The Number of Languages with Maximum State Complexity

Abstract

Champarnaud and Pin (1989) found that the minimal deterministic automaton of a language $L\subset \Sigma ^n$, where $\Sigma =\{0,1\}$, has at most

$$ \sum _{i=0}^n \min (2^i, 2^{2^{n-i}}-1) $$

states, and for each n there exists L attaining this bound. Câmpeanu and Ho (2004) have shown more generally that the tight upper bound for $\Sigma$ of cardinality k and for complete automata is

$$ \frac{k^r-1}{k-1} + \sum _{j=0}^{n-r}(2^{k^j}-1) + 1 $$

where $r=\min \{m:k^m\ge 2^{k^{n-m}}-1\}$. (In these results, requiring totality of the transition function adds 1 to the state count.) Câmpeanu and Ho’s result can be viewed as concerning functions $f:[k]^n\rightarrow [2]$ where $[k]=\{0,\dots ,k-1\}$ is a set of cardinality k. We generalize their result to arbitrary function $f:[k]^n\rightarrow [c]$ where c is a positive integer.

Let $O_i$ be the number of functions from $[b^{i}]$ to $[c^{b^{n-i}}]$ that are onto $[c^{b^{n-i}}-1]$. Câmpeanu and Ho stated that it is very difficult to determine the number of maximum-complexity languages. Here we show that it is equal to $O_i$, for the least i such that $O_i>0$.

For monotone languages a tightness result seems harder to obtain. However, we show that the following upper bound is attained for all $n\le 10$.

$$ \sum _{i=0}^n \min (2^i, M(n-i)-1), $$

where M(k) is the kth Dedekind number.

1 Introduction

The function $+$ on $\mathbb Z/5\mathbb Z$ may seem rather complicated as functions on that set go. On the other hand, $f(x,y,z)=x+y+z$ mod 5 is less so, in that we can decompose it as $(x+y)+z$, so that after seeing x and y, we need not remember the pair (x, y) but only their sum. Out of the $5^{5^3}$ ternary functions on a 5-element set, at most $5^{2\cdot 5^2}$ can be decomposed as $(x*_1y)*_2 z$ for some binary functions $*_1$, $*_2$. In Sect. 2 we make precise a sense in which such are not the most complicated ternary functions. We do this by extending a result of Câmpeanu and Ho [3] to functions taking values in a set of size larger than two.

Rising to an implicit challenge posed by Câmpeanu and Ho, we give a formula for the number of maximally complex languages in Sect. 2.2.

A motivation from finance will be felt in Sects. 3 and 4. The complexity of financial securities came into focus with the 2008 financial crisis. While Arora et al. [1] obtained NP-hardness results for the pricing of a security, here we look at the automatic complexity associated with executing a given trading strategy. The possibility of exercising early leads to a less complex option in our sense, as is easy to see. Thus we shall restrict attention to options which are European insofar as they can only be exercised at the final time n.

2 Complexity of Languages and Operations

Definition 2.1

A deterministic finite automaton (DFA) [9] M is a 5-tuple, $(Q, \Sigma , \delta , q_0, F)$, where

If $\delta$ is not required to be total then we speak of a partial deterministic finite automaton (PDFA).

Definition 2.2

Let $\Sigma = \{0,1\}$, let $n \in \mathbb {Z}^+$ and $X\subseteq \Sigma ^{\le n}$. Define $A_{-}(X)$ to be the minimum |Q| over all PDFAs $M = (Q,\Sigma , \delta , q_0, F)$ for which L(M), the language recognized by M, equals X. We call a PDFA $M = (Q,\Sigma , \delta , q_0, F)$ minimal for X if

$$\begin{aligned} |Q| =A_{-}(X). \end{aligned}$$

2.1 Operations

Champarnaud and Pin [4] obtained the following result.

Theorem 2.3

([4, Theorem 4]). The minimal PDFA of a language $L\subset \{0,1\}^n$ has at most

$$ \sum _{i=0}^n \min (2^i, 2^{2^{n-i}}-1) $$

states, and for each n there exists L attaining this bound.

