A Crevice on the Crane Beach: Finite-Degree Predicates

TL;DR

This paper explores the expressive power of first-order logic over words with finite-degree predicates, demonstrating its locality properties and the conditions under which it satisfies the Crane Beach Property, contrasting it with more expressive logics.

Contribution

It shows that FO with order, MSB$_0$, and finite-degree predicates has the Crane Beach Property, revealing its locality and expressive limitations compared to FO with arbitrary predicates.

Findings

FO[<, Fin] satisfies the Crane Beach Property.

FO[<, Fin] exhibits locality despite expressive power.

FO[<, Fin] can express a wide variety of languages.

Abstract

First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB$_0$, and the finite-degree predicates: FO[Arb] = FO[<, MSB$_0$, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB$_0$]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.

Equations

$$\text{1) }00000\to 00111;$$

$$\text{3) }10000\to 10111;$$

$$\exists x\psi (x)⇝$$

$$\displaystyle\qquad\exists x\Big{[}\mathsf{work}(x)\land\bigvee_{1\leq i\leq 4}\Big{[}(\exists y)[\mathsf{trans}^{(i)}(x,y)]\land\psi(x^{(i)})\big{]}\Big{]}\enspace,$$

$$⟨\mathbb{N},\vec{x}:=\overrightarrow{a_i},\vec{y}:=\overrightarrow{b_M}⟩\models \phi \text{iff}i\in M.$$

$$\phi \equiv (\exists x)(\forall y)[\psi ],\text{with }\psi \equiv \mathbf{a}(x)\wedge (x<y\to \mathbf{b}(y)),$$

$$(\forall^{\textsf{B}}y)[\mathbf{b}(y)]\lor(\exists^{\textsf{B}}x)\Big{[}\mathbf{a}(x)\land(\forall^{\textsf{B}}y)[\psi]\Big{]}\enspace.$$

$$\phi \equiv (Q_1{x}_1)(Q_2{x}_2)\cdots (Q_k{x}_k)[\psi ],$$

$${\mathcal{L}}^m({\mathcal{R}}^n(p))=\mathcal{L}({\mathcal{L}}^{m-1}({\mathcal{R}}^{n-1}(p^{\prime }))),$$

$${\mathcal{L}}^m({\mathcal{R}}^n(p))\ge \mathcal{L}(\mathcal{R}(p^{\prime }))=q.$$

$$\mathsf{bounds}(\epsilon )$$

$$\mathsf{bounds}(t_1{t}_2\cdots t_i)$$

$$\text{mbox}\{\begin{array}{ll}({\mathcal{R}}^n(\ell ),r)& \text{if }{t}_i=\text{Alice}\\ ({\mathcal{L}}^n(\ell ),{\mathcal{R}}^n(r))& \text{if }{t}_i=\text{Neutral}\\ (\ell ,{\mathcal{L}}^n(r))& \text{if }{t}_i=\text{Bob}\end{array}$$

$$\text{mbox}\text{with }(\ell ,r)=\mathsf{bounds}(t_1{t}_2\cdots t_{i-1})\text{ and }n=2^{k-i}.$$

$${\mathcal{L}}^{2^{k-(j-1)}}({\mathcal{L}}^{2^{k-(j-2)}}(\cdots ({\mathcal{L}}^{2^{k-(i+1)}}({\mathcal{R}}^{2^{k-i}}(\ell )))\cdots )).$$

$$m=2^{k-(i+1)}+\cdots +2^{k-(j-1)}$$

$$<2^{k-i}=n.$$

$$w\models \phi (c)\iff c\le f(n)\wedge c=\text{number of 1’s in }w.$$

$$w\models \phi (a,b,c)\iff a=b+f(c).$$

$$\psi (a,b,c)\equiv (\exists c^{\prime })[F(c,c^{\prime })\wedge a=b+c^{\prime }].\text{qed}$$

