Abstract
In 1936, Post and Turing independently proposed two models of computation that are virtually identical. Turing refers back to these models in his (The word problem in semi-groups with cancellation. Ann. Math. 52, 491–505) and calls them “the logical computing machines introduced by Post and the author”. The virtual identity is not to be viewed as a surprising coincidence, but rather as a natural consequence of the way in which Post and Turing conceived of the steps in mechanical procedures on finite strings. To support our view of the underlying conceptual confluence, we discuss the two 1936 papers, but explore also Post’s work in the 1920s and Turing’s paper (Solvable and unsolvable problems. Sci. News 31, 7–23). In addition, we consider their overlapping mathematical work on the word-problem for semigroups (with cancellation) in Post’s (Recursive unsolvability of a problem of Thue. J. Symb. Log. 12, 1–11) and Turing’s (The word problem in semi-groups with cancellation. Ann. Math. 52, 491–505). We argue that the unity of their approach is of deep significance for the theory of computability.
AMS Classifications: 01A60,03-03,03D03,03D10,03D20, 20M05
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- 1.
The broader history of computability has been described in a number of publications, partly by the participants in the early development in Princeton, for example, [43, 57]. Good discussions are found in, among others [9, 14, 26, 58, 62, 68].
There are many excellent books on recursion/computability theory, but not many that take Post’s approach as fundamental. We just mention [13, 47, 56, 67].
- 2.
Machines that are deterministic are Turing’s a-machines; if a-machines operate only on 0s and 1s they are called computing machines; see [71], p. 232.
- 3.
There are other commonalities in their approaches, e.g., in connection with relative computability, which first appeared in Turing’s dissertation and which played such a key role in Post’s later work. However, here we are focusing on their fundamental conceptual analysis.
- 4.
- 5.
In the early evolution of recursion theory, Gödel’s definition was viewed as being a modification of a proposal of Herbrand’s—because Gödel presented it that way in his Princeton Lectures. In a letter to Jean van Heijenoort in 1964, Gödel reasserted that Herbrand had suggested, in a letter, a definition very close to the one actually presented in [28]. However, the connection of Gödel’s definition to Herbrand’s work is much less direct; that is clear from the two letters that were exchanged between Gödel and Herbrand in 1931. John Dawson found the letters in the Gödel Nachlass in 1986; see [17]. The letters are published in [36]; their intellectual context is discussed in [61].
- 6.
How far removed the considerations of Turing (and Post) were from those of Bernays, who actually wrote Supplement II of Hilbert and Bernays [39], should be clear from two facts: (i) In a letter to Church of 22 April 1937, Bernays judged Turing as “very talented” and his concept of computability as “very suggestive”; (ii) Bernays did not mention Turing’s paper in Supplement II, though he knew Turing’s work very well. Indeed, in his letter to Church Bernays pointed out a few errors in [71]; Church communicated the errors to Turing and, in a letter of 22 May 1937 to Bernays, Turing acknowledged them and suggested that he would write a Correction, [72]!
- 7.
In a very informative letter Church wrote on 8 June 1937 to the Polish logician Pepis, the absoluteness of general recursive functions is indirectly argued for. (Church’s letter and its analysis is found in [59].) Gödel’s claim, with “formality” sharpened in the way we indicated, is an almost immediate consequence of the considerations in Supplement II of Hilbert and Bernays [39].
- 8.
Gödel argues at the bottom of p. 166 that the expressions and finite classes of expressions can be mapped to integers (“Gödel numbering”). Thus, he asserts, a “procedure in the sense we want is nothing else but a function f(x1, …, xr) whose arguments as well as its values are integers and which is such that for any system of integers n1, …, nr the value can actually be calculated”. So a “satisfactory definition of calculable functions” is needed, and that’s what the definition of computable function yields for Gödel.
- 9.
In [10], p. 10 and [69], p. 214, Gödel’s remark “That this really is the correct definition of mechanical [our emphasis] computability was established beyond any doubt by Turing.” is taken as showing that computability of functions is defined here by reference to Turing machines, i.e., that Gödel at this point had taken already Turing’s perspective. That view can be sustained only, if the context we sketched is left completely out of consideration.
- 10.
As to Post’s biography, see [15].
The mathematical and philosophical part of Post’s contributions is discussed with great clarity in [26], pp. 92–98. The unity of their approaches is, however, not recognized; it is symptomatic that neither [75] nor the overlapping work in [54, 74] is even mentioned.
A very comprehensive and illuminating account of Post’s work is found in [79].
- 11.