Theorem 2.3 was generalized by Câmpeanu and Ho [3]:

Theorem 2.4

([3, Corollary 10]). For $k\ge 1$, let $[k]=\{0,\dots ,k-1\}$. Let $l\in \mathbb N$ and let M be a minimal DFA for a language $L\subseteq [k]^l$. Let Q be the set of states of M. Then we have:

• (i) $\#Q\le \frac{k^r-1}{k-1}+\sum _{j=0}^{l-r} (2^{k^j}-1)+1$, where $r=\min \{m\mid k^m\ge 2^{k^{l-m}}-1\}$;

• (ii) there is an M such that the upper bound given by (i) is attained.

Both of these results involve an upper bound which can be viewed as a special case of Theorem 2.7 below.

Definition 2.5

Let b, n, and c be positive integers. We say that a PDFA M accepts a function $f:[b]^n\rightarrow [c]$ if there are $c-1$ many special states $q_1,\dots ,q_{c-1}$ of M such that for all $\vec x\in [b]^n$,

• for $i>0$, $f(\vec x)=i$ iff M on input $\vec x$ ends in state $q_i$; and

• $f(\vec x)=0$ iff M does not end in any of the special states on input $\vec x$.

Definition 2.5 generalizes the case $b=2$ studied by Champarnaud and Pin. We write $A^B$ for the set of all functions from B to A.

Definition 2.6

Let ${[c]}^{{[b]}^n}$ be the set of n-ary functions $f:[b]^n\rightarrow [c]$. Let b and c be positive integers and let $\mathfrak C\subseteq {[c]}^{{[b]}^n}$. The Champarnaud–Pin family of $\mathfrak C$ is the family of sets $\{\mathfrak C_k\}_{0\le k\le n}$, where $\mathfrak C_k\subseteq {[c]}^{{[b]}^{n-k}}$, $0\le k\le n$, given by

$$ \mathfrak C_k = \{g\in [c]^{[b]^{n-k}} : \exists f\in \mathfrak C,\, \vec d\in [b]^k\quad \forall \vec x\quad g(\vec x)=f(\vec d,\vec x)\}. $$

So $\mathfrak C_0=\mathfrak C$, $\mathfrak C_1$ is obtained from $\mathfrak C_0$ by plugging in constants for the first input, and so forth. We write $\mathfrak C_n^-=\{f\in \mathfrak C_n: f\ne 0\}$ in order to throw out the constant zero function. Note that $|\mathfrak C_n^-|\ge |\mathfrak C_n|-1$.

Theorem 2.7

Let b and c be positive integers. Let $\mathfrak C\subseteq [c]^{[b]^n}$. An upper bound on the minimal number of states of PDFAs accepting members of $\mathfrak C$ is given by

$$ \sum _{i=0}^n \min (b^i,|\mathfrak C^-_{i}|). $$

The proof will be apparent from the proof of the next result, which is a generalization of Câmpeanu and Ho’s theorem.

Theorem 2.8

Let b and c be positive integers. For the minimal number of states of PDFAs M accepting functions $f:[b]^n\rightarrow [c]$, the upper bound

$$ \sum _{i=0}^n \min (b^i,c^{b^{n-i}}-1) $$

is attained.

Proof

Let $\log =\log _b$. The critical point for this result is the pair of values (i, k) with $i+k=n$ such that $b^i\le c^{b^k}-1$ (i.e., $i<b^k\log c$) and $b^{i+1}>c^{b^{k-1}}-1$ (i.e., $b^{i+1}\ge c^{b^{k-1}}$, i.e., $(i+1)\ge b^{k-1}\log c$), which can be summarized as

$$\begin{aligned} b^{k-1}\log c-1\le i<b^k\log c. \end{aligned}$$

(1)