$${\text{Bit}}^{\prime }=\{(x,y)\mid (x,y-f(x))\in \text{Bit}\}.$$

Full text

I Introduction

Ajtai [1] and Furst, Saxe, and Sipser [2] showed some 30 years ago that Parity, the language of words over $\{0,1\}$ having an even number of $1$, is not computable by families of shallow circuits, namely AC0 circuits. Since then, a wealth of precise expressiveness properties of AC0 has been derived from this sole result [3, 4]. Naturally aiming at a better understanding of the core reasons behind this lower bound, a continuous effort has been made to provide alternative proofs of {\text{{Parity}}}\notin{\text{AC{}^{0}}}. However, this has been a rather fruitless endeavor, with the notable exception of the early works of Razborov [5] and Smolenski [6] that develop a less combinatorial approach with an algebraic flavor. For instance, Koucký et al. [7] foray into descriptive complexity and use model-theoretic tools to obtain {\text{{Parity}}}\notin{\text{AC{}^{0}}}, but assert that “contrary to [their] original hope, [their] Ehrenfeucht-Fraïssé game arguments are not simpler than classical lower bounds.” More recent promising approaches, especially the topological ones of [8, 9], have yet to yield strong lower bounds.

A different take originated from a conjecture of Lautemann and Thérien, investigated by Barrington et al. [10]: the Crane Beach Conjecture. They noticed that the letter [math] acts as a neutral letter in Parity, i.e., [math] can be added or removed from any word without affecting its membership to the language. If a circuit family recognizes a language with a neutral letter, it seems convincing that the circuits for two given input sizes should look very similar, that is: the circuit family must be highly uniform. It was thus conjectured that all neutral letter languages in AC0 were regular, and this was disproved in [10].

This however sparked an interest in the study of neutral letter languages, in particular from the descriptive complexity view. Indeed, AC0 circuits recognize precisely the languages expressible in \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}], where $\mathcal{A}$

$\mathcal{RB}$

denotes all possible numerical predicates (expressing numerical properties of the positions in a word). Further, as all regular neutral letter languages of \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] are star-free [10], i.e., in $\text{FO}[\leq]$, the Crane Beach Conjecture asked:

Are all neutral letter languages of \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] in $\text{FO}[\leq]$?

Note that this echoes the above intuition on uniformity, since the numerical predicates correspond precisely to the allowed power to compute the circuit for a given input length [11]. The intuition on the logic side is even more compelling: if a letter can be introduced anywhere without impacting membership, then the only meaningful relation that can relate positions is the linear order. However, first-order logic can “count” up to $\log n$ (see, e.g., [12]), meaning that even within a word with neutral letters, \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] can assert some property on the number of nonneutral letters. This is, in essence, why nonregular neutral letter languages can be expressed in \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}].

In the recent years, a great deal of efforts was put into studying the Crane Beach Property in different logics, i.e., whether the definable neutral letter languages are regular. Krebs and Sreejith [13], building on the work of Roy and Straubing [14], show that all first-order logics with monoidal quantifiers and $+$ as the sole numerical predicate have the Crane Beach Property. Lautemann et al. [15] show Crane Beach Properties for classes of bounded-width branching programs, with an algebraic approach relying on communication complexity. Some expressiveness results were also derived from Crane Beach Properties, for instance Lee [16] shows that $\text{FO}[+]$ is strictly included in $\text{FO}[\leq,\times]$ by proving that only the former has the Crane Beach Property. Notably, all these logics are quite far from full \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}], and in that sense, fail to identify the part of the arbitrary numerical predicates that fit the intuition that they are rendered useless by the presence of a neutral letter.

In the present paper, we identify a large class of predicates, the finite-degree predicates, and a predicate MSB0 such that any numerical predicate can be first-order defined using them and the order; in symbols, \mbox{\text{FO}[\leq,{\text{MSB${}_{0}$}},\text{$\mathcal{F}$\kern-1.00006pt\scalebox{0.6}{$\mathcal{IN}$}}]}=\mbox{\text{FO}[\text{$\mathcal{A}$\kern-0.20004pt\scalebox{0.6}{$\mathcal{RB}$}}]}. We show that, strikingly, both \text{FO}[\leq,{\text{MSB{}{0}}}] and \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] have the Crane Beach Property, this latter statement being our main result. Hence showing that some nonregular neutral letter language is not expressible in \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] could be done by showing that MSB0 may be removed from any \text{FO}[\leq,{\text{MSB{}{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] formula expressing it.

