilon } \})\). Note that T is the simplest game as its position is only the empty sequence \(\varvec{\epsilon }\).
Example 21

\(M_{\varvec{2}} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [\hat{q}], [ tt ], [ ff ] \}\);

\(\lambda _{\varvec{2}} : [\hat{q}] \mapsto (\mathsf {O}, \mathsf {Q}, 0), [ tt ] \mapsto (\mathsf {P}, \mathsf {A}, 0), [ ff ] \mapsto (\mathsf {P}, \mathsf {A}, 0)\);

\(\vdash _{\varvec{2}} {\mathop {=}\limits ^{\mathrm {df. }}} \{ (\star , [\hat{q}]), ([\hat{q}], [ tt ]), ([\hat{q}], [ ff ]) \}\);

\(\Delta _{\varvec{2}} {\mathop {=}\limits ^{\mathrm {df. }}} \emptyset \);

\(P_{\varvec{2}} {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref}(\{ [\hat{q}] [ tt ], [\hat{q}] [ ff ] \})\), where \([ tt ]\) and \([ ff ]\) are both justified by \([\hat{q}]\).
Example 22

\(M_{N} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [\hat{q}] \} \cup \{ [n] \mid n \in \mathbb {N} \ \! \}\);

\(\lambda _{N} : [\hat{q}] \mapsto (\mathsf {O}, \mathsf {Q}, 0), [n] \mapsto (\mathsf {P}, \mathsf {A}, 0)\);

\(\vdash _{N} {\mathop {=}\limits ^{\mathrm {df. }}} \{ (\star , [\hat{q}]) \} \cup \{ ([\hat{q}], [n]) \mid n \in \mathbb {N} \ \! \}\);

\(\Delta _{N} {\mathop {=}\limits ^{\mathrm {df. }}} \emptyset \);

\(P_{N} {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref}(\{ [\hat{q}] [n] \mid n \in \mathbb {N} \ \! \})\), where [n] is justified by \([\hat{q}]\).
 1.
It is difficult to define an intrinsic, noninductive, nonaxiomatic notion of ‘effective computability’ of strategies on games generated from N via the construction \(\Rightarrow \) of function space (which will be given shortly) since there is no intensional or lowlevel structure in N (see, e.g., [4] for this point);
 2.
The game N contributes almost nothing new to foundations of mathematics.
Example 23

\(M_{\mathcal {N}} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [\hat{q}], [q], [ yes ], [ no ] \}\);

\(\lambda _{\mathcal {N}} : [\hat{q}] \mapsto (\mathsf {O}, \mathsf {Q}, 0), [q] \mapsto (\mathsf {O}, \mathsf {Q}, 0), [ yes ] \mapsto (\mathsf {P}, \mathsf {A}, 0), [ no ] \mapsto (\mathsf {P}, \mathsf {A}, 0)\);

\(\vdash _{\mathcal {N}} {\mathop {=}\limits ^{\mathrm {df. }}} \{ (\star , [\hat{q}]), ([\hat{q}], [ no ]), ([\hat{q}], [ yes ]), ([q], [ no ]), ([q], [ yes ]), ([ yes ], [q]) \}\);

\(\Delta _{\mathcal {N}} {\mathop {=}\limits ^{\mathrm {df. }}} \emptyset \);

\(P_{\mathcal {N}} {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref}(\{ [\hat{q}] . ([ yes ] . [q])^n . [ no ] \mid n \in \mathbb {N} \ \! \})\), where each noninitial occurrence is justified by the last occurrence.
We need two questions \([\hat{q}]\) and [q], the initial and noninitial ones, respectively, for the axiom E1. Intuitively, a play of \(\mathcal {N}\) proceeds by repeating the following: O asks by a question \([\hat{q}]\) or [q] if P would like to ‘count one more,’ and P replies it by \([ yes ]\) if she would like to, and by \([ no ]\) otherwise. Thus, for each \(n \in \mathbb {N}\), the position \([\hat{q}] . ([ yes ] . [q])^n . [ no ]\) is to represent the number n, where each occurrence of an answer (i.e., \([ yes ]\) or \([ no ]\)) is performed for the last occurrence of a question (i.e., \([\hat{q}]\) or [q]).
The game \(\mathcal {N}\) defines natural numbers in an intuitively natural manner, namely as ‘counting processes,’ where our choice of notation for moves is inessential, i.e., \(\mathcal {N}\) is syntaxindependent. Moreover, we may actually define it intrinsically, i.e., without recourse to the set \(\mathbb {N}\), by specifying its positions inductively: \([\hat{q}] [ no ] \in P_{\mathcal {N}} \wedge ([\hat{q}] . \varvec{s} . [ no ] \in P_{\mathcal {N}} \Rightarrow [\hat{q}] . \varvec{s} . [ yes ] [q] [ no ] \in P_{\mathcal {N}})\). Thus, we may define (rather than represent) natural numbers to be positions of \(\mathcal {N}\) though we will not investigate foundational consequences of this definition in the present paper.
As we shall see, such stepbystep processes underlying natural numbers allow us to define ‘effective computability’ of strategies on natural numbers in an intrinsic, noninductive, nonaxiomatic manner in Sect. 3.1.
Now, let us recall the hiding operation on games:
Definition 24
(Hiding on games [71]) Let \(d \in \mathbb {N} \cup \{ \omega \}\); the \({{{\varvec{d}}}}\)hiding operation\(\mathcal {H}^{{\varvec{d}}}\)on games is defined as follows. Given a game G, the \({{{\varvec{d}}}}\)external game\(\mathcal {H}^d(G)\) of G consists of the dexternal arena \((M_{\mathcal {H}^d(G)}, \lambda _{\mathcal {H}^d(G)}, \vdash _{\mathcal {H}^d(G)}, \Delta _{\mathcal {H}^d(G)})\) of the arena G, and the set \(P_{\mathcal {H}^d(G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \mathcal {H}^d_G(\varvec{s}) \mid \varvec{s} \in P_G \ \! \}\) of positions. A game H is normalized if \(\mathcal {H}^\omega (H) = H\), i.e., if \(M_H\) does not contain any internal moves.
Theorem 25
(Closure of games under hiding [71]) For any game G, \(\mathcal {H}^d(G)\) forms a welldefined game for all \(d \in \mathbb {N} \cup \{ \omega \}\). Moreover, if Open image in new window , then Open image in new window for all \(d \in \mathbb {N} \cup \{ \omega \}\). Furthermore, \(\underbrace{\mathcal {H}^1 \circ \mathcal {H}^1 \cdots \circ \mathcal {H}^1}_i(G) = \mathcal {H}^i(G)\) for all \(i \in \mathbb {N}\).
Convention
Thanks to Theorem 25, the ihiding operation \(\mathcal {H}^i\) on games for each \(i \in \mathbb {N}\) can be thought of as the itimes iteration of the 1hiding operation \(\mathcal {H}^1\), which we call the hiding operation (on games) and write \(\mathcal {H}\) for it.
2.2.3 Constructions on games
Now, let us recall the constructions on games given in [71] with ‘tags’ formalized by outer and inner tags defined in Sect. 2.1. A tag refers to an outer or inner tag.
On the other hand, for readers who are not familiar with game semantics, we first give a rather standard presentation of each construction, which keeps ‘tags’ informal and unspecified, before its formal definition. For this aim, we employ:
Notation
Let S and T be sets, and we write \(S + T\) for their disjoint union. Then, we write \(x \in S + T\) if \(x \in S\) or \(x \in T\), where we cannot have both \(x \in S\) and \(x \in T\) by the implicit ‘tag’ for the disjoint union \(S+T\). Also, given functions \(f : S \rightarrow U\) and \(g : T \rightarrow U\), we write [f, g] for the function \(S + T \rightarrow U\) that maps \(x \in S + T\) to \(f(x) \in U\) if \(x \in S\), and to \(g(x) \in U\) otherwise (n.b., it is generalized to more than two functions in the obvious manner). Moreover, given relations \(R_S \subseteq S \times S\) and \(R_T \subseteq T \times T\), we write \(R_S + R_T\) for the relation on \(S + T\) such that \((x, y) \in R_S + R_T {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (x, y) \in R_S \vee (x, y) \in R_T\).

\(M_{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} M_A + M_B\);

\(\lambda _{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} [\lambda _A, \lambda _B]\);

\(\vdash _{A \otimes B} \ {\mathop {=}\limits ^{\mathrm {df. }}} \ \vdash _A + \ \! \vdash _B\);

\(\Delta _{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} [\Delta _A, \Delta _B]\);

\(P_{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{A \otimes B} \mid \varvec{s} \upharpoonright A \in P_A, \varvec{s} \upharpoonright B \in P_B \ \! \},\) where \(\varvec{s} \upharpoonright A\) (resp. \(\varvec{s} \upharpoonright B\)) denotes the jsubsequence of \(\varvec{s}\) that consists of moves of A (resp. B). As an illustration, recall the example \(N \otimes N\) in the introduction, in which the ‘tags’ are informally written as \((\_)^{[i]}\) (\(i = 0, 1\)).
Let us now give the formal definition of tensor, for which the ‘tags’ \((\_)^{[0]}\) and \((\_)^{[1]}\) are formalized by inner tags \((\_, \mathscr {W})\) and \((\_, \mathscr {E})\), respectively:
Definition 26

\(M_{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(a, \mathscr {W})]_{\varvec{e}} \mid [a]_{\varvec{e}} \in M_A \ \! \} \cup \{ [(b, \mathscr {E})]_{\varvec{f}} \mid [b]_{\varvec{f}} \in M_B \ \! \}\);

\(\lambda _{A \otimes B} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \lambda _A([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {W}; \\ \lambda _B([m]_{\varvec{e}}) &{}\text {otherwise;} \end{array}\right. }\)

\(\star \vdash _{A \otimes B} [(m, X)]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (X = \mathscr {W} \wedge \star \vdash _A [m]_{\varvec{e}}) \vee (X = \mathscr {E} \wedge \star \vdash _B [m]_{\varvec{e}})\);

\([(m, X)]_{\varvec{e}} \vdash _{A \otimes B} [(n, Y)]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{{\mathrm {df. }}}} (X = \mathscr {W} = Y \wedge [m]_{\varvec{e}} \vdash _A [n]_{\varvec{f}}) \vee (X = \mathscr {E} = Y \wedge [m]_{\varvec{e}} \vdash _B [n]_{\varvec{f}})\);

\(\Delta _{A \otimes B} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{{\mathrm {df. }}}} \left\{ \begin{array}{lll} {[}(m', \mathscr {W})]_{\varvec{e}} &{}\text {if }X = \mathscr {W},\text { where } \Delta _A([m]_{\varvec{e}}) = [m']_{\varvec{e}}; \\ {[}(m'', \mathscr {E})]_{\varvec{e}} &{}\text {otherwise, where } \Delta _B([m]_{\varvec{e}}) = [m'']_{\varvec{e}}; \end{array}\right. \)

\(P_{A \otimes B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{A \otimes B} \mid \varvec{s} \upharpoonright \mathscr {W} \in P_A, \varvec{s} \upharpoonright \mathscr {E} \in P_B \ \! \}\), where \(\varvec{s} \upharpoonright X\) is the jsubsequence of \(\varvec{s}\) that consists of moves of the form \([(m, X)]_{\varvec{e}}\) yet changed into \([m]_{\varvec{e}}\) (\(X \in \{ \mathscr {W}, \mathscr {E} \}\)).
Example 27

\(M_{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} M_{\mathcal {H}^\omega (A)} + M_B\);

\(\lambda _{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} [\overline{\lambda _{\mathcal {H}^\omega (A)}}, \lambda _B]\), where \(\overline{\lambda _{\mathcal {H}^\omega (A)}} {\mathop {=}\limits ^{\mathrm {df. }}} \langle \overline{\lambda _{\mathcal {H}^\omega (A)}^\textsf {OP}}, \lambda _{\mathcal {H}^\omega (A)}^\textsf {QA}, \lambda _{\mathcal {H}^\omega (A)}^\mathbb {N} \rangle \), and \(\overline{\lambda _{G}^\textsf {OP}} (m) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \mathsf {P} \ &{} \text {if }\lambda _{G}^\textsf {OP} (m) = \textsf {O}; \\ \textsf {O} \ &{}\text {otherwise} \end{array}\right. }\) for any game G;

\(\star \vdash _{A \multimap B} m {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \star \vdash _B m\);

\(m \vdash _{A \multimap B} n \ (m \ne \star ) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (m \vdash _{\mathcal {H}^\omega (A)} n) \vee (m \vdash _B n) \vee (\star \vdash _B m \wedge \star \vdash _{\mathcal {H}^\omega (A)} n)\);

\(\Delta _{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} [\emptyset , \Delta _B]\) (n.b., note that \(\Delta _{\mathcal {H}^\omega (A)} = \emptyset \));

\(P_{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{\mathcal {H}^\omega (A) \multimap B} \mid \varvec{s} \upharpoonright \mathcal {H}^\omega (A) \in P_{\mathcal {H}^\omega (A)}, \varvec{s} \upharpoonright B \in P_B \ \! \}\).
Note that the domain A must be normalized into \(\mathcal {H}^\omega (A)\) since otherwise the linear implication \(A \multimap B\) may not satisfy the axiom DP2. It conceptually makes sense too for the roles of P and O in A are exchanged, and thus P should not be able to ‘see’ internal moves of A. Note also that \(A \multimap B\) is almost \(A \otimes B\) if A is normalized except the switch of the roles in A; dually to \(A \otimes B\), only P can switch between A and B during a play of \(A \multimap B\) (see [3] for the proof). Surprisingly, this simple point changes \(A \otimes B\) into \(A \multimap B\).
Similarly to tensor, the formal definition of linear implication is as follows:
Definition 28

\(M_{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(a, \mathscr {W})]_{\varvec{e}} \mid [a]_{\varvec{e}} \in M_{\mathcal {H}^\omega (A)} \ \! \} \cup \{ [(b, \mathscr {E})]_{\varvec{f}} \mid [b]_{\varvec{f}} \in M_B \ \! \}\);

\(\lambda _{A \multimap B} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \overline{\lambda _{\mathcal {H}^\omega (A)}}([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {W}, \text { where } \overline{\lambda _{\mathcal {H}^\omega (A)}}\text { is defined above;} \\ \lambda _B([m]_{\varvec{e}}) &{}\text {otherwise;} \end{array}\right. }\)

\(\star \vdash _{A \multimap B} [(m, X)]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} X = \mathscr {E} \wedge \star \vdash _B [m]_{\varvec{e}}\);

\([(m, X)]_{\varvec{e}} \vdash _{A \multimap B} [(n, Y)]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \begin{aligned} &{}(X = \mathscr {W} = Y \wedge [m]_{\varvec{e}} \vdash _{\mathcal {H}^\omega (A)} [n]_{\varvec{f}}) \\ &{}\vee (X = \mathscr {E} = Y \wedge [m]_{\varvec{e}} \vdash _B [n]_{\varvec{f}}) \\ &{}\vee (X = \mathscr {E} \wedge Y = \mathscr {W} \wedge \star \vdash _B [m]_{\varvec{e}} \wedge \star \vdash _{\mathcal {H}^\omega (A)} [n]_{\varvec{f}}); \end{aligned} \end{array}\right. }\)