$$ b^{k-1}\log c\le i+1\le b^k\log c. $$

We shall define a set A of k-ary functions of size $(c^{b^{k-1}})/b$ which when using the b many transitions (substitutions for say $p_1$) maps onto each of the $c^{b^{k-1}}$ many $k-1$-ary functions $\alpha$. This will suffice if

$$ c^{b^{k-1}}/b\le b^i $$

which does hold for all b by (1). The construction is similar to that of [3, Figure 1 and Theorem 8]; we shall be slightly more explicit than they were. Let $s=c^{b^{k-1}}-1$. Let $f_0,\dots ,f_{s-1}$ the set of all nonzero $k-1$-ary functions. As s may not be divisible by b, let us write $s=qb+r$ with quotient $q\ge 0$ and remainder $0\le r<b$. For j with $0\le j\le q-1$, let $g_j$ be given by

$$ g_j(i, \vec x)=f_{jb+i}(\vec x) $$

for each $i\in [b]$ and $\vec x\in [b]^{k-1}$. Let $g_q$ be given by $g_q(i,\vec x)=f_{qb+i}(\vec x)$ for each $0\le i\le r-1$, and let $g_q(i,\vec x)$ be arbitrary for $r\le i<b$. Finally, extend the set of functions $g_0,\dots ,g_q$ to $b^i$ many k-ary functions in an arbitrary way, obtaining functions $h_\sigma$ for $\sigma \in [b]^i$. Then our function attaining the bound is given by

$$ H(\sigma ,\tau )=h_\sigma (\tau ). $$

   $\square$

When $b=2$ and c is larger, Theorem 2.8 corresponds to automatic complexity of equivalence relations on binary strings as studied in [6]. When $b=c$, we have the case of n-ary operations on a given finite set, which is of great interest in universal algebra.

2.2 The Number of Maximally Complex Languages

Definition 2.9

Let b and c be positive integers and let $0\le i\le n$. Let $O_i=O_i^{(b,c,n)}$ be the number of functions from $[b^{i}]$ to $[c^{b^{n-i}}]$ that are onto $[c^{b^{n-i}}-1]$. That is, functions $f:[b^{i}] \rightarrow [c^{b^{n-i}}]$ such that for each $y\in [c^{b^{n-i}}-1]$ there is an $x\in [b^i]$ with $f(x)=y$.

Câmpeanu and Ho lamented that it seemed very difficult to count the number of maximum-complexity languages. Here we show

Theorem 2.10

Let b and c be positive integers and let $n\ge 0$. The number of maximum complexity functions $f:[b]^n\rightarrow [c]$ is $O_i$, where $0\le i\le n$ is minimal such that $O_i>0$.

Proof

Champarnaud and Pin, and Câmpeanu and Ho, and the present authors in Theorem 2.8, all found a maximal complexity by explicitly exhibiting the general automaton structure of a maximal-complexity language: we start with states corresponding to binary strings and end with strings corresponding to Boolean functions, and there is a crossover point in the middle where, in order that all states be used, we need an onto function exactly as specified in the definition of $O_i$. The crossover point occurs for the least i such that $O_i>0$, which is when the value of the minimum of $(b^i,c^{b^{n-i}}-1)$ switches from the first to the second coordinate. The number of such functions is then the number of such onto functions. Since we do not require totality and do not use a state for output 0 (“reject”) we omit the constant 0 Boolean function in the range of our onto maps.   $\square$

Note that the number of onto functions is well known in terms of Stirling numbers of the second kind. Let $O_{m,n}$ be the number of onto functions from [m] to [n]. Then

where , the number of equivalence relations on [m] with n equivalence classes, is a Stirling number of the second kind.