His work was published belatedly and only partially in [3]. In addition to the completeness question, Bernays also investigated the independence of the axioms of PM. He discovered that the associativity of disjunction is actually provable from the remaining axioms (that were then shown to be independent of each other).
- 12.
Tag systems may be characterized as normal systems in which for its productions \(gP \Rightarrow P\bar{g}\):
-
1.
All of the gs are of the same length;
-
2.
the \(\bar{g}\) s corresponding to a given g depend only on its initial symbol;
-
3.
if a given g occurs on the left in one of the productions, then so do all other strings of the same length having the same initial symbol as that g.
Post discusses Tag in the introductory section of Post [51] and in Sect. 3 of Post [50].
-
1.
- 13.
- 14.
- 15.
Gödel [36] pp. 169, 171.
The notes that were exchanged between Gödel and Post are in this volume of Gödel’s Collected Works.
- 16.
We neglect in our discussion “states of mind” of the computor. Here is the reason why. Turing argues in Sect. 9.I that the number of these states is bounded, and Post calls this Turing’s “finite number of mental states” assumption. However, in Sect. 9,III Turing argues that the computor’s state of mind [“mental state”] can be replaced in favor of “a more physical and definite counterpart of it”. In a sense then, the essential components of a computation have been fully externalized; see [13], p. 6, how this is accomplished through the concept of an “instantaneous description”.
- 17.
In footnote 8, p. 105 of Post [49], he criticizes masking the identification of recursiveness and effective calculability under a definition as Church had done. This, Post continues, “hides the fact that a fundamental discovery in the limitations of the mathematizing power of Homo Sapiens has been made and blinds us to the need of its continual verification.”
- 18.
Post pointed this out in a number of different places: (1) Urquhart on p. 643 of Urquhart [79] quotes from Post’s 1938 notebook and discusses, with great sensitivity, “an internal reason for Post’s failure to stake his claim to the incompleteness and undecidability results in time” on pp. 630–633; (2) in [50], p. 377, Post refers to the last paragraph of [49] and writes: “However, should Turing’s finite number of mental states hypothesis... bear under adverse criticism, and an equally persuasive analysis be found for all humanly possible modes of symbolization, then the writer’s position, while still tenable in an absolute sense, would become largely academic.”
- 19.
This problem later turned out to be a useful tool for obtaining unsolvability theorems regarding Noam Chomsky’s hierarchy of formal languages, which by the way, was itself based quite explicitly on Post production systems. See also the extended discussion of the correspondence problem in [79], p. 648.
- 20.
It is interesting that Markov’s proof [46] of the unsolvability of Thue’s problem, which was quite independent of Post’s, did use normal systems.
- 21.
Of course this is to be understood in relation to the Church-Turing Thesis.
- 22.
In the formulation of Turing [71], the tape is infinite in only one direction and a machine’s operations are specified by quintuples allowing for a change of symbol together with a motion to the left or right as a single step. Of course this difference is not significant.
- 23.
Post works with the closely related unsolvability of the problem of whether a particular distinguished symbol will ever appear on the tape, because unlike the halting problem, it appears explicitly in [71]. But dissatisfied with the rigor of Turing’s treatment, Post outlines his own proof of that fact. He also includes a careful critique pointing out how a convention that Turing had adopted for convenience in constructing examples had been permitted to undermine some of the key proofs. Anyhow, for the present application to Thue’s problem, Post begins by deleting all quadruples for which the distinguished symbol is the third symbol of the four, thus changing the unsolvability to one of halting.
- 24.
Turing’s proof was published in the prestigious Annals of Mathematics. Eight years later an analysis and critique of Turing’s paper [4] was published in the same journal. Boone found the proof to be essentially correct, but needing corrections and expansions in a number of the details.
- 25.
Gandy, in [25], uses particular discrete dynamical systems to analyze “machines”, i.e., discrete mechanical devices, and “proves” a mechanical thesis (M) corresponding to Turing’s thesis. Dershowitz and Gurevich in [22] give an axiomatization and claim to prove Church’s Thesis. (For a discussion of this claim, see [63].) In the context of our discussion here, one can say that Gandy and Dershowitz & Gurevich introduce very general models of computations—“Gandy Machines” in Gandy’s case, “Abstract State Machines” in Dershowitz and Gurevich’s case—and reduce them to Turing machine computations.
- 26.
A cut in O (determined by x) is a partition of O into two non-empty parts O1 and O2, such that all the elements of O1 stand in the relation R to all the elements of O2 (and x is taken to be in O1).
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Davis, M., Sieg, W. (2015). Conceptual Confluence in 1936: Post and Turing. In: Sommaruga, G., Strahm, T. (eds) Turing’s Revolution. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22156-4_1
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