The proof for the Crane Beach Property of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] relies on a communication complexity argument different from that of [15]. It is also unrelated to the database collapse techniques of [10] (succinctly put, no logic with the Crane Beach Property has the so-called independence property, i.e., can encode arbitrary large sets). We will show that in fact \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] does have the independence property. This provides, to the best of our knowledge, the first example of a logic that exhibits both the independence and the Crane Beach properties.

The aforementioned counting property of \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] led to the conjecture [10, 16] that a logic has the Crane Beach Property if and only if it cannot count beyond a constant. To the best of our knowledge, neither of the directions is known; we show however that \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] can only count up to a constant, by showing that it cannot even express very restricted forms of the addition. This adds evidence to the “if” direction of the conjecture.

Structure of the paper. In Section II, we introduce the required notions, although some familiarity with language theory and logic on words is assumed (see, e.g., [4]). In Section III, we show that \mbox{\text{FO}[\leq,{\text{MSB${}_{0}$}},\text{$\mathcal{F}$\kern-1.00006pt\scalebox{0.6}{$\mathcal{IN}$}}]}=\mbox{\text{FO}[\text{$\mathcal{A}$\kern-0.20004pt\scalebox{0.6}{$\mathcal{RB}$}}]}. In Section IV, we present a simple proof, relying on a much harder result from [10], that \text{FO}[\leq,{\text{MSB{}_{0}}}] has the Crane Beach Property. The failing of the aforementioned collapse technique for \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] is shown in Section V. We tackle the Crane Beach Property of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}], our main result, in Section VI, after the necessary tools have been developed. Finally, in Section VII, we focus on the counting inabilities of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}].

Previous works. Finite-degree predicates were introduced by the second author in [17], in the context of two-variable logics. Therein, it is shown that the two-variable fragment of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] has the Crane Beach Property, and, even stronger, that the neutral letter languages expressible with $k$ quantifier alternations can be expressed without the finite-degree predicates with the same amount of quantifier alternations. The techniques used in [17] are specific to two-variable logics, relying heavily on the fact that each quantification depends on a single previously quantified variable. We thus stress that the communication complexity argument developed in Section VI is unrelated to [17].

The fact that two sets of predicates can both verify the Crane Beach Property while their union does not has already been witnessed in [10]. Indeed, letting $\mathcal{M}$

$\mathcal{ON}$

be the set of monoidal numerical predicates, the Property holds for both $\text{FO}[\leq,+]$ and \text{FO}[\leq,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}}] but fails for \text{FO}[\leq,+,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}}], although this latter class is less expressive than \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] (this can be shown using the same proof as [7, Proposition 5]).

II Preliminaries

II-A Generalities

We write $\mathbb{N}=\{0,1,2,\ldots\}$ for the set of nonnegative numbers. For $n\in\mathbb{N}$, we let $[n]=\{0,1,\ldots,n-1\}$. A function $f\colon\mathbb{N}\to\mathbb{N}$ is nondecreasing if $m>n$ implies $f(m)\geq f(n)$.

An alphabet $A$ is a finite set of letters (symbols), and we write $A^{*}$ for the set of finite words. For $u=u_{0}u_{1}\cdots u_{n-1}$, the length $n$ of $u$ is denoted $|u|$. We write $\varepsilon$ for the empty word and $A^{\leq k}$ for words of length $\leq k$.

II-B Logic on words

For an alphabet $A$, let $\sigma_{A}$ be the vocabulary $\{\mathbf{a}\mid a\in A\}$ of unary letter predicates. A (finite) word $u=u_{0}u_{1}\cdots u_{n-1}\in A^{*}$ is naturally associated with the structure over $\sigma_{A}$ with universe $[n]$ and with $\mathbf{a}$ interpreted as the set of positions $i$ such that $u_{i}=a$, for any $a\in A$. A numerical predicate is a $k$-ary relation symbol together with an interpretation in $[n]^{k}$ for each possible universe size $n$. Given a formula $\varphi$ that relies on some numerical predicates and a word $u$, we write $u\models\varphi$ to mean that $\varphi$ is true of the $\sigma_{A}$-structure for $u$ augmented with the interpretations of the numerical predicates for the universe of size $|u|$. A formula $\varphi$ thus defines or expresses the language $\{u\in A^{*}\mid u\models\varphi\}$.