Note also that the number of functions from [a] to [b] that are onto the first $b-1$ elements of [b] is, in terms of the number m of elements going to the not-required element,

$$ \sum _{m=0}^{a-(b-1)} \left( {\begin{array}{c}a\\ m\end{array}}\right) O_{a-m,b-1}. $$

Example 2.11

When $n=3$ and $b=c=2$, we have that $O_i$ is the number of functions from $2^i$ to $2^{2^{3-i}}$ that are onto $2^{2^{3-i}}-1$. In this case, $O_1=0$. However, $O_2$ is the number of functions from 4 to 4 that are onto 3. This is

$$ \sum _{m=0}^{4-(4-1)} \left( {\begin{array}{c}4\\ m\end{array}}\right) O_{4-m,4-1} = O_{4,3} + 4\, O_{3,3} = 36 + 24 = 60. $$

These 60 languages are shown in Table 1.

Table 1. All possible sets Z with $A\_(Z) =7$.

2.3 Polynomial-Time Algorithm

It is perhaps worth pointing out that there is a polynomial-time algorithm for finding the minimal automaton of Boolean functions, based on essentially the Myhill–Nerode theorem [5, 10]. In this subsection we detail that somewhat.

Definition 2.12

Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L. Define a relation $R_L$ on strings by the rule that $x R_L y$ if there is no distinguishing extension for x and y.

As is well known, $R_L$ is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes.

Theorem 2.13

(Myhill–Nerode). A language L is regular if and only if $R_L$ has a finite number of equivalence classes. Moreover, the number of states in the smallest deterministic finite automaton (DFA) recognizing L is equal to the number of equivalence classes in $R_L$. In particular, there is a unique DFA with minimum number of states.

The difference is that for us we require $|xz|\le n$ and $|yz|\le n$, see Listing 1.

3 Monotone Boolean Functions

The main theoretical results of the paper are in Sect. 2. The present, longer section deals with a more computational and exploratory investigation: what happens if we try to prove that the natural upper bound on complexity is attained in restricted settings such as monotone functions?

Definition 3.1

An isotone map is a function $\varphi$ with $a\le b\implies \varphi (a)\le \varphi (b)$.

Fig. 1.

Isotone 1:1 map from $\mathbf {2}^4$ to $F_3^-$.

The Online Encyclopedia of Integer Sequences (OEIS) has a tabulation of Dedekind numbers, i.e., the number M(n) of monotone functions [12], which is also the number of elements of the free distributive lattice on n generators and the number of antichains of subsets of [n].

Definition 3.2

For an integer $n\ge 0$, $F_n$ is the set of monotone Boolean functions of n variables (equivalently, the free distributive lattice on n generators, allowing 0 and 1 to be included), and $F_n^-=F_n\setminus \{0\}$ where 0 is the constant 0 function.

The illustrative case $n=3$ is shown in Fig. 4.

Theorem 3.3

The minimal automaton of a monotone language $L\subset \{0,1\}^n$ has at most

$$ \sum _{i=0}^n \min (2^i, |F_{n-i}|-1) $$

states. This bound is attained for $n\le 10$.

Proof

The upper bound follows from Theorem 2.7. The sharpness results are obtained in a series of theorems tabulated in Table 2.   $\square$

Thinking financially, an option is monotone if whenever s is pointwise dominated by t and $s\in L$ then $t\in L$, where L is the set of exercise situations for the option. This is the case for common options like call options or Asian average-based options and makes financial sense if a rise in the underlying is always desirable and always leads to a higher option value.

Table 2. Maximum complexity of monotone securities.

Example 3.4

(Asian option; Shreve [11, Exercise 1.8]). This is the example that in part motivates our looking at monotone options. Let $n=3$ and consider a starting capital $S_0=4$, up-factor $u=2$, down-factor $d=\frac{1}{2}$. Let $Y_i=\sum _{k=0}^i S_k$. The payoff at time $n=3$ is $(\frac{1}{4}Y_3-4)^+$. To fit this example into our framework in the present paper, let us look at which possibilities lead to exercising, i.e., $\frac{1}{4}Y_3-4>0$ or $Y_3>16$. Computation shows that the set of exercise outcomes is $\{011, 100, 101, 110, 111\}$. The complexity is 6 (Fig. 3), so it is maximally complex for a monotone option.