\(\Delta _{A \multimap B}([(b, \mathscr {E})]_{\varvec{f}}) {\mathop {=}\limits ^{\mathrm {df. }}} [(b', \mathscr {E})]_{\varvec{f}}\), where \(\Delta _B([b]_{\varvec{f}}) = [b']_{\varvec{f}}\);

\(P_{A \multimap B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{\mathcal {H}^\omega (A) \multimap B} \mid \varvec{s} \upharpoonright \mathscr {W} \in P_{\mathcal {H}^\omega (A)}, \varvec{s} \upharpoonright \mathscr {E} \in P_B \ \! \}\), where pointers in \(\varvec{s}\) from initial occurrences of A to those of B are deleted in \(\varvec{s} \upharpoonright \mathscr {W}\) and \(\varvec{s} \upharpoonright \mathscr {E}\).
Example 29

\( M_{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} M_A + M_B\);

\( \lambda _{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} [\lambda _A, \lambda _B]\);

\( \vdash _{A \& B} \ {\mathop {=}\limits ^{\mathrm {df. }}} \ \vdash _A + \ \! \vdash _B\);

\( \Delta _{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} [\Delta _A, \Delta _B]\);

\( P_{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{A \& B} \mid (\varvec{s} \upharpoonright A \in P_A \wedge \varvec{s} \upharpoonright B = \varvec{\epsilon }) \vee (\varvec{s} \upharpoonright A = \varvec{\epsilon } \wedge \varvec{s} \upharpoonright B \in P_B) \ \! \}\).
Definition 30

\( M_{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(a, \mathscr {W})]_{\varvec{e}} \mid [a]_{\varvec{e}} \in M_A \ \! \} \cup \{ [(b, \mathscr {E})]_{\varvec{f}} \mid [b]_{\varvec{f}} \in M_B \ \! \}\);

\( \lambda _{A \& B} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \lambda _A([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {W}; \\ \lambda _B([m]_{\varvec{e}}) &{}\text {otherwise;} \end{array}\right. }\)

\( \star \vdash _{A \& B} [(m, X)]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (X = \mathscr {W} \wedge \star \vdash _A [m]_{\varvec{e}}) \vee (X = \mathscr {E} \wedge \star \vdash _B [m]_{\varvec{e}})\);

\( [(m, X)]_{\varvec{e}} \vdash _{A \& B} [(n, Y)]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (X = \mathscr {W} = Y \wedge [m]_{\varvec{e}} \vdash _A [n]_{\varvec{f}}) \vee (X = \mathscr {E} = Y \wedge [m]_{\varvec{e}} \vdash _B [n]_{\varvec{f}})\);

\( \Delta _{A \& B} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} \left\{ \begin{array}{ll} {[}(m',\mathscr {W})]_{\varvec{e}} &{}\text {if }X = \mathscr {W},\text { where } \Delta _A([m]_{\varvec{e}}) = [m']_{\varvec{e}}; \\ {[}(m'', \mathscr {E})]_{\varvec{e}} &{}\text {otherwise, where } \Delta _B([m]_{\varvec{e}}) = [m'']_{\varvec{e}}; \end{array}\right. \)

\( P_{A \& B} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{A \& B} \mid (\varvec{s} \upharpoonright \mathscr {W} \in P_A \wedge \varvec{s} \upharpoonright \mathscr {E} = \varvec{\epsilon }) \vee (\varvec{s} \upharpoonright \mathscr {W} = \varvec{\epsilon } \wedge \varvec{s} \upharpoonright \mathscr {E} \in P_B) \ \! \}\).
For the cartesian closed bicategory\(\mathcal {DG}\) of dynamic games and strategies defined in [71], however, we have to generalize the construction \( C \multimap A \& B\) on normalized games A, B and C, where & precedes \(\multimap \), because we need to pair strategies \(\sigma : L\) and \(\tau : R\) such that Open image in new window and Open image in new window , and the ambient game of the pairing \(\langle \sigma , \tau \rangle \) would be such a generalization of \( C \multimap A \& B\).

\(M_{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} M_C + (M_L {\setminus } M_C) + (M_R {\setminus } M_C)\);

\(\lambda _{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} [\overline{\lambda _C}, \lambda _L \downharpoonright M_C, \lambda _R \downharpoonright M_C]\);

\(m \vdash _{\langle L, R \rangle } n {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} m \vdash _L n \vee m \vdash _R n\);

\(\Delta _{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} [\Delta _L, \Delta _R]\);

\( P_{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{L \& R} \mid (\varvec{s} \upharpoonright L \in P_L \wedge \varvec{s} \upharpoonright B = \varvec{\epsilon }) \vee (\varvec{s} \upharpoonright A = \varvec{\epsilon } \wedge \varvec{s} \upharpoonright R \in P_R) \ \! \}\)
Note that the pairing \(\langle L, R \rangle \) does not depend on the choice of the normalized games A, B and C such that Open image in new window and Open image in new window . Also, we have \( \langle C \multimap A, C \multimap B \rangle = C \multimap A \& B\); in particular, \(\langle T \multimap A, T \multimap B\rangle \) and \( A \& B\) coincide up to tags; pairing of games generalizes this phenomenon in the sense that Open image in new window holds (see [71] for the proof), where the ‘tags’ for the disjoint union \(M_{\langle L, R \rangle } = M_C + (M_L {\setminus } M_C) + (M_R {\setminus } M_C)\) must be formulated in such a way that establishes the subgame relation Open image in new window

Adding no tags on external moves of the form \([(c, \mathscr {W})]_{\varvec{e}}\) of L or R, where \([c]_{\varvec{e}}\) must be a move of C by the definition of \(\multimap \) (Definition 28);

Changing external moves of the form \([(a, \mathscr {E})]_{\varvec{f}}\) of L, where \([a]_{\varvec{f}}\) must be a move of A by the definition of \(\multimap \), into \([((a, \mathscr {W}), \mathscr {E})]_{\varvec{f}}\);

Changing external moves of the form \([(b, \mathscr {E})]_{\varvec{g}}\) of R, where \([b]_{\varvec{g}}\) must be a move of B by the definition of \(\multimap \), into \([((b, \mathscr {E}), \mathscr {E})]_{\varvec{g}}\);

Changing internal moves \([l]_{\varvec{h}}\) of L into \([(l, \mathscr {S})]_{\varvec{h}}\);

Changing internal moves \([r]_{\varvec{k}}\) of R into \([(r, \mathscr {N})]_{\varvec{k}}\).

\( att _{\langle L, R \rangle }(m) = att _{\langle L, R \rangle }(n) \vee att _{\langle L, R \rangle }(m) = C \vee att _{\langle L, R \rangle }(n) = C\);

\( peel _{\langle L, R \rangle }(m) \vdash _L peel _{\langle L, R \rangle }(n) \vee peel _{\langle L, R \rangle }(m) \vdash _R peel _{\langle L, R \rangle }(n)\).
Definition 31

\(\centerdot M_{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(c, \mathscr {W})]_{\varvec{e}} \mid [(c, \mathscr {W})]_{\varvec{e}} \in M_L^{\mathsf {Ext}} \cup M_R^{\mathsf {Ext}}, [c]_{\varvec{e}} \in M_C \ \! \}\) \( \cup \{ [((a, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \mid [(a, \mathscr {E})]_{\varvec{f}} \in M_L^{\mathsf {Ext}}, [a]_{\varvec{f}} \in M_A \ \! \}\) \(\cup \{ [((b, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \mid [(b, \mathscr {E})]_{\varvec{g}} \in M_R^{\mathsf {Ext}}, [b]_{\varvec{g}} \in M_B \ \! \}\) \(\cup \{ [(l, \mathscr {S})]_{\varvec{h}} \mid [l]_{\varvec{h}} \in M_L^{\mathsf {Int}} \} \cup \{ [(r, \mathscr {N})]_{\varvec{k}} \mid [r]_{\varvec{k}} \in M_R^{\mathsf {Int}} \ \! \};\)

\(\lambda _{\langle L, R \rangle } ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \overline{\lambda _C} ([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {W}; \\ \lambda _A([a]_{\varvec{e}}) &{}\text {if }X = \mathscr {E},\text { and }m \text { is of the form } (a, \mathscr {W}); \\ \lambda _B([b]_{\varvec{e}}) &{}\text {if }X = \mathscr {E}, \text { and } m \text { is of the form }(b, \mathscr {E}); \\ \lambda _L([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {S}; \\ \lambda _R([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {N}; \end{array}\right. }\)

\(\star \vdash _{\langle L, R \rangle } [(m, X)]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} X = \mathscr {E} \wedge (\exists [a]_{\varvec{e}} \in M_A^{\mathsf {Init}} . \ \! m = (a, \mathscr {W}) \vee \exists [b]_{\varvec{e}} \in M_B^{\mathsf {Init}} . \ \! m = (b, \mathscr {E}))\);

\([(m, X)]_{\varvec{e}} \vdash _{\langle L, R \rangle } [(n, Y)]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} ( att _{\langle L, R \rangle }([(m, X)]_{\varvec{e}}) = att _{\langle L, R \rangle }([(n, Y)]_{\varvec{f}}) \vee att _{\langle L, R \rangle }([(m, X)]_{\varvec{e}}) = C \vee att _{\langle L, R \rangle }([(n, Y)]_{\varvec{f}}) = C) \wedge ( peel _{\langle L, R \rangle }([(m, X)]_{\varvec{e}}) \vdash _L peel _{\langle L, R \rangle }([(n, Y)]_{\varvec{f}}) \vee peel _{\langle L, R \rangle }([(m, X)]_{\varvec{e}}) \vdash _R peel _{\langle L, R \rangle }([(n, Y)]_{\varvec{f}}))\), where the map \( peel _{\langle L, R \rangle } : M_{\langle L, R \rangle } \rightarrow M_L \cup M_R\) is given by \([(c, \mathscr {W})]_{\varvec{e}} \mapsto [(c, \mathscr {W})]_{\varvec{e}}\), \([((a, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \mapsto [(a, \mathscr {E})]_{\varvec{f}}\), \([((b, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \mapsto [(b, \mathscr {E})]_{\varvec{g}}\), \([(l, \mathscr {S})]_{\varvec{h}} \mapsto [l]_{\varvec{h}}\), \([(r, \mathscr {N})]_{\varvec{k}} \mapsto [r]_{\varvec{k}}\), and the map \( att _{\langle L, R \rangle } : M_{\langle L, R \rangle } \rightarrow \{ L, R, C \}\) by \([(c, \mathscr {W})]_{\varvec{e}} \mapsto C\), \([((a, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \mapsto L\), \([((b, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \mapsto R\), \([(l, \mathscr {S})]_{\varvec{h}} \mapsto L\), \([(r, \mathscr {N})]_{\varvec{k}} \mapsto R\);

\(\Delta _{\langle L, R \rangle } ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} \left\{ \begin{array}{ll} {[}(l', \mathscr {S})]_{\varvec{e}} &{}\text {if }X = \mathscr {S}, \text { where } \Delta _L([l]_{\varvec{e}}) = [l']_{\varvec{e}}; \\ {[}(r', \mathscr {N})]_{\varvec{e}} &{}\text {if }X = \mathscr {N}, \text { where } \Delta _R([r]_{\varvec{e}}) = [r']_{\varvec{e}}; \end{array}\right. \)

\(P_{\langle L, R \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{\langle L, R \rangle } \mid (\varvec{s} \upharpoonright L \in P_L \wedge \varvec{s} \upharpoonright B = \varvec{\epsilon }) \vee (\varvec{s} \upharpoonright R \in P_R \wedge \varvec{s} \upharpoonright A = \varvec{\epsilon }) \ \! \}\), where \(\varvec{s} \upharpoonright L\) (resp. \(\varvec{s} \upharpoonright R\)) is the jsubsequence of \(\varvec{s}\) that consists of moves x such that \( peel _{\langle L, R \rangle }(x) \in M_L\) (resp. \( peel _{\langle L, R \rangle }(x) \in M_R\)) yet changed into \( peel _{\langle L, R \rangle }(x)\), and \(\varvec{s} \upharpoonright B\) (resp. \(\varvec{s} \upharpoonright A\)) is the jsubsequence of \(\varvec{s}\) that consists of moves \([((b, \mathscr {E}), \mathscr {E})]_{\varvec{g}}\) with \([b]_{\varvec{g}} \in M_B\) (resp. \([((a, \mathscr {W}), \mathscr {E})]_{\varvec{f}}\) with \([a]_{\varvec{f}} \in M_A\)) yet changed into \([b]_{\varvec{g}}\) (resp. \([a]_{\varvec{f}}\)).
Example 32

\(M_{!A} {\mathop {=}\limits ^{\mathrm {df. }}} M_A \times \mathbb {N}\);

\(\lambda _{!A} : (a, i) \mapsto \lambda _A(a)\);

\(\star \vdash _{!A} (a, i) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \star \vdash _A a\);

\((a, i) \vdash _{!A} (a', j) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} i = j \wedge a \vdash _A a'\);

\(\Delta _{!A} : (a, i) \mapsto (\Delta _A(a), i)\);

\(P_{!A} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{!A} \mid \forall i \in \mathbb {N} . \ \! \varvec{s} \upharpoonright i \in P_A \ \! \}\), where \(\varvec{s} \upharpoonright i\) is the jsubsequence of \(\varvec{s}\) that consists of moves (a, i) yet changed into a.
Definition 33

Open image in new window , where \(\Delta _A([m]_{\varvec{e}}) = [m']_{\varvec{e}}\);

\(P_{!A} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{!A} \mid \forall \varvec{f} \in \mathcal {T} . \ \! \varvec{s} \upharpoonright \varvec{f} \in P_A \wedge (\varvec{s} \upharpoonright \varvec{f} \ne \varvec{\epsilon } \Rightarrow \forall \varvec{g} \in \mathcal {T} . \ \! \varvec{s} \upharpoonright \varvec{g} \ne \varvec{\epsilon } \Rightarrow ede (\varvec{f}) \ne ede (\varvec{g})) \ \! \}\), where \(\varvec{s} \upharpoonright \varvec{f}\) is the jsubsequence of \(\varvec{s}\) that consists of moves of the form Open image in new window yet changed into \([m]_{\varvec{e}}\).
Thus, our exponential !A is essentially a slight modification of the one in [37, 51] which generalizes moves \([m]_{\ell ^{i} \hbar \varvec{e}}\) to Open image in new window , where \([m]_{\varvec{e}} \in M_A\), \(i \in \mathbb {N}\) and \(\varvec{f} \in \mathcal {T}\). By the condition on positions of !A, an element \(\varvec{f}\) in an outer tag Open image in new window that represents a natural number \(i \in \mathbb {N}\), i.e., \( ede (\varvec{f}) = (i)\), is unique in each \(\varvec{s} \in P_{!A}\).
Notation
We often write \(A \Rightarrow B\) for the linear implication \(!A \multimap B\) for any games A and B, which we call the implication or function space from A to B. The constructions \(\multimap \) and \(\Rightarrow \) are both right associative.
Example 34
Similarly to the case of pairing, exponential is generalized in [71]: Given a game G such that Open image in new window for some normalized games A and B, there is the promotion\(G^{\dagger }\) of G such that Open image in new window In fact, promotion is a generalization of exponential because \((!T \multimap B)^{\dagger }\) and !B coincide up to tags for any normalized game B; see [71] for the proof. Promotion of games is defined in [71] because a morphism \(A \rightarrow B\) in the bicategory \(\mathcal {DG}\) is a strategy \(\phi : G\) such that Open image in new window and therefore, it is necessary to take a generalized promotion \(\phi ^{\dagger }\) (Definition 57) for composition of strategies in \(\mathcal {DG}\), whose ambient game is \(G^{\dagger }\).