II-C Classes of formulas

We let $\mathcal{A}$

$\mathcal{RB}$

be the set of all numerical predicates. Given a set \mathcal{N}\subseteq\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}, we write $\text{FO}[\mathcal{N}]$ for the set of first-order formulas built using the symbols from $\mathcal{N}\cup\sigma_{A}$, for any alphabet $A$. Similarly, $\text{MSO}[\mathcal{N}]$ denotes monadic second-order formulas built with those symbols. We further define the quantifiers Maj  and $\exists^{\equiv}_{i}$, for $i\in\mathbb{N}$, that will only be used in discussions:

• •

$u\models({\text{Maj\;}}x)[\varphi(x)]$ iff there is strict majority of positions $i\in[|u|]$ such that $\langle u,x:=i\rangle\models\varphi$;

• •

$u\models(\exists^{\equiv}_{i})[\varphi(x)]$ iff the number of positions $i\in[|u|]$ verifying $\langle u,x:=i\rangle\models\varphi$ is a multiple of $i$.

We will write $\text{MAJ}[\mathcal{N}]$ and $\text{FO+MAJ}[\mathcal{N}]$ with the obvious meanings. Further, $\text{FO+MOD}[\mathcal{N}]$ allows all the quantifiers $\exists^{\equiv}_{i}$ in $\text{FO}[\mathcal{N}]$ formulas.

II-D On numerical predicates

The most ubiquitous numerical predicate here will be the binary order predicate $\leq$. The predicate that zeroes the most significant bit (MSB) of a number will also be important: (m,n)\in{\text{MSB{}_{0}}} iff $n=m-2^{\lfloor\log m\rfloor}$. Note that both predicates do not depend on the universe size, and we single out this concept:

Definition 1**.**

A $k$-ary numerical predicate $P$ is unvaried if there is a set $E\subseteq\mathbb{N}^{k}$ such that the interpretation of $P$ on universes of size $n$ is $E\cap[n]^{k}$. In this case, we identify $P$ with the set $E$. It is varied otherwise.111The relevance of this concept has been noted in previous works (e.g.,  [10]), but was left unnamed. The second author used in [17] the terms (non)uniform, an unfortunate coinage in this context. We prefer here the less conflicting terms (un)varied. We write \text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}^{\text{u}} for the set of unvaried numerical predicates.

Naturally, any varied predicate can be converted to an unvaried one by turning the universe length into an argument and quantifying the maximum position; this implies in particular that \mbox{\text{FO}[\text{$\mathcal{A}$\kern-0.20004pt\scalebox{0.6}{$\mathcal{RB}$}}]}=\mbox{\text{FO}[\text{$\mathcal{A}$\kern-0.20004pt\scalebox{0.6}{$\mathcal{RB}$}}^{\text{u}}]}. This is however not entirely innocuous, as will be discussed in Section VII.

We will rely on the following class of unvaried predicates, generalizing a definition of [17] (see also the older notion of “finite formula” [18]):

Definition 2**.**

An unvaried predicate $P\subseteq\mathbb{N}^{k}$ is of finite degree222The name stems from the fact that the hypergraph defined by $P$, with edges of size $k$, is of finite degree. if for all $n\in\mathbb{N}$, $n$ appears in a finite number of tuples in $P$. We write $\mathcal{F}$

$\mathcal{IN}$

for the class of such predicates.