For $n=3$ we are looking at isotone functions from $\{0,1\}$ to the family of monotone functions on two variables p and q. For the Asian option in Example 3.4 $\{0,1\}$ are mapped to $\{p\wedge q, 1\}$. For the majority function, $\{0,1\}$ are mapped to $\{p\wedge q, p\vee q\}$ (Fig. 3).

The sets $\{p\wedge q,p\vee q\}$ and $\{p\wedge q, 1\}$ both have the desirable property (from the point of view of increasing the complexity) that by substitution we obtain a full set of nonzero monotone functions in one fewer variables, in this case $\{p,1\}$.

Fig. 2.

Adequacy in the proof that $2^4\rightarrow 19\rightrightarrows 5$.

Fig. 3.

Asian option, and European call option (corresponding to the majority function).

Definition 3.5

Let us say that a set of monotone functions on variables $p_1,\dots ,p_n$ is adequate if by substitutions of values for $p_1\in \{0,1\}$ they contain all monotone nonzero functions on $p_2,\dots ,p_n$. If one value for $p_1$ suffices then we say strongly adequate.

Let us write $\mathbf 2^i$ for the set $\{0,1\}^i$ with the product ordering.

Definition 3.6

If there is an embedding of $\mathbf {2}^i$ into $F_j^-$ ensuring adequacy onto $F_{j-1}^-$, in the sense that we map into $F_{j-1}$ (so self-loops may be used in the automaton), and we map onto $F_{j-1}^-$, then we write

$$ 2^i\rightarrow |F_j^-|\rightrightarrows |F_{j-1}^-|. $$

It is crucial to note that in Sect. 2, adequacy was automatic: the concept of function is much more robust than that of a monotone function, meaning that functions can be combined in all sorts of ways and remain functions. As an example of the unusual but convenient notation of Definition 3.6, we have:

Theorem 3.7

There is an embedding of $\mathbf {2}^2$ into $F_3^-$ ensuring adequacy onto $F_2^-$. In symbols,

$$ 4\rightarrow 19\rightrightarrows 5. $$

Proof

We use formulas of the form $(r\wedge b)\vee a$ with $a\le b$, as follows:

Theorem 3.8

There is an embedding of $\mathbf {2}^4$ into $F_4^-$ ensuring adequacy onto $F_3^-$:

$$ 16\rightarrow 167\rightrightarrows 19 $$

Proof

We make sure to hit p, q, r as follows: $(r\wedge b)\vee a_i$, $1\le i\le 2$, where $a_1<a_2\le b$, and $a_i,b\in F_3^-$, with $b\in T$. Here T is the top cube in $F_3^-$,

$$ T=\{b\in F_3^-: \text {maj}\le b\le p\vee q\vee r\} $$

$$\begin{aligned} = \left\{ \begin{array}{l} (p\wedge q)\vee (p\wedge r)\vee (q\wedge r),\, \\ p\vee (q\wedge r),\quad q\vee (p\wedge r),\quad r\vee (p\wedge q), \\ p\vee q,\quad p\vee r,\quad q\vee r,\quad p\vee q\vee r \end{array}\right\} . \end{aligned}$$

Let $\psi :\{0,1\}^3\rightarrow T$ be an isomorphism. Not that $b\not \in \{\hat{0},p,q,r\}$. And $\{a_1,a_2\}\subset \{b,p,q,r,\hat{0}\}$ where $\hat{0}$ is $p\wedge q\wedge r$, the least element of $F_3^-$. Let

$$ (a_1,a_2)= {\left\{ \begin{array}{ll} (\hat{0},b) &{} \text {if } b \text { bounds none of } p,q,r;\\ (p,b)\text { or }(\hat{0},p) &{} \text {if say } b>p;\\ \end{array}\right. } $$

By Lemma above, $(r\wedge \psi (x))\vee a_i \le (r\wedge \psi (y))\vee c_i$ iff $x\le y$ and $a_i\le c_i$.   $\square$

We can consider whether $u\rightarrow v\rightrightarrows w$ whenever the numbers are of the form $2^m$, $|F_n^-|\in \{1,2,5,19,167,\dots \}$, $|F_{n-1}^-|$, and $u\le v$ and $w\le 2u$ (as u increases, being 1:1 becomes harder but being adequate becomes easier). In the case of strong adequacy witnessed by $p=p_0$ we write simply $u\rightarrow v\rightarrow _{p_0} w$; this can only happen when $w\le u$.