\(M_{G^{\dagger }} {\mathop {=}\limits ^{\mathrm {df. }}} ((M_G {\setminus }M_{!A})\times \mathbb {N})+ \{ (a, \langle i, k \rangle ) \mid (a, k) \in M_{!A}, i \in \mathbb {N} \}\);

\(\lambda _{G^{\dagger }} : ((m, i) \in (M_G {\setminus } M_{!A}) \times \mathbb {N}) \mapsto \lambda _G(m), (a, \langle i, k \rangle ) \mapsto \lambda _G(a, k)\);

\(\star \vdash _{G^{\dagger }} (m, i) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \star \vdash _G m\) for all \(i \in \mathbb {N}\);
 $$\begin{aligned}&(m, i) \vdash _{G^{\dagger }} (n, j) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} (i = j \wedge m, n \in M_G {\setminus } M_{!A} \wedge m \vdash _G n) \\&\quad \vee (i = j \wedge m \vdash _A n) \vee (m \in M_G {\setminus } M_{!A} \wedge (n, j) \in M_{!A} \wedge m \vdash _G (n, j) \wedge \exists k \in \mathbb {N} . \ \! j = \langle i, k \rangle ; \end{aligned}$$

\(\Delta _{G^{\dagger }} : (m, i) \mapsto (\Delta _G(m), i)\);

\(P_{G^{\dagger }} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{G^{\dagger }} \mid \forall i \in \mathbb {N} . \ \! \varvec{s} \upharpoonright i \in P_G \ \! \}\), where \(\varvec{s} \upharpoonright i\) is the jsubsequence of \(\varvec{s}\) that consists of moves of the form (m, i) with \(m \in M_G {\setminus } M_{!A}\) or \((a, \langle i, k \rangle )\) with \(a \in M_A \wedge k \in \mathbb {N}\) yet changed into m and (a, k), respectively.

We duplicate moves of G coming from !A, i.e., ones of the form Open image in new window for each \(\varvec{g} \in \mathcal {T}\);

We duplicate moves of G coming from B, i.e., ones of the form \([(b, \mathscr {E})]_{\varvec{e}}\), as Open image in new window for each \(\varvec{g} \in \mathcal {T}\);

We duplicate internal moves \([m]_{\varvec{e}}\) of G as Open image in new window for each \(\varvec{g} \in \mathcal {T}\)
Then, the labeling function, the enabling relation and the dummy function of \(G^{\dagger }\) are again defined by pattern matching on inner tags in the obvious manner, for which like the case of pairing we use peeling and attributes just for convenience. Also, positions of \(G^{\dagger }\) are given by a straightforward generalization of those of exponential defined in Definition 33. Formally, we define promotion of games as follows:
Definition 35

\(\lambda _{G^{\dagger }} {\mathop {=}\limits ^{\mathrm {df. }}} \lambda _G \circ peel _{G^{\dagger }}\), where \( peel _{G^{\dagger }}\) is the function \(M_{G^{\dagger }} \rightarrow M_G\) that maps Open image in new window , Open image in new window , Open image in new window ;

\(x \vdash _{G^{\dagger }} y {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} att _{G^{\dagger }}(x) = att _{G^{\dagger }}(y) \wedge peel _{G^{\dagger }}(x) \vdash _G peel _{G^{\dagger }}(y)\), where \( att _{G^{\dagger }}\) is the function \(M_{G^{\dagger }} \rightarrow \mathcal {T}\) that maps Open image in new window , Open image in new window , Open image in new window ;

Open image in new window , where \(\Delta _G([m]_{\varvec{e}}) = [m']_{\varvec{e}}\);

\(P_{G^{\dagger }} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{G^{\dagger }} \mid \forall \varvec{g} \in \mathcal {T} . \ \! \varvec{s} \upharpoonright \varvec{g} \in P_G \wedge (\varvec{s} \upharpoonright \varvec{g} \ne \varvec{\epsilon } \Rightarrow \forall \varvec{h} \in \mathcal {T} . \ \! \varvec{s} \upharpoonright \varvec{h} \ne \varvec{\epsilon } \Rightarrow ede (\varvec{g}) \ne ede (\varvec{h})) \ \! \}\), where \(\varvec{s} \upharpoonright \varvec{g}\) is the jsubsequence of \(\varvec{s}\) that consists of moves x such that \( att _{G^{\dagger }}(x) = \varvec{g}\) yet changed into \( peel _{G^{\dagger }}(x)\).

\(M_{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} M_J + M_K\), where let \((\_)^{[0]}\) (resp. \((\_)^{[1]}\)) be the ‘tag’ on B in J (resp. K);

\(\lambda _{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} [\lambda _J \downharpoonright M_{B^{[0]}}, \lambda ^{+\mu }_J \upharpoonright M_{B^{[0]}}, \lambda ^{+\mu }_K \upharpoonright M_{B^{[1]}}, \lambda _K \downharpoonright M_{B^{[1]}}]\), where \(\lambda _G^{+ \mu } {\mathop {=}\limits ^{\mathrm {df. }}} \langle \lambda _G^{\mathsf {OP}}, \lambda _G^{\mathsf {QA}}, n \mapsto \lambda _G^{\mathbb {N}} (n) + \mu \rangle \) (G is J or K), and \(\mu {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Max}(\mu (J), \mu (K)) + 1\);

\(\star \vdash _{J \ddagger K} m {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \star \vdash _K m\);

\(m \vdash _{J \ddagger K} n \ (m \ne \star ) {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} m \vdash _J n \vee m \vdash _K n \vee (\star \vdash _{B^{[1]}} m \wedge \star \vdash _{B^{[0]}} n)\);

\(\Delta _{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} [\Delta _J, \Delta _K] \upharpoonright M_{J \ddagger K}\);

\(P_{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {J}_{J \ddagger K} \mid \varvec{s} \upharpoonright J \in P_J, \varvec{s} \upharpoonright K \in P_K, \varvec{s} \upharpoonright B^{[0]}, B^{[1]} \in pr _B \ \! \}\), where \( pr _B {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in P_{B^{[0]} \multimap B^{[1]}} \mid \forall \varvec{t} \preceq {\varvec{s}}. \ \mathsf {Even}(\varvec{t}) \Rightarrow \varvec{t} \upharpoonright B^{[0]} = \varvec{t} \upharpoonright B^{[1]} \ \! \}\).
Concatenation corresponds to composition without hiding in the introduction, and it plays a central role in [71]. We shall see later that concatenation \(\sigma \ddagger \tau \) of strategies\(\sigma : J\) and \(\tau : K\) (Definition 60), where Open image in new window and Open image in new window for some normalized games A, B and C, forms a welldefined strategy on the game \(J \ddagger K\), and also \(\mathcal {H}^\omega (\sigma ) : A \multimap B\), \(\mathcal {H}^\omega (\tau ) : B \multimap C\) and Open image in new window whence the composition (with hiding) \(\mathcal {H}^\omega (\sigma ) ; \mathcal {H}^\omega (\tau )\) of \(\mathcal {H}^\omega (\sigma )\) and \(\mathcal {H}^\omega (\tau )\) (Definition 60) satisfies \(\mathcal {H}^\omega (\sigma ) ; \mathcal {H}^\omega (\tau ) : A \multimap C\) (for Open image in new window implies \(\phi : H\) for any strategy \(\phi \) and games G and H; see [71] for the proof).
Note that Open image in new window is just the familiar relation \(\sigma ; \tau : A \multimap C\) when \(\sigma : J = A \multimap B\) and \(\tau : K = B \multimap C\); concatenation is to capture a generalization of this phenomenon.

We do not change moves of A or C, i.e., ones of the form \([(a, \mathscr {W})]_{\varvec{e}} \in M_J^{\mathsf {Ext}}\) or \([(c, \mathscr {E})]_{\varvec{f}} \in M_K^{\mathsf {Ext}}\);

We change moves of \(B^{[0]}\) in J, i.e., external ones of the form \([(b, \mathscr {E})]_{\varvec{g}}\), into \([((b, \mathscr {E}), \mathscr {S})]_{\varvec{g}}\);

We change moves of \(B^{[1]}\) in K, i.e., external ones of the form \([(b, \mathscr {W})]_{\varvec{g}}\), into \([((b, \mathscr {W}), \mathscr {N})]_{\varvec{g}}\);

We change internal moves \([m]_{\varvec{l}}\) of J into \([(m, \mathscr {S})]_{\varvec{l}}\);

We change internal moves \([n]_{\varvec{r}}\) of K into \([(n, \mathscr {N})]_{\varvec{r}}\).
Then, the labeling function, the enabling relation and the dummy function of \(J \ddagger K\) are defined by the obvious pattern matching on inner tags, and positions of \(J \ddagger K\) are defined as usual. Formally, concatenation of games is defined as follows, where it should be clear how the peeling \( peel _{J \ddagger K}\) and the attributes \( att _{J \ddagger K}\) work:
Definition 36
 $$\begin{aligned}&M_{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(a, \mathscr {W})]_{\varvec{e}} \mid [(a, \mathscr {W})]_{\varvec{e}} \in M_J^{\mathsf {Ext}}, [a]_{\varvec{e}} \in M_A \ \! \} \\&\quad \cup \{ [(c, \mathscr {E})]_{\varvec{f}} \mid [(c, \mathscr {E})]_{\varvec{f}} \in M_K^{\mathsf {Ext}}, [c]_{\varvec{f}} \in M_C \ \! \} \\&\quad \cup \{ [((b, \mathscr {E}), \mathscr {S})]_{\varvec{g}} \mid [(b, \mathscr {E})]_{\varvec{g}} \in M_J^{\mathsf {Ext}}, [b]_{\varvec{g}} \in M_B \ \! \} \\&\quad \cup \{ [((b, \mathscr {W}), \mathscr {N})]_{\varvec{g}} \mid [(b, \mathscr {W})]_{\varvec{g}} \in M_K^{\mathsf {Ext}}, [b]_{\varvec{g}} \in M_B \ \! \} \\&\quad \cup \{ [(m, \mathscr {S})]_{\varvec{l}} \mid [m]_{\varvec{l}} \in M_J^{\mathsf {Int}} \ \! \} \cup \{ [(n, \mathscr {N})]_{\varvec{r}} \mid [n]_{\varvec{r}} \in M_K^{\mathsf {Int}} \ \! \}; \end{aligned}$$

\(\lambda _{J \ddagger K}([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \lambda _J^{+\mu }([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {S} \wedge \exists [b]_{\varvec{e}} \in M_B . \ \! [m]_{\varvec{e}} = [(b, \mathscr {E})]_{\varvec{e}} \in M_J^{\mathsf {Ext}}; \\ \lambda _J([m]_{\varvec{e}}) &{}\text {if } X = \mathscr {W} \vee (X = \mathscr {S} \wedge [m]_{\varvec{e}} \in M_J^{\mathsf {Int}}); \\ \lambda _K^{+\mu }([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {N} \wedge \exists [b]_{\varvec{e}} \in M_B . \ \! [m]_{\varvec{e}} = [(b, \mathscr {W})]_{\varvec{e}} \in M_K^{\mathsf {Ext}}; \\ \lambda _K([m]_{\varvec{e}}) &{}\text {if }X = \mathscr {E} \vee (X = \mathscr {N} \wedge [m]_{\varvec{e}} \in M_K^{\mathsf {Int}}) \end{array}\right. }\) where \(\lambda _J^{+ \mu }\) and \(\lambda _K^{+ \mu }\) are as defined above;

\(\star \vdash _{J \ddagger K} [(m, X)]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} X = \mathscr {E} \wedge \star \vdash _C [m]_{\varvec{e}}\);

\([(m, X)]_{\varvec{e}} \vdash _{J \ddagger K} [(n, Y)]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} {\left\{ \begin{array}{ll} \begin{aligned} &{} ( att _{J \ddagger K}([(m, X)]_{\varvec{e}}) = J = att _{J \ddagger K}([(n, Y)]_{\varvec{f}}) \\ &{} \ \wedge peel _{J \ddagger K}([(m, X)]_{\varvec{e}}) \vdash _J peel _{J \ddagger K}([(n, Y)]_{\varvec{f}})) \\ &{}\vee ( att _{J \ddagger K}([(m, X)]_{\varvec{e}}) = K = att _{J \ddagger K}([(n, Y)]_{\varvec{f}}) \\ &{} \ \ \ \ \wedge peel _{J \ddagger K}([(m, X)]_{\varvec{e}}) \vdash _K peel _{J \ddagger K}([(n, Y)]_{\varvec{f}})) \\ &{}\vee (X = \mathscr {N} \wedge Y = \mathscr {S} \wedge \exists [b]_{\varvec{e}}, [b']_{\varvec{f}} \in M_B^{\mathsf {Init}} . \\ &{} \ \ \ \ \ m = (b, \mathscr {W}) \wedge n = (b, \mathscr {E})) \end{aligned} \end{array}\right. }\) where the function \( att _{J \ddagger K} : M_{J \ddagger K} \rightarrow \{ J, K \}\) is defined by \([(a, \mathscr {W})]_{\varvec{e}} \mapsto J\), \([(m, \mathscr {S})]_{\varvec{l}} \mapsto J\), \([((b, \mathscr {E}), \mathscr {S})]_{\varvec{g}} \mapsto J\), \([(c, \mathscr {E})]_{\varvec{f}} \mapsto K\), \([(n, \mathscr {N})]_{\varvec{r}} \mapsto K\), \([((b, \mathscr {W}), \mathscr {N})]_{\varvec{g}} \mapsto K\), and the function \( peel _{J \ddagger K} : M_{J \ddagger K} \rightarrow M_J \cup M_K\) by \([(a, \mathscr {W})]_{\varvec{e}} \mapsto [(a, \mathscr {W})]_{\varvec{e}}\), \([(c, \mathscr {E})]_{\varvec{f}} \mapsto [(c, \mathscr {E})]_{\varvec{f}}\), \([((b, \mathscr {E}), \mathscr {S})]_{\varvec{g}} \mapsto [(b, \mathscr {E})]_{\varvec{g}}\), \([((b, \mathscr {W}), \mathscr {N})]_{\varvec{g}} \mapsto [(b, \mathscr {W})]_{\varvec{g}}\), \([(m, \mathscr {S})]_{\varvec{l}} \mapsto [m]_{\varvec{l}}\), \([(n, \mathscr {N})]_{\varvec{r}} \mapsto [n]_{\varvec{r}}\);

\(\Delta _{J \ddagger K} ([(m, X)]_{\varvec{e}}) {\mathop {=}\limits ^{\mathrm {df. }}} \left\{ \begin{array}{ll} { [}(m', \mathscr {S})]_{\varvec{e}} &{}\text {if }X = \mathscr {S} \text { and } \Delta _J([m]_{\varvec{e}}) = [m']_{\varvec{e}}; \\ {[}(m'', \mathscr {N})]_{\varvec{e}} &{}\text {if }X = \mathscr {N}\text { and } \Delta _K([m]_{\varvec{e}}) = [m'']_{\varvec{e}}; \\ {[}((b, \mathscr {W}), \mathscr {N})]_{\varvec{e}} &{}\text {if }X = \mathscr {S}, \Delta _J([m]_{\varvec{e}}) \uparrow \text { and }m = (b, \mathscr {E}); \\ {[}((b, \mathscr {E}), \mathscr {S})]_{\varvec{e}} &{}\text {if }X = \mathscr {N}, \Delta _K([m]_{\varvec{e}}) \uparrow \text { and }m = (b, \mathscr {W}); \end{array}\right. \)