Note that this does not imply that there is a $N$ that bounds the number of appearance for all $n$’s. Some examples:

• •

MSB0 is not a finite-degree predicate, as, e.g., (2^{n},0)\in{\text{MSB{}_{0}}} for any $n$, hence [math] appears infinitely often;

• •

Any unvaried monadic numerical predicate is of finite degree, this implies in particular that any language over a unary alphabet is expressed by a \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] formula;

• •

The graph of any nondecreasing unbounded function $f\colon\mathbb{N}\to\mathbb{N}$ defines a finite-degree predicate, since $f^{-1}(n)$ is a finite set for all $n$;

• •

The order, sum, and multiplication are not of finite degree;

• •

One can usually “translate” unvaried predicates to make them finite degree; for instance, the predicate true of $(x,y)$ if $y-x<x<y$ is of finite degree, see also the proof of Proposition 4.

II-E Crane Beach Property

A language $L\subseteq A^{*}$ is said to have a neutral letter if there is a $e\in A$ such that adding or deleting $e$ from a word does not change its membership to $L$. Following [15], we say that a logic has the Crane Beach Property if all the neutral letter languages it defines are regular. We further say that it has the strong Crane Beach Property if all the neutral letter languages it defines can be defined using order as the sole numerical predicate.

III \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}] and \text{FO}[\leq,{\text{MSB{}_{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] define the same

languages

In this section, we express all the numerical predicates using only finite-degree ones, MSB0, and the order. The result is a variant of [17, Theorem 3], where it is proven for the two-variable fragment, and on neutral letter languages.

Theorem 1**.**

\text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}]* and \text{FO}[\leq,{\text{MSB{}_{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] define the same languages.*

Proof.

We show that any \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}^{\text{u}}] language is definable in \text{FO}[\leq,{\text{MSB{}_{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}].

The main idea is to divide the set of word positions in four contiguous zones and have the variables range over only the second zone, called the work zone. Given an input of length $\ell=2^{n}$, the set of positions $[\ell]$ is divided in four zones of equal size $2^{n-2}$; if the input length is not a power of $2$, then we apply the same split as the closest greater power of two, leaving the third and fourth zone possibly smaller than the first two.

As an example, suppose that the word size is $\ell=11110$ (here and in the following, we write numbers in binary). The four zones of $[\ell]$ will be:

[TABLE]

The work zone has two salient properties: 1. Checking that a number $k\in[\ell]$ belongs to it amounts to checking that $k$ has exactly one greater power of two; in particular, two work-zone positions share the same MSB; 2. Any number in $[\ell]$ outside the work zone can be obtained by replacing the MSB of a number in the work zone with some other bits (0, 10, and 11, for the first, third, and fourth zone, respectively); we call this a translation to a zone, e.g., in our example above, $10101$ is the translation of $01101$ to the third zone.

More formally, we can define a formula $\mathsf{work}(x)$ which is true iff $x$ belongs to the work zone, by expressing that there is exactly one power of two strictly greater than $x$, using the monadic predicate true on powers of two. Moreover, we can define formulas $\mathsf{trans}^{(i)}(x,y)$, $1\leq i\leq 4$, which are true if $x$ is in the work zone and $y$ is its translation to the $i$-th zone; let us treat the case $i=3$, the others being similar. The formula $\mathsf{trans}^{(3)}(x,y)$ is true if $y$ is obtained by replacing the MSB of $x$ with $10$, this is expressed using MSB0 by finding $z$ such that {\text{MSB{}{0}}}(x,z) holds and then checking that $y$ is the first value $z^{\prime}$ strictly greater than $x$ such that {\text{MSB{}{0}}}(z^{\prime},z) holds.

The strategy will then be to: 1. Quantify over the work zone only; 2. Modify the predicates to internally change the MSBs according to which zone the variables were supposed to belong; 3. Compute the translations of the variables for the letter predicates. Step 1 relies on $\mathsf{work}$ and $\mathsf{trans}^{(i)}$, Step 2 transforms all numerical predicates to finite-degree ones, and Step 3 simply uses $\mathsf{trans}^{(i)}$.

Let \varphi\in\mbox{\text{FO}[\text{$\mathcal{A}$\kern-0.20004pt\scalebox{0.6}{$\mathcal{RB}$}}^{\text{u}}]}. Step 1. We rewrite $\varphi$ with annotated variables; with $x$ a variable, we write $x^{(i)}$, $1\leq i\leq 4$, to mean “$x$ translated to zone $i$”—as all the variables will be quantified in the work zone, this is well defined. The following rewriting is then performed:

[TABLE]

and mutatis mutandis for $\forall$.