Theorem 3.9

We have the following adequacy calculations:

• 1. $2^0\rightarrow 2\rightarrow 1$

• 2. $2^1\rightarrow 2\rightarrow 1$

• 3. $2^0\rightarrow 5\rightrightarrows 2$

• 4. $2^1\rightarrow 5\rightarrow 2$

• 5. $2^2\rightarrow 5\rightarrow 2$

We omit the trivial proof of Theorem 3.9.

Theorem 3.10

$2^3\rightarrow 19\rightrightarrows 5$ and $2^4\rightarrow 19\rightrightarrows 5$.

Proof

The map in Fig. 1 is onto $F_3^-\setminus \{p,q,r\}$ so it works. As shown in Fig. 2, if we restrict that map to the top cube, mapping onto $T\cup \{1\}\setminus \{p\vee q\vee r\}$, and set $r=0$ then we map onto $F_2^-$.   $\square$

Lemma 3.11

Let $a_1,a_2,b_1,b_2$ be Boolean functions of p, q, r and let

$$ f_{a_ib_i}(p,q,r,s) = [s\wedge b_i]\vee [\lnot s\wedge a_i]. $$

Then $f_{a_1b_1}\le f_{a_2b_2}\iff a_1\le a_2$ and $b_1\le b_2$.

Proof

By definition,

$$ f_{a_1b_1}\le f_{a_2b_2}\iff [p_4\wedge b_1]\vee [\lnot p_4\wedge a_1] \le [p_4\wedge b_2]\vee [\lnot p_4\wedge a_2]. $$

Clearly, $a_1\le a_2$ and $b_1\le b_2$ implies this, so we just need the converse. If $a_1\not \le a_2$ then any assignment that makes $p_4$ false, $a_1$ true, and $a_2$ false will do. Similarly if $b_1\not \le b_2$ then any assignment that makes $p_4$ true, $b_1$ true, and $b_2$ false will do.

   $\square$

Theorem 3.12

There is an injective isotone map from $\mathbf {2}^6$ into $F_4$, and in fact

$$ 2^6\rightarrow 167\rightrightarrows 19. $$

Proof

We start with a monotone version of the simple equation $2^{2^n}=(2^{2^{n-1}})^2$. Namely, a pair of monotone functions g, h of $n-1$ variables, with $g\le h$, gives another monotone function via

Now consider elements a of the bottom hypercube in $F_3$ and b of the top hypercube in $F_3$ in Fig. 2. So we must have $a\le b$ since the bottom is below the top (and $a=b$ can happen since the two hypercubes overlap in the majority function). Let $f_{ab} = [p_4\wedge b]\vee [\lnot p_4\wedge a]$. Since $a\le b$, $f_{ab}$ is monotonic. By Lemma 3.11, these functions $f_{ab}$ are ordered as $\mathbf {2}^6=\mathbf {2}^3\times \mathbf {2}^3$.

Finally, in order to ensure adequacy we modify this construction to reach higher in $F_4^-$, replacing the top cube in the lower half by a cube formed from the upper half. In more detail, consider $(r\wedge b)\vee a$ with $a\le b$ from $F_3^-$, where the a’s are chosen from the bottom cube of $F_3$, and the b’s from the top cube, except that when a is the top of the bottom cube we let b be the top cube with the top replaced by 1, and when a is the bottom of the bottom cube we let b be the cube

$$ \{p,q,r,p\wedge q, p\wedge r, q\wedge r, p\vee q, p\vee r, q\vee r, p\wedge q\wedge r, p\vee q\vee r\}. $$

   $\square$

Fig. 4.