\(P_{J \ddagger K} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {J}_{J \ddagger K} \mid \varvec{s} \upharpoonright J \in P_J, \varvec{s} \upharpoonright K \in P_K, \varvec{s} \upharpoonright B^{[0]}, B^{[1]} \in pr _B \ \! \}\), where \(\varvec{s} \upharpoonright J\) (resp. \(\varvec{s} \upharpoonright K\)) is the jsubsequence of \(\varvec{s}\) that consists of moves m such that \( att _{J \ddagger K}(m) = J\) (resp. \( att _{J \ddagger K}(m) = K\)) yet changed into \( peel _{J \ddagger K}(m)\), \(\varvec{s} \upharpoonright B^{[0]}, B^{[1]}\) is the jsubsequence of \(\varvec{s}\) that consists of moves of \(B^{[0]}\) or \(B^{[1]}\), i.e., moves \([((b, X), Y)]_{\varvec{e}}\) such that \([b]_{\varvec{e}} \in M_B \wedge ((X = \mathscr {E} \wedge Y = \mathscr {S}) \vee (X = \mathscr {W} \wedge Y = \mathscr {N}))\) yet changed into \([(b, \overline{X})]_{\varvec{e}}\), for which \(\overline{\mathscr {E}} {\mathop {=}\limits ^{\mathrm {df. }}} \mathscr {W}\) and \(\overline{\mathscr {W}} {\mathop {=}\limits ^{\mathrm {df. }}} \mathscr {E}\), and \( pr _B {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{t} \in P_{B \multimap B} \mid \forall \varvec{u} \preceq {\varvec{t}}. \ \mathsf {Even}(\varvec{u}) \Rightarrow \varvec{u} \upharpoonright \mathscr {W} = \varvec{u} \upharpoonright \mathscr {E} \ \! \}\).
Example 37
Finally, let us recall the rather trivial currying\(\Lambda \) and uncurrying\(\Lambda ^{\circleddash }\) [6] of games. Roughly, currying \(\Lambda \) generalizes the map \(A \otimes B \multimap C \mapsto A \multimap (B \multimap C)\), and uncurrying \(\Lambda ^\circleddash \) does the inverse \(A \multimap (B \multimap C) \mapsto A \otimes B \multimap C\), where A, B and C are arbitrary normalized games. Note that the games \(A \otimes B \multimap C\) and \(A \multimap (B \multimap C)\) coincide up to ‘tags,’ and therefore, the currying and the uncurrying operations boil down to the trivial manipulations on ‘tags.’
Nevertheless, we formalize such manipulations of ‘tags’ here. For their simplicity, let us skip their informal definitions and just present the formal ones:
Definition 38
 $$\begin{aligned}&M_{\Lambda (G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [(a, \mathscr {W})]_{\varvec{e}} \mid [((a, \mathscr {W}), \mathscr {W})]_{\varvec{e}} \in M_G^{\mathsf {Ext}}, [a]_{\varvec{e}} \in M_A \ \! \} \\&\cup \{ [((b, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \mid [((b, \mathscr {E}), \mathscr {W})]_{\varvec{f}} \in M_G^{\mathsf {Ext}}, [b]_{\varvec{f}} \in M_B \ \! \} \\&\cup \{ [((c, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \mid [(c, \mathscr {E})]_{\varvec{g}} \in M_G^{\mathsf {Ext}}, [c]_{\varvec{g}} \in M_C \ \! \} \cup \{ [(m, \mathscr {N})]_{\varvec{h}} \mid [m]_{\varvec{h}} \in M_G^{\mathsf {Int}} \ \! \}; \end{aligned}$$

\(\lambda _{\Lambda (G)}(x) {\mathop {=}\limits ^{\mathrm {df. }}} \lambda _G( peel _{\Lambda (G)}(x))\) for all \(x \in M_{\Lambda (G)}\), where the map \( peel _{\Lambda (G)} : M_{\Lambda (G)} \rightarrow M_G\) is given by \([(a, \mathscr {W})]_{\varvec{e}} \mapsto [((a, \mathscr {W}), \mathscr {W})]_{\varvec{e}}\), \([((b, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \mapsto [((b, \mathscr {E}), \mathscr {W})]_{\varvec{f}}\), \([((c, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \mapsto [(c, \mathscr {E})]_{\varvec{g}}\), \([(m, \mathscr {N})]_{\varvec{h}} \mapsto [m]_{\varvec{h}}\);

\(\star \vdash _{\Lambda (G)} [m]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \exists [c]_{\varvec{e}} \in M_C^{\mathsf {Init}} . \ \! m = ((c, \mathscr {E}), \mathscr {E})\);

\([m]_{\varvec{e}} \vdash _{\Lambda (G)} [n]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} peel _{\Lambda (G)}([m]_{\varvec{e}}) \vdash _{G} peel _{\Lambda (G)}([n]_{\varvec{f}})\);

\(\Delta _{\Lambda (G)} : [(m, \mathscr {N})]_{\varvec{e}} \mapsto [(m', \mathscr {N})]_{\varvec{e}}\), where \(\Delta _G : [m]_{\varvec{e}} \mapsto [m']_{\varvec{e}}\);

\(P_{\Lambda (G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{\Lambda (G)} \mid peel _{\Lambda (G)}^*(\varvec{s}) \in P_G \ \! \}\), where pointers in \( peel _{\Lambda (G)}^*(\varvec{s})\) are inherited from the ones in \(\varvec{s}\) in the obvious sense.
It is easy to see that Open image in new window if Open image in new window which is a generalization of the equation \(\Lambda (A \otimes B \multimap C) = A \multimap (B \multimap C)\).
Definition 39
 $$\begin{aligned}&M_{\Lambda ^\circleddash (H)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [((a, \mathscr {W}), \mathscr {W})]_{\varvec{e}} \mid [(a, \mathscr {W})]_{\varvec{e}} \in M_H^{\mathsf {Ext}}, [a]_{\varvec{e}} \in M_A \ \! \} \\&\quad \cup \{ [((b, \mathscr {E}), \mathscr {W})]_{\varvec{f}} \mid [((b, \mathscr {W}), \mathscr {E})]_{\varvec{f}} \in M_H^{\mathsf {Ext}}, [b]_{\varvec{f}} \in M_B \ \! \} \\&\quad \cup \{ [(c, \mathscr {E})]_{\varvec{g}} \mid [((c, \mathscr {E}), \mathscr {E})]_{\varvec{g}} \in M_H^{\mathsf {Ext}}, [c]_{\varvec{g}} \in M_C \ \! \} \cup \{ [(m, \mathscr {S})]_{\varvec{h}} \mid [m]_{\varvec{h}} \in M_H^{\mathsf {Int}} \ \! \}; \end{aligned}$$

\(\lambda _{\Lambda ^\circleddash (H)}(x) {\mathop {=}\limits ^{\mathrm {df. }}} \lambda _H( peel _{\Lambda ^\circleddash (H)}(x))\) for all \(x \in M_{\Lambda ^\circleddash (H)}\), where the map \( peel _{\Lambda ^\circleddash (H)} : M_{\Lambda ^\circleddash (H)} \rightarrow M_H\) is defined by \([((a, \mathscr {W}), \mathscr {W})]_{\varvec{e}} \mapsto [(a, \mathscr {W})]_{\varvec{e}}\), \([((b, \mathscr {E}), \mathscr {W})]_{\varvec{f}} \mapsto [((b, \mathscr {W}), \mathscr {E})]_{\varvec{f}}\), \([(c, \mathscr {E})]_{\varvec{g}} \mapsto [((c, \mathscr {E}), \mathscr {E})]_{\varvec{g}}\), \([(m, \mathscr {S})]_{\varvec{h}} \mapsto [m]_{\varvec{h}}\);

\(\star \vdash _{\Lambda ^\circleddash (H)} [m]_{\varvec{e}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \exists [c]_{\varvec{e}} \in M_C^{\mathsf {Init}} . \ \! m = (c, \mathscr {E})\);

\([m]_{\varvec{e}} \vdash _{\Lambda ^\circleddash (H)} [n]_{\varvec{f}} {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} peel _{\Lambda ^\circleddash (H)}([m]_{\varvec{e}}) \vdash _{H} peel _{\Lambda ^\circleddash (H)}([n]_{\varvec{f}})\);

\(\Delta _{\Lambda ^\circleddash (H)} : [(m, \mathscr {S})]_{\varvec{h}} \mapsto [(m', \mathscr {S})]_{\varvec{h}}\), where \(\Delta _H : [m]_{\varvec{h}} \mapsto [m']_{\varvec{h}}\);

\(P_{\Lambda ^\circleddash (H)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \varvec{s} \in \mathscr {L}_{\Lambda ^\circleddash (H)} \mid peel _{\Lambda ^\circleddash (H)}^*(\varvec{s}) \in P_H \ \! \}\), where pointers in \( peel _{\Lambda ^\circleddash (H)}^*(\varvec{s})\) are inherited from the ones in \(\varvec{s}\).
Dually to currying, Open image in new window holds if Open image in new window which generalizes the equation \(\Lambda ^\circleddash (A \multimap (B \multimap C)) = A \otimes B \multimap C\). It is also easy to see that \(\Lambda \) and \(\Lambda ^\circleddash \) are inverses to each other on normalized games, i.e., \(\Lambda \circ \Lambda ^\circleddash (A \multimap (B \multimap C)) = A \multimap (B \multimap C)\) and \(\Lambda ^\circleddash \circ \Lambda (A \otimes B \multimap C) = A \otimes B \multimap C\) for any normalized games A, B and C. Note also that currying \(\Lambda (G)\) and the uncurrying \(\Lambda ^\circleddash (H)\) both do not depend on the choice of the normalized games A, B and C such that Open image in new window and Open image in new window
The following are two of the technical highlights of [71], where it is straightforward to see that the additional structure of dummy functions and the strengthened axiom DP2 are preserved under the constructions on games introduced so far (and therefore the two results still hold).
Theorem 40
(Constructions on games [71]) Games are closed under \(\otimes \), \(\multimap \),&, \(\langle \_, \_ \rangle \), !, \((\_)^{\dagger }\), \(\ddagger \), \(\Lambda \) and \(\Lambda ^{\circleddash }\). Moreover, the subgame relation is preserved under these constructions, i.e., Open image in new window if Open image in new window for all \(i \in I\), where \(\clubsuit _{i \in I}\) is either \(\otimes \), \(\multimap \),&, \(\langle \_, \_ \rangle \), !, \((\_)^{\dagger }\), \(\ddagger \), \(\Lambda \) or \(\Lambda ^{\circleddash }\) (and thus, I is either \(\{ 1 \}\) or \(\{ 1, 2 \}\)).
Lemma 41
 1
Open image in new window if \(\clubsuit _{i \in I}\) is not \(\ddagger \);
 2
\(\mathcal {H}^d (G_1 \ddagger G_2) = \mathcal {H}^d(G_1) \ddagger \mathcal {H}^d(G_2)\) if \(\mathcal {H}^d(G_1 \ddagger G_2)\) is not yet normalized, and Open image in new window otherwise, where Open image in new window and Open image in new window for some normalized game B.
2.3 Strategies
Next, let us recall another central notion of strategies.
2.3.1 Strategies
Our strategies are the dynamic variant introduced in [71]. However, there is nothing special in the definition: A dynamic strategy on a (dynamic) game is a strategy on the game in the conventional sense [6, 51], i.e.,
Definition 42

(S1) It is nonempty and evenprefixclosed (i.e., \(\varvec{s}mn \in \sigma \Rightarrow \varvec{s} \in \sigma )\);

(S2) It is deterministic (i.e., \(\varvec{s}mn, \varvec{s'}m'n' \in \sigma \wedge \varvec{s}m = \varvec{s'}m' \Rightarrow \varvec{s}mn = \varvec{s'}m'n'\)).
Convention
Henceforth, strategies refer to dynamic strategies by default.
Notation
Henceforth, we often indicate the form of tags of moves \([m_{X_1 X_2 \dots X_k}]_{\varvec{e}}\) of a game G informally by \([G_{X_1 X_2 \dots X_k}]_{\varvec{e}}\).
Example 43
Example 44
Example 45
Next, let us recall two constraints on strategies: innocence and well bracketing. One of the highlights of HOgames [38] is to establish a onetoone correspondence between terms of PCF in a certain \(\eta \)long normal form, known as PCF Böhm trees [8], and innocent, wellbracketed strategies (on games modeling types of PCF). That is, the two conditions limit the codomain of the interpretation of PCF, i.e., the category of HOgames, in such a way that the interpretation becomes full.
Roughly, a strategy is innocent if its computation depends only on the Pview of each oddlength position (rather than the entire position), and well bracketed if every ‘questionanswering’ by the strategy is done in the ‘lastquestionfirstanswered’ fashion. Formally:
Definition 46
(Innocence of strategies [6, 38, 51]) A strategy \(\sigma : G\) is innocent if \(\varvec{s}mn, \varvec{t} \in \sigma \wedge \varvec{t} m \in P_G \wedge \lceil \varvec{t} m \rceil _G = \lceil \varvec{s} m \rceil _G \Rightarrow \varvec{t}mn \in \sigma \wedge \lceil \varvec{t}mn \rceil _G = \lceil \varvec{s}mn \rceil _G\).
Definition 47
(Well bracketing of strategies [6, 38, 51]) A strategy \(\sigma : G\) is well bracketed if, whenever \(\varvec{s} q \varvec{t} a \in \sigma \), where q is a question that justifies an answer a, every question in \(\varvec{t'}\) defined by \(\lceil \varvec{s} q \varvec{t} \rceil _G = \lceil \varvec{s} q \rceil _G . \varvec{t'}\)^{12} justifies an answer in \(\varvec{t'}\).
The bijective correspondence holds also for the game model [6], on which our games and strategies are based. Moreover, it corresponds to modeling states and control operators to relaxing innocence and well bracketing in the model; in this sense, the two conditions characterize ‘purely functional’ languages [6].
Note that innocence and well bracketing have been imposed on strategies in order to establish full abstraction and/or definability [15], but neither is our main concern in the present paper. However, we would like P to be able to collect a bounded number of ‘relevant’ moves from each oddlength position in an ‘effective’ fashion; for this point, it is convenient to focus on innocent strategies since it then suffices for P to trace back the chain of justifiers. In fact, we shall define our notion of ‘effective computability’ only on innocent strategies in Sect. 3.
On the other hand, we do not impose well bracketing on strategies (thus, control operators are ‘effective’ in our sense); nevertheless, we shall consider only strategies modeling terms of PCF in the present work, which are all well bracketed.
Remark
We conjecture that it is possible to define ‘effectivity’ of noninnocent strategies in a fashion similar to the case of innocent ones defined in Sect. 3.1. For this, however, we need to modify the procedure for P to collect a bounded number of moves from each oddlength position (defined in Sect. 3.1) so that she may refer to moves outside of Pviews, which is left as future work.
Convention
From now on, a strategy refers to an innocent strategy by default. We may clearly regard innocent strategies \(\sigma : G\) as (partial) view functions\(f_\sigma : \lceil P_G^{\mathsf {Odd}} \rceil _G \rightharpoonup M_G\) with the pointer structure implicit (see [51] for the details); we shall freely exchange the treerepresentation \(\sigma \) and the function representation \(f_\sigma \), and often write \(\sigma \) for \(f_\sigma \).
As in the case of games, we now define the hiding operation on strategies. Note that an evenlength position is not necessarily preserved under the hiding operation on jsequences. For instance, let \(\varvec{s} m n \varvec{t}\) be an evenlength position of a game G such that \(\varvec{s} m\) (resp. \(\varvec{t} n\)) consists of external (resp. internal) moves only. By IEswitch on G, m is an Omove, and so \(\mathcal {H}^\omega (\varvec{s} m n \varvec{t}) = \varvec{s} m\) is of oddlength. Thus, we define:
Definition 48
The following beautiful theorem in a sense implies that the above definition is a reasonable one. Also, it induces the hiding functor\(\mathcal {H}^\omega \) from the bicategory \(\mathcal {DG}\) of dynamic games and strategies to the category \(\mathcal {G}\) of static games and strategies [71].
Theorem 49
(Hiding theorem [71]) If \(\sigma : G\), then \(\mathcal {H}^d(\sigma ) : \mathcal {H}^d(G)\) for all \(d \in \mathbb {N} \cup \{ \omega \}\), and \(\underbrace{\mathcal {H}^1 \circ \mathcal {H}^1 \cdots \circ \mathcal {H}^1}_i (\sigma ) = \mathcal {H}^i(\sigma ) : \mathcal {H}^i(G)\) for all \(i \in \mathbb {N}\).
Convention
Thanks to Theorem 49, the ihiding operation \(\mathcal {H}^i\) on strategies for each \(i \in \mathbb {N}\) can be thought of as the itimes iteration of the 1hiding operation \(\mathcal {H}^1\), which we call the hiding operation (on strategies) and write \(\mathcal {H}\) for it.
It is straightforward to see that normalized games (resp. strategies) are equivalent to static games (resp. strategies) given in [6]; see [71] for the details.
2.3.2 Constructions on strategies
Next, let us review standard constructions on strategies [6, 51], for which we need to adopt our tags. Having introduced our formalization of ‘tags’ for constructions on games in Sect. 2.2.3, let us just present formalized constructions on strategies (without standard, informal versions) as it should be clear enough.
First, the following derelictions just ‘copycat’ the last occurrence of an Omove:
Definition 50
Example 51
Next, as in the case of tensor of games, we have:
Definition 52
Intuitively, the tensor \(\sigma \! \otimes \! \tau : A \otimes B \multimap C \otimes D\) plays by \(\sigma \) if the last occurrence of an Omove belongs to \(A \multimap C\), and by \(\tau \) otherwise.
Example 53
The next one is the pairing in the category \(\mathcal {G}\) of static games and strategies [6]:
Definition 54
However, for the bicategory \(\mathcal {DG}\) of dynamic games and strategies [71], we need the following generalization (for the reason explained right before Definition 31):
Definition 55
It is clearly a generalization of static pairing; consider the case where \(L = C \multimap A\) and \(R = C \multimap B\).
Convention
Henceforth, pairing of strategies refers to the generalized one.
Example 56
Next, let us recall promotion of strategies:
Definition 57
As stated before, [71] generalizes promotion of strategies (for the reason explained before Definition 35) as follows:
Definition 58
It is clearly a generalization of static promotion; consider the case \(G = \ !A \multimap B\).
Convention
Henceforth, promotion of strategies refers to the generalized one.
Example 59
Now, let us recall a central construction of strategies in [71], which reformulates composition (with hiding) of strategies as follows:
Definition 60
We also write \(\tau \circ \sigma \) for \(\sigma ; \tau \). If \(J = A \multimap B\), \(K = B \multimap C\), then our composition Open image in new window coincides with the standard one [6, 38, 51]; see [71] for the details. In this sense, our composition generalizes the standard one, and moreover it is decomposed into concatenation plus hiding.
Example 61
Finally, we recall the currying and the uncurrying of strategies:
Definition 62
Theorem 63
(Constructions on strategies [71]) Derelictions are welldefined strategies, and strategies are closed under \(\otimes \), \(\langle \_, \_ \rangle \), \((\_)^{\dagger }\), \(\ddagger \), \(\Lambda \) and \(\Lambda ^\circleddash \).
Lemma 64
 1
\(\mathcal {H}^d(\spadesuit _{i \in I}\sigma _i) = \spadesuit _{i \in I} \mathcal {H}^d(\sigma _i)\) if \(\spadesuit _{i \in I}\) is not \(\ddagger \);
 2
\(\mathcal {H}^d (\sigma _1 \ddagger \sigma _2) = \mathcal {H}^d(\sigma _1) \ddagger \mathcal {H}^d(\sigma _2)\) if \(\mathcal {H}^d(\sigma _1 \ddagger \sigma _2)\) is not yet normalized, and \(\mathcal {H}^d(\sigma _1 \ddagger \sigma _2) = \mathcal {H}^d(\sigma _1) ; \mathcal {H}^d(\sigma _2)\) otherwise.
3 Viable strategies
We have defined our games and strategies in the previous section. In this main section of the present paper, we introduce a novel notion of ‘effective’ or viable strategies, and show that viable (dynamic)^{14} strategies subsume all computations of the programming language PCF [58, 61], and thus they are Turing complete in particular.
In Sect. 3.1, we define viability of strategies and show that it is preserved under all the constructions on strategies defined in Sect. 2.3.2. We then describe various examples of viable strategies in Sect. 3.2, based on which we finally prove in Sect. 3.3 that viable (dynamic) strategies may interpret all terms of PCF.
3.1 Viable strategies
The idea of viable strategies is as follows. First, it seems necessary to restrict the number of previous occurrences which P is allowed to look at (to calculate the next Pmove) to a bounded one since the number of oddlength positions of a game can be infinite, e.g., consider the game \(\mathcal {N}\) (Example 23).^{15} Fortunately, to model the language PCF, it turns out that strategies only need to read off at most the last three occurrences of each Pview (and possibly a few initial or internal moves, which are easily identified as well) as we shall see, which is clearly ‘effective’ in an informal sense. Thus, it remains to formulate how strategies ‘effectively’ compute the next Pmove from such a bounded number of previous occurrences. Note that (as already mentioned) computation of internal Omoves should be done by P, but it is rather trivial by the axiom DP2 (Definition 18), and therefore, we shall omit it for brevity. Note also that we shall focus on innocent strategies as a means to narrow down previous occurrences to be concerned with.^{16}
As the set \(\pi _1(M_G)\) is finite for any game G (Definition 8), innocent strategies that are finitary in the sense that their view functions are finite seem sufficient at first glance. However, to model fixedpoint combinators in PCF, strategies need to initiate new threads unboundedly many times [7, 38] (Example 75); also, ‘effective’ strategies have to be closed under promotion (Definition 57) for modeling PCF, in which possible outer tags are infinitely many. Thus, finitary strategies are not strong enough.
Then, how can we give a stronger notion of ‘effectivity’ of the next Pmove from (a bounded number of) previous occurrences solely in terms of games and strategies? Our solution, which is the main achievement of the present work, is to define a strategy \(\sigma : G\) to be ‘effective’ or viable (Definition 70) if it is ‘describable’ by a finitary strategy, called an instruction strategy for \(\sigma \) (Definition 68), on the instruction game for G (Definition 65); see Sect. 1.7 for an illustration of the idea.
Having explained the idea of viable strategies, let us proceed in the present section to make it mathematically precise.
Notation
Given a game G, we assign a symbol \(\mathsf {m}\) to each \(m \in \pi _1(M_G)\), for which we may assume that these symbols are pairwise distinct for the set \(\pi _1 (M_G)\) is finite, and define \(\mathsf {Sym}(\pi _1 (M_G)) {\mathop {=}\limits ^{\mathrm {df. }}} \{ \mathsf {m} \mid m \in \pi _1(M_G) \}\). Also, we assign symbols to elements of outer tags by the map \(\mathscr {C} : \ell \mapsto \prime \), \(\hbar \mapsto \sharp \), Open image in new window , Open image in new window , and define \(\mathsf {T} {\mathop {=}\limits ^{\mathrm {df. }}} \{ \prime , \sharp , \langle , \rangle \}\). (N.b., these symbols are technically not necessary at all; they are just for conceptual clarity.)
Definition 65

\(M_{\mathcal {G}(M_G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ [\hat{q}], [q], [\square ], [\checkmark ] \} \cup \{ [\mathsf {m}] \mid \mathsf {m} \in \mathsf {Sym}(\pi _1(M_G)) \} \cup \{ [\mathsf {e}] \mid \mathsf {e} \in \mathsf {T} \ \! \}\), where the elements of \(M_{\mathcal {G}(M_G)}\) are assumed to be pairwise distinct;

\(\lambda _{\mathcal {G}(M_G)} : [\hat{q}] \mapsto (\mathsf {O}, \mathsf {Q}, 0)\), \([q] \mapsto (\mathsf {O}, \mathsf {Q}, 0)\), \([\square ] \mapsto (\mathsf {P}, \mathsf {A}, 0)\), \([\checkmark ] \mapsto (\mathsf {P}, \mathsf {A}, 0)\), \([\mathsf {m}] \mapsto (\mathsf {P}, \mathsf {A}, 0)\), \([\mathsf {e}] \mapsto (\mathsf {P}, \mathsf {A}, 0)\);

\(\vdash _{\mathcal {G}(M_G)} {\mathop {=}\limits ^{\mathrm {df. }}} \{ (\star , [\hat{q}]), ([\hat{q}], [\square ]), ([q], [\checkmark ]) \} \cup \{ ([\hat{q}], [\mathsf {m}]) \mid \mathsf {m} \in \mathsf {Sym}(\pi _1(M_G)) \} \cup \{ ([\mathsf {m}], [q]) \mid \mathsf {m} \in \mathsf {Sym}(\pi _1(M_G)) \} \cup \{ ([q], [\mathsf {e}]) \mid \mathsf {e} \in \mathsf {T} \ \! \} \cup \{ ([\mathsf {e}], [q]) \mid \mathsf {e} \in \mathsf {T} \ \! \}\);

\(\Delta _{\mathcal {G}(M_G)} {\mathop {=}\limits ^{\mathrm {df. }}} \emptyset \);

\(P_{\mathcal {G}(M_G)} {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref} (\{ [\hat{q}] [\square ] \} \cup \{ [\hat{q}] [\mathsf {m}] [q] [\mathscr {C}(e_1)] [q] [\mathscr {C}(e_2)] \dots [q] [\mathscr {C}(e_k)] [q] [\checkmark ] \mid \mathsf {m} \in \mathsf {Sym}(\pi _1(M_G)), e_1 e_2 \dots e_k \in \mathcal {T} \ \! \})\), where each noninitial occurrence is justified by the last occurrence.
Convention
We loosely call games of the form \( \mathcal {G}(M_G) \& \mathcal {G}(M_G) \& \mathcal {G}(M_G) \Rightarrow \mathcal {G}(M_G)\), where G is some game, instruction games as well. For brevity, we write ‘tags’ for instruction games \( \mathcal {G}(M_G) \& \mathcal {G}(M_G) \& \mathcal {G}(M_G) \Rightarrow \mathcal {G}(M_G)\)informally, i.e., \( \mathcal {G}(M_G)^{[0]} \& \mathcal {G}(M_G)^{[1]} \& \mathcal {G}(M_G)^{[2]} \Rightarrow \mathcal {G}(M_G)^{[3]}\), \(\hat{q}^{[0]}\), \(q^{[1]}\), \(\sharp ^{[2]}\), \(\checkmark ^{[3]}\), etc.
The positions \([\hat{q}] [\mathsf {m}] [q] [\mathscr {C}(e_1)] [q] [\mathscr {C}(e_2)] \dots [q] [\mathscr {C}(e_k)] [q] [\checkmark ]\) and \([\hat{q}_G] [\square ]\) of \(\mathcal {G}(M_G)\) are to represent the move \([m]_{e_1 e_2 \dots e_k} \in M_G\) (if it exists) and ‘no move,’ respectively. Note that pointers in these positions are all trivial, and thus, we usually omit them.
Notation
However, there remain two problems. The first one is the pairing \(\langle \sigma , \tau \rangle : \langle L, R \rangle \) of strategies \(\sigma : L\) and \(\tau : R\) such that Open image in new window and Open image in new window for some normalized games A, B and C: Because moves of C are common to \(\sigma \) and \(\tau \), the last three occurrences of each Pview may not suffice; the pairing \(\langle \sigma , \tau \rangle \) needs to know whether A or B the first occurrence of each position of \(\langle L, R \rangle \) belongs to. Also, the occurrence becomes no longer initial as soon as the pairing is postconcatenated; thus, it does not suffice to trace the first occurrence of each position. We shall overcome this point by collecting necessary information as states (Definition 67).
The second one is how to ‘effectively’ calculate the ‘relevant’ (and finite) part of outer tags represented by moves occurring in an instruction game (for the number of all outer tags is infinite). For this point, we introduce the notion of mviews:
Definition 66
(Mviews) Let G be a game, and assume \(\varvec{s} \in P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}\), where we omit tags in \(\varvec{s}\) in the present definition for brevity. Pairing each occurrence of \(\langle \) in \(\varvec{s}\) with the most recent yet unpaired occurrence of \(\rangle \) in \(\varvec{s}\), one component of such a pair is called the mate of the other. The depth of an occurrence of \(\langle \) in \(\varvec{s}\) is the number of previous occurrences of \(\langle \) in the same component game \(\mathcal {G}(M_G)\) whose mate does not occur before that occurrence; the depth of an occurrence of \(\rangle \) in \(\varvec{s}\) is the depth of its mate in \(\varvec{s}\). The matching view (mview) \(\llbracket \varvec{s} \rrbracket _G^d\) of \(\varvec{s}\) up to depth \(d \in \mathbb {N}\) is the subsequence of \(\varvec{s}\) that consists of occurrences of \(\langle \) or \(\rangle \) of depth \(\leqslant d\).
It is clearly ‘effective’ to calculate the mview of a given position of an instruction game in an informal sense. For instance, deterministic pushdown automata [35, 48, 64] may compute mviews into the stack, where we assume that positions of games are written on the input tape, in the obvious manner. We may even dispense with a stack by embedding the depth d of each occurrence of \(\langle \) or \(\rangle \) by the dtimes iteration of \(q . \prime \) right after the occurrence in positions of instruction games (for which we need to slightly modify the notion of instruction games accordingly). Nevertheless, for simplicity, we shall not specify a method for the calculation of mviews.
Notation
We are now ready to make the notion of ‘describable by a finitary strategy’ precise:
Definition 67

\(\mathcal {S}_{\mathcal {A}} \subseteq \pi _1(M_G)^*\) is a finite set, whose elements are called states;

\(\partial _{\varvec{m}} (\varvec{t}x) {\mathop {=}\limits ^{\mathrm {df. }}} (\varvec{t}x \! \downharpoonright \mathcal {A}_{\varvec{m}}, \llbracket \varvec{t}x \rrbracket _G^{\Vert \mathcal {A}_{\varvec{m}}\Vert })\) for all \( \varvec{t}x \in P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G) \& \varvec{2}}^{\mathsf {Odd}}\)