The lattice $F_3$ of all monotone Boolean functions in three variables p, q, r.

Theorem 3.13

$2^5\rightarrow 167\rightrightarrows 19$.

Proof

A small modification of Theorem 3.12; only use bottom, top and two intermediate “cubes” within the cube.   $\square$

Open Problem. For $n=11$ we need to determine whether the following holds, which has so far proved too computationally expensive:

$$ 2^7\rightarrow 167\rightrightarrows 19? $$

That is, is there an isotone map from the 128-element lattice $\mathbf {2}^7$ into $F_4^-$, the set of nonzero monotone functions in variables p, q, r, s, such that upon plugging in constants for p, we cover all of $F_3^-$, the set of nonzero monotone functions in q, r, s?

4 Early-Monotone Functions and Complete Simple Games

In this section we take the financial ideas from Sect. 3 one step further, by noting that the Asian option (Example 3.4) has the added property that earlier bits matter more. In economics terms, we have what is called a complete simple game: there is a set of goods linearly ordered by intrinsic value. You get some of the goods and there are thresholds for how much value you need to win.

Definition 4.1

Let $e_i\in \{0,1\}^n$ be defined by $e_i(j)=1$ if and only if $j=i$. An n-ary Boolean function f is early if for all $0\le i< j < n$ and all $y\in \{0,1\}^n$ with $y(i)=y(j)=0$, if $f(y+e_j)=1$ then $f(y+e_i)=1$.

The number of early (not necessarily monotone) functions starts

$$ 2,4,12,64,700, 36864,\dots $$

If a function is early and monotone we shall call it early-monotone. Early-monotonicity encapsulates an idea of time-value-of-money; getting paid now is better than next week, getting promoted now is better than next decade, etc.

In the early context one needs the map from $2^m$ into the early functions to be “early”, i.e., the function mapped to by 100 should dominate the one mapped to by 010 etc. That is, the map must be order-preserving from $2^m$ with the majorization lattice order into the complete simple games.

The number of early-monotone functions on n variables, including zero, is

$$ 2,3,5,10,27,119,1173,\dots $$

which appears in OEIS A132183 as the number of “regular” Boolean functions in the terminology of Donald Knuth. He describes them also as the number of order ideals (or antichains) of the binary majorization lattice with $2^n$ points.

Definition 4.2

The binary majorization lattice $E_n$ is the set $\{0,1\}^n$ ordered by $(a_1,\dots ,a_n)\le (b_1,\dots ,b_n)$ iff $a_1+\dots +a_k\le b_1+\dots +b_k$ for each k.

The lattice $E_5$ for $n=5$ is illustrated in [7, Fig. 8, Volume 4A, Part 1]. The basic properties of this lattice are discussed in [7, Exercise 109 of Section 7.1.1]. The majorization order is obtained by representing e.g. 1101 as $(1,2,4,\infty )$, showing where the kth 1 appears (the $\infty$ signifying that there is no fourth 1 in 1101), and ordering these tuples by majorization. OEIS cites work of Stefan Bolus [2] who calls the “regular” functions complete simple games [8], a term from the economics and game theory literature. There, arbitrary monotone functions are called simple games, and“complete” refers to the fact that the positions have a complete linear ordering (in the finance application, earlier positions are most valuable). Figure 5 shows that in the complete-simple-games setting we have

for a total maximal complexity of 47 for complete simple games at $n=8$. This contrasts with Theorem 3.8 which shows that for arbitrary simple games the complexity can reach 58 at $n=8$.

Fig. 5.

(a) The majorization lattice $E_4$ on 4 variables. (b) The 27 complete simple games $C_4$ on 4 variables. The symbol $\vee$ denotes an element that is join-reducible. and denote the image under the first map $E_4\rightarrow C_4$ and in particular denotes some elements sufficient for the second map $C_4\rightarrow C_3$ to be onto. (c) The 10 complete simple games $C_3$ on 3 variables. (Color figure online)