(Q)\([m]_{\varvec{e}} \in M_G^{\mathsf {Init}} \Rightarrow \mathcal {Q}_{\mathcal {A}}(m) = \top \).
Remark
Note that it does not make a difference if each stalgorithm \(\mathcal {A} {:}{:} G\) focuses on the Pviews of each \( \varvec{t}x \in P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G) \& \varvec{2}}^{\mathsf {Odd}}\) for \(\lceil \varvec{t} x \rceil = \varvec{t} x\) by the trivial pointers.
Definition 68
Convention
Given an stalgorithm \(\mathcal {A} {:}{:} G\), each instruction strategy \(\mathcal {A}_{\varvec{m}}^\circledS \) has to specify pointers in \(P_G\), which in the present work are always either the last or the third last occurrence, or the justifier of the second occurrence of the Pview of each oddlength position of G (as we shall see); it is why \(\mathcal {A}_{\varvec{m}}^\circledS \) is on the game \( \mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G) \& \varvec{2}\), not the instruction game \(\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\), so that it may specify the ternary choice on the justifiers in the component game \(\varvec{2}\) (by \( tt \), \( ff \) or ‘no answer’). However, since justifiers in \(P_G\) occurring in this paper are all obvious ones, we henceforth regard \(\mathcal {A}_{\varvec{m}}\) and \(\mathcal {A}_{\varvec{m}}^\circledS \) as \(\mathcal {A}_{\varvec{m}} : \partial _{\varvec{m}} (P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}^{\mathsf {Odd}}) \rightharpoonup M_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}\) and \(\mathcal {A}_{\varvec{m}}^\circledS : \mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\), respectively, keeping the justifiers implicit.
Remark
Since an stalgorithm \(\mathcal {A} {:}{:} G\) refers to mviews only occasionally, we regard each \(\mathcal {A}_{\varvec{m}}\) as a partial function \(\{ \varvec{t}x \! \downharpoonright \mathcal {A}_{\varvec{m}} \mid \varvec{t}x \in P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}^{\mathsf {Odd}} \} \rightharpoonup M_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}\) in most cases. Accordingly, \(\mathcal {A}_{\varvec{m}}^\circledS \) is mostly a strategy on the game \(\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\) whose partial function representation \(\mathcal {A}_{\varvec{m}}\) is finite.
Thus, an instruction strategy is a strategy on the game \(\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\), where G is a game, that is finitary in the sense that it is representable by a finite partial function, and so it is clearly ‘effective’ in an informal sense. We shall see that the number 3 on \(\mathcal {G}(M_G)^3\) is the least number to achieve Turing completeness in Sect. 3.3. As already mentioned, our idea is to utilize such an instruction strategy as a ‘description’ of a strategy on G, which may be ‘effectively’ read off:
Definition 69
Clearly, \(\mathcal {A} {:}{:} G \Rightarrow \mathsf {st}(\mathcal {A}) : G\) holds. We are now ready to define the central notion of the present work, namely ‘effective computability’ of strategies:
Definition 70
(Viability of strategies) A strategy \(\sigma : G\) is viable if there exists an stalgorithm \(\mathcal {A} {:}{:} G\) that realizes \(\sigma \), i.e., \(\mathsf {st}(\mathcal {A}) = \sigma \).
That is, a strategy \(\sigma : G\) is viable if there is a finitary strategy on \(\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\) that ‘describes’ the computation of \(\sigma \). The terms realize and realizability come from mathematical logic, in which a realizer refers to some computational information that ‘realizes’ the constructive truth of a mathematical statement [65].
 1
Given \(\varvec{s}a \in P_G^{\mathsf {Odd}}\), P calculates the current state \(\varvec{m} {\mathop {=}\limits ^{\mathrm {df. }}} \mathcal {Q}_{\mathcal {A}}^\star (\lceil \varvec{s}a \rceil )\) and the last (up to) three moves \(\lceil \varvec{s}a \rceil \! \downharpoonright 3\) of the Pview; if \(\varvec{m} \not \in \mathcal {S}_{\mathcal {A}}\), then she stops, i.e., the next move is undefined;
 2
Otherwise, she composes \((\underline{\lceil \varvec{s}a \rceil \! \downharpoonright 3}_3)^{\dagger }\) with \(\mathcal {A}_{\varvec{m}}^\circledS \), calculating \(\mathcal {A}_{\varvec{m}}^\circledS \circ (\underline{\lceil \varvec{s}a \rceil \! \downharpoonright 3}_3)^{\dagger }\);
 3
Finally, she reads off the next move \(\mathcal {M}(\mathcal {A}_{\varvec{m}}^\circledS \circ (\underline{\lceil \varvec{s}a \rceil \! \downharpoonright 3}_3)^{\dagger })\) (and its justifier) and performs that move (with the pointer).
Note that there are two kinds of processes in viable strategies \(\sigma : G\). The first one is the process of \(\sigma \) per se whose atomic steps are \((\varvec{s} a \in P_G^{\mathsf {Odd}}) \mapsto \varvec{s}a . \sigma (\lceil \varvec{s}a \rceil )\), and the second one is the process of an stalgorithm \(\mathcal {A}\) realizing \(\sigma \) whose atomic steps are \(\varvec{t} x \in P_{\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)}^{\mathsf {Odd}} \mapsto \varvec{t} x . \mathcal {A}_{\varvec{m}} \circ \partial _{\varvec{m}}(\varvec{t} x)\), where \(\varvec{m}\) is the current state. The former is abstract and highlevel, while the latter is symbolic and lowlevel. In this manner, we have achieved a mathematical formulation of highlevel and lowlevel computational processes and ‘effective computability’ of the former in terms of the latter (as promised in the introduction).
Henceforth, in order to establish Theorem 76, which is a key result for the main theorem (Theorem 81), we shall focus on the following stalgorithms:
Definition 71
 1.
The symbol \(\square \) does not occur in \(\mathcal {A}_{\varvec{m}}\) for any \(\varvec{m} \in \mathcal {S}_{\mathcal {A}}\);
 2.
It does not refer to any input outer tag when it computes an inner element, i.e., if \( \hat{q}^{[3]} . \varvec{s} . \mathsf {n}^{[3]} \in \mathcal {A}^\circledS _{\varvec{m}} : \mathcal {G}(M_G)^{[0]} \& \mathcal {G}(M_G)^{[1]} \& \mathcal {G}(M_G)^{[2]} \Rightarrow \mathcal {G}(M_G)^{[3]}\), where \(\varvec{m} \in \mathcal {S}_{\mathcal {A}}\) and \(n \in \pi _1(M_G)\), then \(q^{[0]}\), \(q^{[1]}\) and \(q^{[2]}\) do not occur in \(\varvec{s}\);
 3.
If it refers to an input outer tag, then it must belongs to the last move of the current Pview of G, i.e., if q occurs as a Pmove in some \(\varvec{s} \in \mathcal {A}^\circledS _{\varvec{m}}\), where \(\varvec{m} \in \mathcal {S}_{\mathcal {A}}\), then the ‘tag’ on the move is \((\_)^{[2]}\).
Convention
Henceforth, stalgorithms refer to standard ones by default. Of course, standardness is closed under all constructions on stalgorithms (see the proof of Theorem 76), and stalgorithms given below are all standard.
Example 72
Example 73
Example 74
Clearly \(\mathsf {st}(\mathcal {A}( pred )) = pred \), establishing viability of \( pred \). Also, it is easy to see that \(\mathcal {A}( pred )\) is standard.
Example 75

After an opening occurrence \([a_\mathscr {E}]_{\varvec{g}}\), \( fix _A\) copies it and performs the move Open image in new window with the pointer toward the initial occurrence \([a_\mathscr {E}]_{\varvec{g}}\);

If O initiates a new thread in the inner implication by a move Open image in new window , then \( fix _A\) copies it and launches a new thread in the outer implication by performing the move Open image in new window with the pointer toward the justifier of the second last occurrence of the current Pview;

If O performs a move Open image in new window (resp. Open image in new window , Open image in new window , \([a''_{\mathscr {E}}]_{\varvec{f}}\)) in an existing thread, then \( fix _A\) copies it and performs the move Open image in new window (resp. \([a''_{\mathscr {E}}]_{\varvec{f}}\), Open image in new window , Open image in new window ) in the dual thread, i.e., the thread to which the third last occurrence of the current Pview belongs, with the pointer toward the third last occurrence.

If the rightmost component is of the form \((\underline{[a_\mathscr {E}]_{\varvec{f}}})^{\dagger }\), then \(\mathcal {A}( fix _A)_{m}^\circledS \) recognizes it by the inner tag \(\mathscr {E}\), and calculates the next move Open image in new window once and for all for the inner element \(a_{\mathscr {EW}}\) and ‘digitbydigit’ for the outer tag Open image in new window (by giving Open image in new window and copying \(\varvec{f}\));

If the rightmost component is of the form Open image in new window , then the leftmost component (which corresponds to the third last occurrence of the current Pview of the game \((A \Rightarrow A) \Rightarrow A\)) is of the form \((\underline{[a'_\mathscr {E}]_{\varvec{f}}})^{\dagger }\), and thus \(\mathcal {A}( fix _A)_{m}^\circledS \) recognizes it by the inner tags \(\mathscr {EW}\) and \(\mathscr {E}\), and calculates the next move \([a_\mathscr {E}]_{\varvec{f}}\) once and for all for the inner element \(a_\mathscr {E}\) and ‘digitbydigit’ for the outer tag \(\varvec{f}\) (by ignoring Open image in new window and copying \(\varvec{f}\));

If the rightmost component is of the form Open image in new window , then \(\mathcal {A}( fix _A)_{m}^\circledS \) calculates the next move Open image in new window similarly to the above case yet with the help of mviews for the outer tag (see Sect. 3.2 for the details);

If the rightmost component is of the form Open image in new window , then \(\mathcal {A}( fix _A)_{m}^\circledS \) calculates the next move Open image in new window in a similar manner to the above case with the help of mviews for the outer tag (see Sect. 3.2 for the details);
We now turn to establishing a key theorem, which states that viability of strategies is preserved under all the constructions on strategies defined in Sect. 2.3.2:
Theorem 76
(Preservation of viability) Viable strategies are closed under tensor \(\otimes \), pairing \(\langle \_, \_ \rangle \), promotion \((\_)^{\dagger }\), concatenation \(\ddagger \), currying \(\Lambda \) and uncurrying \(\Lambda ^\circleddash \) if the underlying stalgorithms are standard, where the standardness is also preserved.
Proof
It is clear that pairing of strategies may be handled in a completely similar manner; currying and uncurrying are even simpler. Thus, we skip the proof for them.
 1.Let \(\varvec{t} [m]_{\varvec{\tilde{e}}} [n]_{\varvec{e}} \in \varphi \) and \(\varvec{t'} [m]_{\varvec{\tilde{e}'}} [n]_{\varvec{e'}} \in \varphi ^{\dagger }\) such that \(\mathcal {Q}_{\mathcal {A}(\varphi ^{\dagger })}^\star (\lceil \varvec{t'}[m]_{\varvec{\tilde{e}}} \rceil ) = \varvec{s}\) and \(\varvec{t'} [m]_{\varvec{\tilde{e}'}} [n]_{\varvec{e'}} \upharpoonright \varvec{\hat{e}} = \varvec{t} [m]_{\varvec{\tilde{e}}} [n]_{\varvec{e}}\), where the initial move that hereditarily justifies \([n]_{\varvec{e'}}\) in \(\varphi ^{\dagger }\) is of the form Open image in new window . Let us describe how \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) calculates the representation \(\mathsf {n} . \mathscr {C}^*(\varvec{e'})\) of the next move \([n]_{\varvec{e'}}\) by a case analysis on m and n:

If m and n both belong to A, which \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) may recognize by the method described below, then \(\varvec{\tilde{e}}\), \(\varvec{e}\), \(\varvec{\tilde{e}'}\) and \(\varvec{e'}\) are, respectively, of the form Open image in new window , Open image in new window , Open image in new window and Open image in new window .^{20} Then, with the help of mviews, \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) calculates the first half \(\mathsf {n} . \langle \langle \mathscr {C}^*(\varvec{\hat{e}}) \rangle \sharp \) of \(\mathsf {n} . \mathscr {C}^*(\varvec{e'})\) by simulating the computation of \(\mathsf {n}\) by \(\mathcal {A}(\varphi )_{\varvec{s}}\) and referring to \(\mathscr {C}^*(\varvec{\tilde{e}'})\), and computes the remaining half \(\langle \mathscr {C}^*(\varvec{g}) \rangle \rangle \sharp \mathscr {C}^*(\varvec{h})\) by simulating the computation of \(\mathscr {C}^*(\varvec{e})\) by \(\mathcal {A}(\varphi )_{\varvec{s}}\) (inserting \(\rangle \) before \(\sharp \)). Clearly, \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) is standard.

If m and n belong to A and B, respectively, then \(\varvec{\tilde{e}}\), \(\varvec{e}\), \(\varvec{\tilde{e}'}\) and \(\varvec{e'}\) are of the form Open image in new window , Open image in new window and Open image in new window , respectively. Again, with the help of mviews, \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) calculates the first half \(\mathsf {n} . \langle \mathscr {C}^*(\varvec{\hat{e}}) \rangle \sharp \) of \(\mathsf {n} . \mathscr {C}^*(\varvec{e'})\) by simulating the computation of \(\mathsf {n}\) by \(\mathcal {A}(\varphi )_{\varvec{s}}\) and referring to \(\mathscr {C}^*(\varvec{\tilde{e}'})\), and then computes the remaining half \(\mathscr {C}^*(\varvec{h})\) by simulating the computation of \(\mathscr {C}^*(\varvec{e})\) by \(\mathcal {A}(\varphi )_{\varvec{s}}\). Again, \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) is clearly standard.

The remaining two cases are completely analogous to the above cases.

 2.
It remains to stipulate how \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) distinguishes the above four cases. Assume \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} \in \mathcal {A}(\varphi )_{\varvec{s}}\) when \(\mathcal {A}(\varphi )_{\varvec{s}}\) has computed the inner element n. Note that \(\varvec{v}\) has enough information to identify \(\mathsf {n}\), and every move occurring in \(\varvec{v}\) has a ‘tag’ \((\_)^{[0]}\), \((\_)^{[1]}\) or \((\_)^{[2]}\) (but not \((\_)^{[3]}\)). Thus, by simulating this computation of \(\mathsf {n}\) by \(\mathcal {A}(\varphi )_{\varvec{s}}\) but replacing \(\mathsf {n}^{[3]}\) with \(\hat{q}^{[2]}\) to learn about \(\mathsf {m}\), \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) may recognize the current case out of the four described above. Specifically, if \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \mapsto x\) is the first step for \(\mathcal {A}(\varphi )_{\varvec{s}}\) to compute \(\varvec{e}\), then correspondingly \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) computes as \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \mapsto v_1\), \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} v_1 v_2 \mapsto v_3\), ..., \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \mapsto \hat{q}^{[2]}\), \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} \mapsto x'\), where \(x'\) is the first step for \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) to compute \(\varvec{e'}\); and if \(\mathcal {A}(\varphi )_{\varvec{s}}\) next computes \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} x y \mapsto z\), then correspondingly \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) computes as \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} x' y' \mapsto v_1\), \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} x' y' v_1 v_2 \mapsto v_3\), ..., \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} x' y' \varvec{v} \mapsto \hat{q}^{[2]}\), \(\hat{q}^{[3]} \varvec{v} \mathsf {n}^{[3]} q^{[3]} \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} x' y' \varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]} \mapsto z'\), where \(y', z'\) are the second and the third steps for \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) to compute \(\varvec{e'}\), and so on. That is, \(\mathcal {A}(\varphi ^{\dagger })_{\varvec{s}}\) remembers the current case by inserting the sequence \(\varvec{v} \hat{q}^{[2]} \mathsf {m}^{[2]}\) (of length \(\leqslant 8\) for \(\mathcal {A}(\varphi )\) is standard) between each computational step. Note that \(\mathcal {A}(\varphi ^{\dagger })\) remains to be standard.
3.2 Examples of viable strategies
This section presents various examples of a viable strategy realized by a standard stalgorithm. These strategies except fixedpoint strategies \( fix _A\) are actually finitary; thus, we need the notion of viable strategies only for promotion \((\_)^{\dagger }\) and \( fix _A\), both of which are necessary for the proof of Turing completeness in Sect. 3.3.
Example 77
It is straightforward to see that \(\mathsf {st}(\mathcal {A}( der _A)) = der _A\) holds, showing viability of \( der _A\). Also, \(\mathcal {A}( der _A)\) is clearly standard.
Example 78
Example 79
Example 80
With the help of mviews, there is clearly a finite table \(\mathcal {A}( fix _A)_m\) that implements \(\mathcal {A}( fix _A)^\circledS _m\). It is then not hard to see that \(\mathsf {st}(\mathcal {A}( fix _A)) = fix _A\) holds, showing that \( fix _A\) is viable. Also, it is easy to see that \(\mathcal {A}( fix _A)\) is standard.
3.3 Turing completeness
In the last two sections, we have seen through examples that each ‘atomic’ strategy definable by PCF [6, 71] is viable, and it is realized by a standard stalgorithm. In addition, Theorem 76 shows that constructions on strategies preserve this property. From these two facts, our main theorem immediately follows:
Theorem 81
(Main theorem) Every normalized strategy \(\sigma : S_\sigma \) definable by PCF has a viable strategy \(\phi _\sigma : D_\sigma \) that satisfies Open image in new window
Proof
 1.
\((\sigma : S_\sigma ) \in \mathcal {PCF}\) if \(\sigma : S_\sigma \) is PCFatomic, i.e., \( der _A : A \Rightarrow A\), \( zero _A : A \Rightarrow \mathcal {N}\), \( succ : \mathcal {N} \Rightarrow \mathcal {N}\), \( pred : \mathcal {N} \Rightarrow \mathcal {N}\), \( zero ? : \mathcal {N} \Rightarrow \varvec{2}\), \( case _{A} : A \& A \& \varvec{2} \Rightarrow A\) or \( fix _A : (A \Rightarrow A) \Rightarrow A\), where A is generated from \(\mathcal {N}\), \(\varvec{2}\) and/or T, by & and/or \(\Rightarrow \) (n.b., the construction of A is ‘orthogonal’ to that of \(\sigma : S_\sigma \));
 2.
\((\Lambda (\sigma ) : A \Rightarrow (B \Rightarrow C)) \in \mathcal {PCF}\) if \( (\sigma : A \& B \Rightarrow C) \in \mathcal {PCF}\) for some normalized games A, B and C, where note that Open image in new window ;
 3.
\( (\langle \varphi , \psi \rangle : C \Rightarrow A \& B) \in \mathcal {PCF}\) if \((\varphi : C \Rightarrow A), (\psi : C \Rightarrow B) \in \mathcal {PCF}\) for some normalized games A, B and C;
 4.
\((\iota ^{\dagger } ; \kappa : A \Rightarrow C) \in \mathcal {PCF}\) if \((\iota : A \Rightarrow B), (\kappa : B \Rightarrow C) \in \mathcal {PCF}\) for some normalized games A, B and C
We then assign a strategy \(\phi _\sigma : D_\sigma \) to each \((\sigma : S_\sigma ) \in \mathcal {PCF}\) along with the above construction of \(\sigma : S_\sigma \) as follows: 1. \(D_\sigma {\mathop {=}\limits ^{\mathrm {df. }}} S_\sigma \) and \(\phi _\sigma {\mathop {=}\limits ^{\mathrm {df. }}} \sigma \) (where \(\sigma : S_\sigma \) is PCFatomic); 2. \(D_{\Lambda (\sigma )} {\mathop {=}\limits ^{\mathrm {df. }}} \Lambda (D_\sigma )\) and \(\phi _{\Lambda (\sigma )} {\mathop {=}\limits ^{\mathrm {df. }}} \Lambda (\phi _\sigma )\); 3. \(D_{\langle \varphi , \psi \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} \langle D_\varphi , D_\psi \rangle \) and \(\phi _{\langle \varphi , \psi \rangle } {\mathop {=}\limits ^{\mathrm {df. }}} \langle \phi _\varphi , \phi _\psi \rangle \); 4. \(D_{\iota ^{\dagger } ;\, \kappa } {\mathop {=}\limits ^{\mathrm {df. }}} D_\iota ^{\dagger } \ddagger D_\kappa \) and \(\phi _{\iota ^{\dagger } ;\, \kappa } {\mathop {=}\limits ^{\mathrm {df. }}} \phi _\iota ^{\dagger } \ddagger \phi _\kappa \).
We have shown in the previous sections that PCFatomic strategies are all viable and realized by standard stalgorithms. Also, the constructions \(\Lambda \), \(\langle \_, \_ \rangle \) and \((\_)^{\dagger } \ddagger (\_)\) on strategies have been shown to preserve viability, more specifically, realizability by standard stalgorithms, of strategies in Theorem 76. Thus, we may conclude that \(\phi _\sigma : D_\sigma \) is viable (and realized by a standard stalgorithm) for each \((\sigma : S_\sigma ) \in \mathcal {PCF}\).
 1.
\(\mathcal {H}^\omega (\phi _\sigma ) = \mathcal {H}^\omega (\sigma ) = \sigma \) and \(\mathcal {H}^\omega (D_\sigma ) = \mathcal {H}^\omega (S_\sigma ) = S_\sigma \) if \(\sigma : S_\sigma \) is PCFatomic since in this case \(\sigma \) and \(S_\sigma \) are both normalized;
 2.
\(\mathcal {H}^\omega (\phi _{\Lambda (\sigma )}) = \mathcal {H}^\omega (\Lambda (\phi _\sigma )) = \Lambda (\mathcal {H}^\omega (\phi _\sigma )) = \Lambda (\sigma )\) and Open image in new window ;
 3.
\(\mathcal {H}^\omega (\phi _{\langle \varphi , \psi \rangle }) = \mathcal {H}^\omega (\langle \phi _\varphi , \phi _\psi \rangle ) = \langle \mathcal {H}^\omega (\phi _\varphi ), \mathcal {H}^\omega (\phi _\psi ) \rangle = \langle \varphi , \psi \rangle \) and Open image in new window ;
 4.
\(\mathcal {H}^\omega (\phi _{\iota ^{\dagger } ;\, \kappa }) = \mathcal {H}^\omega (\phi _\iota ^{\dagger } \ddagger \phi _\kappa ) = \mathcal {H}^\omega (\phi _\iota )^{\dagger } ; \mathcal {H}^\omega (\phi _\kappa ) = \iota ^{\dagger } ; \kappa \) and Open image in new window
Since PCF is Turing complete [32, 50], this result particularly implies:
Corollary 82
(Turing completeness) Every partial recursive function \(f : \mathbb {N}^k \rightharpoonup \mathbb {N}\), where \(k \in \mathbb {N}\) and \(\mathbb {N}^k {\mathop {=}\limits ^{\mathrm {df. }}} \underbrace{\mathbb {N} \times \mathbb {N} \dots \times \mathbb {N}}_{k}\), has a viable strategy \(\phi _f : D_f\) such that Open image in new window , where \( \mathcal {N}^k {\mathop {=}\limits ^{\mathrm {df. }}} \underbrace{\mathcal {N} \& \mathcal {N} \dots \& \mathcal {N}}_k\), and \(\mathcal {H}^\omega (\langle \underline{n_1}, \underline{n_2}, \dots , \underline{n_k} \rangle ^{\dagger } \ddagger \phi _f) \simeq \underline{f(n_1, n_2, \dots , n_k)}\) for all \((n_1, n_2, \dots , n_k) \in \mathbb {N}^k\).^{21}
Proof
Let \(\mathsf {x_1 : N, x_2 : N, \dots , x_k : N \vdash F : N}\) be a term of PCF that implements a given partial recursive function \(f : \mathbb {N}^k \rightharpoonup \mathbb {N}\), i.e., \(\mathsf {F [n_1/x_1, n_2/x_2, \dots , n_k/x_k]}\) evaluates to \(\mathsf {f(n_1, n_2, \dots , n_k)}\) if \(f(n_1, n_2, \dots , n_k)\) is defined, and diverges otherwise, for all \(n_i \in \mathbb {N}\) (\(i = 1, 2, \dots , k\)), where \(\mathsf {n_i : N}\) is the \(n_i\)th numeral, and \(\mathsf {F [n_1/x_1, n_2/x_2, \dots , n_k/x_k]}\) is the result of substituting \(\mathsf {n_i}\) for \(\mathsf {x_i}\) in \(\mathsf {F}\) for \(i = 1, 2, \dots , k\) (see, e.g., [32, 50] for Turing completeness of PCF). Then, there exists a normalized strategy \(\sigma _f : \mathcal {N}^k \Rightarrow \mathcal {N}\) in \(\mathcal {PCF}\) that interprets \(\mathsf {F}\) in the standard game semantics of PCF [6]. By Theorem 81, there exists a viable strategy \(\phi _f : D_f\) such that \(\mathcal {H}^\omega (\phi _f) = \sigma _f\) and Open image in new window Hence, \(\mathcal {H}^\omega (\langle \underline{n_1}, \underline{n_2}, \dots , \underline{n_k} \rangle ^{\dagger } \ddagger \phi _f) = \mathcal {H}^\omega (\langle \underline{n_1}, \underline{n_2}, \dots , \underline{n_k} \rangle )^{\dagger } ; \mathcal {H}^\omega (\phi _f) = \langle \underline{n_1}, \underline{n_2}, \dots , \underline{n_k} \rangle ^{\dagger } ; \sigma _f \simeq \underline{f(n_1, n_2, \dots , n_k)}\), where the last Kleene equality \(\simeq \) is by the adequacy [8] of the standard game semantics of PCF. \(\square \)
Remark
Crucially, there is clearly a partial recursive function \(f : \mathbb {N}^k \rightharpoonup \mathbb {N}\) such that \(\sigma _f\) is not viable (but \(\phi _f\)is) by the finitary nature of tables for stalgorithms.
As our gamesemantic model of computation is Turing complete, some of the wellknown theorems in computability theory [16, 60] are immediately generalized (in the sense that they are not restricted to computation on natural numbers):
Corollary 83
 1
 2
\(\langle \{ \varvec{\epsilon } \}, \tau _1, \tau _2, \dots , \tau _m \rangle ^{\dagger } \ddagger \phi _{\sigma _1, \sigma _2, \dots , \sigma _n} \simeq \langle \sigma _1, \sigma _2, \dots , \sigma _n, \tau _1, \tau _2, \dots , \tau _m \rangle ^{\dagger } \ddagger \phi \) for any strategies \(\tau _j : T \Rightarrow B_j\) (\(j = 1, 2, \dots , m\)).
Proof
We define \(\phi _{\sigma _1, \sigma _2, \dots , \sigma _n} {\mathop {=}\limits ^{\mathrm {df. }}} \underbrace{\Lambda ^\circleddash (\cdots \Lambda ^\circleddash }_m(\langle \sigma _1, \sigma _2, \dots , \sigma _n \rangle ^{\dagger } \ddagger \underbrace{\Lambda ( \cdots \Lambda }_m(\phi ) \cdots )) \cdots )\), where the proof of Theorem 76 describes how to ‘effectively’ obtain a standard stalgorithm that realizes \(\phi _{\sigma _1, \sigma _2, \dots , \sigma _n}\) in an informal sense.^{22} Note that Corollary 82 implies that it is in fact a generalization of the conventional smn theorem [16]. \(\square \)
Corollary 84
(Generalized FRT) Given a viable strategy \(\varphi : D\) realized by a standard stalgorithm such that Open image in new window for some normalized game A, there is another viable strategy \(\sigma _\varphi : D_\varphi \) realized by a standard stalgorithm such that Open image in new window and \(\mathcal {H}^\omega (\sigma _\varphi ^{\dagger } \ddagger \varphi ) = \mathcal {H}^\omega (\sigma _\varphi ) : T \Rightarrow A\).
Proof
Just define \(D_\varphi {\mathop {=}\limits ^{\mathrm {df. }}} (T \Rightarrow D)^{\dagger } \ddagger ((A \Rightarrow A) \Rightarrow A)\) and \(\sigma _\varphi {\mathop {=}\limits ^{\mathrm {df. }}} (\varphi _T)^{\dagger } \ddagger fix _A\), where \(\varphi _T : T \Rightarrow D\) is \(\varphi \) up to tags. Again, Corollary 82 implies that it is in fact a generalization of the conventional first recursion theorem (FRT) [16]. \(\square \)
Finally, let us show that the converse of the main theorem (i.e., Theorem 81) also holds for classical computation because it would then give further naturality and/or reasonability of our definition of ‘effectivity,’ viz. viability, of strategies:
Theorem 85
(Conservativeness) Any viable strategy \(\phi : G\) with Open image in new window , where \(k \in \mathbb {N}\), can be simulated by a partial recursive function \(f_\phi : \mathbb {N}^k \rightharpoonup \mathbb {N}\) in the sense that \(\mathcal {H}^\omega (\langle \underline{n_1}, \underline{n_2}, \dots , \underline{n_k} \rangle ^{\dagger } \ddagger \phi ) \simeq \underline{f_\phi (n_1, n_2, \dots , n_k)}\) for all \(n_1, n_2, \dots , n_k \in \mathbb {N}\).
Proof
(sketch) This theorem is not as surprising as Theorem 81 for one may just employ the Church–Turing thesis [16]. Nevertheless, let us give a proof sketch for the theorem, by which ardent readers may fill in the details if they wish to.
The idea is to simulate a given viable strategy \(\phi : G\) by a 6tape TM \(\mathscr {M}\) by writing down an input on the first tape, the entire history of previous occurrences of moves, i.e., each position during a play, on the second tape, and the last three occurrences of each Pview as well as the next Pmove on the third tape, where the fourth, the fifth and the sixth tapes are used for auxiliary computations (specified below).
On the other hand, the last three occurrences of the Pview of each oddlength position (in the format described above yet without identifiers), their identifiers and mviews (when the moves \([m]_{e_1 e_2 \dots e_k}\) are encoded as strategies \(\underline{[m]_{e_1 e_2 \dots e_k}}\)) are written, respectively, on the third, the fourth and the fifth tapes, where each occurrence of a move, an identifier or an mview is separated again by \(\mathsf {\$}\).
 1.
Copy the last occurrence \([m_1]_{\varvec{e^{(1)}}}\) of the current Pview on the second tape onto the initial cells of the third tape, calculate its identifier and mview, possibly utilizing the sixth tape as a working tape, and write them down on the fourth and the fifth tapes, respectively, in the obvious manner, erasing all contents of the sixth tape after the calculation;
 2.
Locate the second last occurrence \([m_2]_{\varvec{e^{(2)}}}\) of the current Pview on the second tape by the identifier associated with the occurrence \([m_1]_{\varvec{e^{(1)}}}\) and then execute the same computation as the one on \([m_1]_{\varvec{e^{(1)}}}\), where the new contents prefixed with \(\mathsf {\$}\) on the third, the fourth and the fifth tapes are concatenated to the existing contents;
 3.
Similarly, locate the third last occurrence \([m_3]_{\varvec{e^{(3)}}}\) of the current Pview on the second tape (which is easy as it locates next to \([m_2]_{\varvec{e^{(2)}}}\)) and execute the same computation on it (so that the third, the fourth and the fifth tapes contain all information of the last three occurrences of the Pview);
 4.
With the contents on the third, the fourth and the fifth tapes, compute the next Pmove \([m]_{\varvec{e}}\) and the identifier of its justifier, write them down on the second tape and erase all contents on the third, the fourth and the fifth tapes.
Remark
Theorem 85 does not hold for higherorder computation because TMs cannot take additional inputs from O during the course of computation. Of course, one may consider generalized TMs that may interact with O, but then it is no longer TMs in the usual sense; in fact, this idea naturally leads to the gamesemantic model of computation developed in the present paper.
Also, as an immediate corollary of Theorem 85, we obtain Corollary 89, for which we first need some auxiliary concepts:
Definition 86
(Standard strategies on PCFgames) A PCFgame is a normalized game constructed from \(\mathcal {N}\), \(\varvec{2}\) and/or T, via & and/or \(\Rightarrow \). A strategy \(\sigma \) on a PCFgame G is standard if the substance (Definition 6) of the last move of the jsubsequence \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) of each maximal (with respect to \(\preceq \)) element \(\varvec{s} \in \sigma \) that consists of moves of the same component game \(\mathcal {N}^{[i]}\) (i.e., moves of \(\mathcal {N}\) with the same inner tag) is \( no \) if \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) is nonempty and of evenlength.
Next, let us introduce the translation \(\mathscr {T}\) that maps PCFgames and standard strategies on them to the corresponding conventional games and strategies on them, respectively, simply by replacing \(\mathcal {N}\) with N:
Definition 87

\(\mathscr {T}(T) {\mathop {=}\limits ^{\mathrm {df. }}} T\), \(\mathscr {T}(\varvec{2}) {\mathop {=}\limits ^{\mathrm {df. }}} \varvec{2}\) and \(\mathscr {T}(\mathcal {N}) {\mathop {=}\limits ^{\mathrm {df. }}} N\);

\( \mathscr {T}(A \& B) {\mathop {=}\limits ^{\mathrm {df. }}} \mathscr {T}(A) \& \mathscr {T}(B)\);

\(\mathscr {T}(A \Rightarrow B) {\mathop {=}\limits ^{\mathrm {df. }}} \mathscr {T}(A) \Rightarrow \mathscr {T}(B)\).
 1
Replacing the jsubsequence \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) of each maximal element \(\varvec{s} \in \sigma \) that consists of moves of the same component game \(\mathcal {N}^{[i]}\) with the jsequence \([\hat{q}] [n]\) (1.a. by deleting all occurrences of [q] or \([ yes ]\), and replacing [no] with [n]) if \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) is the maximal element of \(\underline{n}\) (Example 43), and with the jsequence \([\hat{q}]\) (1.b. by deleting all noninitial occurrences) otherwise (i.e., if \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) is of oddlength), where we omit tags for brevity; let us write \(\mathscr {T}(\varvec{s})\) for the resulting jsequence obtained from \(\varvec{s}\);
 2
Defining \(\mathscr {T}(\sigma ) {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref}(\{ \mathscr {T}(\varvec{s}) \mid \varvec{s} \in \sigma , \varvec{s} \text { is maximal in }\sigma \ \! \})^{\mathsf {Even}}\).
Remark
By the specific methods 1.a and 1.b in the step 1 of the translation \(\sigma \mapsto \mathscr {T}(\sigma )\), the relative order between occurrences in different component games of \(\mathscr {T}(G)\) is automatically determined (and thus \(\mathscr {T}(\sigma )\) is unambiguous). Note also that the strategy \(\sigma : G\) is required to be standard since otherwise some jsubsequence \(\varvec{s} \upharpoonright \mathcal {N}^{[i]}\) is \([\hat{q}] [ yes ] ([q] [ yes ])^n\) for some \(n \in \mathbb {N}\), and thus \(\mathscr {T}\) on \(\sigma \) would not be well defined (n.b., for instance, how to translate a nonstandard strategy \(\sigma : \mathcal {N}^{[0]} \Rightarrow \mathcal {N}^{[1]}\) that contains the maximal position \([\hat{q}]^{[1]} [\hat{q}]^{[0]} [ yes ]^{[0]} [ no ]^{[1]}\)?).
Definition 88
(Intrinsic equivalence [6, 7, 38]) The intrinsic equivalence\(\simeq _G\) on strategies on a game G is defined by \(\sigma \simeq _G \tilde{\sigma } {\mathop {\Leftrightarrow }\limits ^{\mathrm {df. }}} \forall \tau : G \Rightarrow \Sigma . \ \! \tau \circ \sigma _T^{\dagger } = \tau \circ \tilde{\sigma }_T^{\dagger }\) for all \(\sigma , \tilde{\sigma } : G\), where \(\sigma _T, \tilde{\sigma }_T : T \Rightarrow G\) are \(\sigma \) and \(\tilde{\sigma }\) up to tags, respectively, and \(\Sigma \) is the game given by \(M_\Sigma {\mathop {=}\limits ^{\mathrm {df. }}} \{ [\hat{q}], [a] \}\), where a is any element with \(a \ne \hat{q}\), \(\lambda _\Sigma : [\hat{q}] \mapsto (\mathsf {O}, \mathsf {Q}, 0), [a] \mapsto (\mathsf {P}, \mathsf {A}, 0)\), \(\vdash _\Sigma {\mathop {=}\limits ^{\mathrm {df. }}} \{ (\star , [\hat{q}]), ([\hat{q}], [a]) \}\), \(\Delta _\Sigma {\mathop {=}\limits ^{\mathrm {df. }}} \emptyset \) and \(P_\Sigma {\mathop {=}\limits ^{\mathrm {df. }}} \mathsf {Pref}(\{ [\hat{q}] [a] \})\).
See [7, 38, 51] for the proof that shows \(\simeq _G\) is in fact an equivalence relation on strategies on a given game G. We are finally ready to establish:
Corollary 89
(Universality) Let A be the dynamic game semantics of a type \(\mathsf {A}\) of PCF (following [71]). Given any viable strategy \(\alpha : A\) such that \(\mathcal {H}^\omega (\alpha ) : \mathcal {H}^\omega (A)\) is standard, \(\mathscr {T}(\mathcal {H}^\omega (\alpha )) : \mathscr {T}(\mathcal {H}^\omega (A))\) coincides with the conventional game semantics \(\tilde{\alpha } : \mathscr {T}(\mathcal {H}^\omega (A))\) of some term of PCF up to intrinsic equivalence [6].
Proof
(sketch) Applying the proof of Theorem 85, we may see that \(\mathscr {T}(\mathcal {H}^\omega (\alpha )) : \mathscr {T}(\mathcal {H}^\omega (A))\) is recursive, i.e., each move performed by \(\mathscr {T}(\mathcal {H}^\omega (\alpha ))\) is computable by a TM; thus, it is the conventional game semantics \(\tilde{\alpha } : \mathscr {T}(\mathcal {H}^\omega (A))\) of some term of PCF up to intrinsic equivalence by the universality theorem of [7, 38].
4 Conclusion and future work
The present work has given a novel notion of ‘computability,’ namely viability of strategies. Due to its intrinsic, noninductive, nonaxiomatic nature, it can be seen as a fundamental investigation of ‘effective’ computation beyond classical computation, where note that viability of strategies makes sense universally, i.e., regardless of the underlying games (e.g., games do not have to correspond to types of PCF).
Furthermore, our gamesemantic model of computation formulates both highlevel and lowlevel computational processes and defines ‘computability’ of the former in terms of the latter, which sheds new light on the very notion of computation. For instance, strategies \(\underline{n} : T \Rightarrow \mathcal {N}\) may be seen as the definition of natural numbers, and thus, a viable strategy of the form \(\phi : \mathcal {N}^k \Rightarrow \mathcal {N}\) can be regarded as highlevel computation on natural numbers, not on their representations, and (the table of) an stalgorithm that realizes \(\phi \) can be seen as its symbolic implementation.
There are various directions for further work. First, we need to analyze the exact computational power of viable strategies, in comparison with other known notion of higherorder computability [50]. Also, as an application, the present framework may give an accurate measure for computational complexity [47], where note that the work on dynamic games and strategies [71] has already given such a measure via internal moves, but the present work may refine it further since two single steps in a game G may take different numbers of steps in the instruction game \(\mathcal {G}(M_G)^3 \Rightarrow \mathcal {G}(M_G)\). Moreover, it is of theoretical interest to see which theorems in computability theory can be generalized by the present framework in addition to the smn and the first recursion theorems. However, the most imminent future work is perhaps, by exploiting the flexibility of game semantics, to enlarge the scope of the present work (i.e., not only the language PCF) in order to establish a computational model of various logics and programming languages. We are particularly interested in how to apply our approach to noninnocent strategies.
This question seems highly interesting from a theoretical perspective.^{23}Is the class of all \((n+1)\)viable strategies strictly larger than that of all nviable strategies for each \(n \in \mathbb {N}\)?
We shall aim to answer these questions as future work as well.Does the hierarchy, if it exists, correspond to any known hierarchy (perhaps in computability theory or proof theory)?
Acknowledgements
The author acknowledges the financial support from Funai Overseas Scholarship, and also he is grateful to Samson Abramsky and Robin Piedeleu for fruitful discussions.
Footnotes
 1.
This idea is similar to that of \( denotational \)\( semantics \) of programming languages [62], but there is an important difference: Denotational semantics interprets programs usually by (extensional) functions, but we are concerned with (intensional) \( processes \).
 2.
It is both technically and conceptually simpler to focus on \( sequential \) games, in which two participants \( alternately\,\, and\,\, separately \) take actions, than to consider more general \( concurrent \) games, in which both participants may be active \( simultaneously \) [2].
 3.
For simplicity, here we focus on \( closed \) terms, i.e., ones with the \( empty \)\( context \).
 4.
The diagrams are only to make it explicit which component game each move belongs to; the two positions are just finite sequences \(q^{[0]} n^{[0]} q^{[1]} m^{[1]}\) and \(q^{[1]} m^{[1]} q^{[0]} n^{[0]}\) equipped with the pointers \(q^{[0]} \leftarrow n^{[0]}\) and \(q^{[1]} \leftarrow m^{[1]}\).
 5.
But they exhibit simple communication between the participants as in the above examples, i.e., game semantics is intensional to some degree but not completely. It is roughly why game semantics has been highly successful in denotational semantics.
 6.
More precisely, they are the \( last \)\( occurrences \)\( of \)Pmoves; however, for convenience we shall keep using this abuse of terminology in the rest of the introduction.
 7.
This choice is far from canonical; it should also work to employ another representation of natural numbers, e.g., the binary one. We have chosen the unary representation for its simplicity.
 8.
This would be seen, if achieved, as a mathematical formalization of Brouwer’s \( intuitionism \) [66], in which ‘(mental) constructions’ are made precise as gamesemantic computational processes, viz. plays by viable dynamic strategies, and moreover extended to \( interactive \) ‘constructions.’
 9.
Strictly speaking, the variants [6, 51] are slightly different, and [71] chooses [6] as the basis, but [51] describes a lot of useful technical details. [6] is chosen in [71] because it combines good points of the two bestknown variants, \(\textit{AJM}\)games [7] and \(\textit{HO}\)games [38] so that it may model linearity as in [5] and also characterize various programming features as in [6] though these points are left as future work in [71].
 10.
We have employed (a representation of) a bijection \(\mathbb {N}^*{\mathop {\rightarrow }\limits ^{\sim }} \mathbb {N}\), rather than \(\mathbb {N} \times \mathbb {N} {\mathop {\rightarrow }\limits ^{\sim }} \mathbb {N}\), for a simple definition of outer tags.
 11.
Although the main focus of the work [71] is to capture the smallstep operational semantics of a programming language by such a stepbystep hiding operation \(\mathcal {H}\), such finegrained steps do not play a main role in the present work. Nevertheless, we keep the structure \(\lambda _G^{\mathbb {N}}\) as it makes sense for a model of computation to be equipped with the stepbystep hiding operation, and it would be interesting as future work to consider ‘effective computability’ of the hiding operation.
 12.
\(\lceil \varvec{s} q \varvec{t} \rceil _G\) must be of the form \(\lceil \varvec{s} q \rceil _G . \varvec{t'}\) by \( (generalized) \) \( visibility \) on \(\varvec{s} q \varvec{t} a\) (Definition 17).
 13.
 14.
As already mentioned in the introduction, viability makes sense for conventional or static strategies as well, but viable static strategies are \( not \) Turing complete.
 15.
Note that this is analogous to the computation of a TM which looks at only one cell of an infinite tape at a time.
 16.
Of course, there might be another way to ‘effectively’ eliminate irrelevant occurrences from the history of previous occurrences; in fact, we need more than Pviews to model languages with \( states \) [6], which is left as future work.
 17.
Strictly speaking, the pairing is up to ‘tags,’ as already explained in Sect. 1.7.
 18.
Note, however, that \( succ \) itself is already finitary; \(\mathcal {A}( succ )^\circledS \) is just for an illustration, and viability of strategies is necessary only for more complex strategies.
 19.
Again, the composition is up to ‘tags,’ as already explained in Sect. 1.7.
 20.
The outer tags which \(\mathcal {A}(\varphi )_{\varvec{s}}\) refers to for computing \(\varvec{e}\) are only \(\varvec{\tilde{e}}\) as \(\mathcal {A}(\varphi )\) is standard, and therefore, it is clearly possible to adjust the computation of \(\mathscr {C}^*(\varvec{e})\) by \(\mathcal {A}(\varphi )\) to the computation of the latter half \(\langle \mathscr {C}^*(\varvec{g}) \rangle \rangle \sharp \mathscr {C}^*(\varvec{h})\) of \(\mathscr {C}^*(\varvec{e'})\) by \(\mathcal {A}(\varphi ^{\dagger })\) given below.
 21.
Recall that \(\langle \_, \_ \rangle \), \((\_)^{\dagger }\) and \(\ddagger \) are \( pairing \), \( promotion \) and \( concatenation \) of strategies given in Sect. 2.3.2.
 22.
It is interesting future work to formalize this informal ‘effective computability’ by certain viable strategies.
 23.
Note that this question would not have arisen in the first place if we had not defined ‘effective computability’ \( solely \)\( in \)\( terms \)\( of \)\( games \)\( and \)\( strategies \).
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