Step 2. We sketch this step for binary numerical predicates. Suppose such a predicate $P$ is used in $\varphi$. For $1\leq i,j\leq 4$, we define the predicate $P^{(i,j)}$ that expects two work-zone positions, translates them to the $i$-th and $j$-th zone, respectively, then checks whether they belong to $P$. Crucially, as the inputs are work-zone positions, $P^{(i,j)}$ immediately rejects if they do not share the same MSB: it is thus a finite-degree predicate. Now every occurrence of $P(x^{(i)},y^{(j)})$ in $\varphi$ can be replaced by $P^{(i,j)}(x,y)$.

Step 3. The only remaining annotated variables appear under letter predicates. To evaluate them, we simply have to retrieve the translated position. Hence each $\mathbf{a}(x^{(i)})$ will be replaced by $(\exists y)[\mathsf{trans}^{(i)}(x,y)\land\mathbf{a}(y)]$, concluding the proof. ∎

Remark*.*

Theorem 1 can be shown to hold also for \text{FO+MAJ}[\leq,{\text{MSB{}{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}], i.e., this logic is equiexpressive with \text{FO+MAJ}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}]. The main modification to the proof is to allow arbitrary quantifications (as opposed to work zone ones only) and compute the work zone equivalent of each position before checking the numerical predicates. This ensures that the number of positions verifying a formula is not changed. Likewise, \text{FO+MOD}[\leq,{\text{MSB{}{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] is equivalent with \text{FO+MOD}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}].

IV \text{FO}[\leq,{\text{MSB{}_{0}}}] has the Crane Beach Property

Following a short chain of rewriting, we will express MSB0 using predicates that appear in [10] and conclude that:

Theorem 2**.**

\text{FO}[\leq,{\text{MSB{}_{0}}}]* has the strong Crane Beach Property.*

Proof.

Let $f\colon\mathbb{N}\to\mathbb{N}$ be defined by $f(n)=2^{\left(\lfloor\log n\rfloor^{2}\right)}$, and let $F\subseteq\mathbb{N}^{2}$ be its graph. Barrington et al. [10, Corollary 4.14] show that $\text{FO}[\leq,+,F]$ has the strong Crane Beach Property; we show that MSB0 can be expressed in that logic. First, the monadic predicate $Q=\{2^{n}\mid n\in\mathbb{N}\}$ is definable in $\text{FO}[\leq,F]$, since $n$ is a power of two iff $f(n-1)\neq f(n)$. Second, given $n\in\mathbb{N}$, the greatest power of two smaller than $n$ is $p=2^{\lfloor\log n\rfloor}$, which is easy to find in $\text{FO}[\leq,Q]$. Finally, {\text{MSB{}_{0}}}(n,m) is true iff $m+p=n$, and is thus definable in $\text{FO}[\leq,+,F]$. ∎

Remark*.*

From Lange [19], $\text{MAJ}[\leq]$ and $\text{FO+MAJ}[\leq,+]$ are equiexpressive, and as MSB0 is expressible using the unary predicate $\{2^{n}\mid n\in\mathbb{N}\}$ and the sum, this shows that \text{MAJ}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] is equiexpressive with \text{FO+MAJ}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}]. Hence \text{MAJ}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] does not have the strong Crane Beach Property.

V \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] has the Independence

Property

In [10], an important tool is introduced to show Crane Beach Properties, relying on the notion of collapse in databases, see [20, Chapter 13] for a modern account. Specifically, let us define an ad-hoc version of the:

Definition 3** (Independence property (e.g., [21])).**

Let $\mathcal{N}$ be a set of unvaried numerical predicates. Let $\vec{x},\vec{y}$ be two vectors of first-order variables of size $k$ and $\ell$, respectively. A formula $\varphi(\vec{x},\vec{y})$ of $\text{FO}[\mathcal{N}]$, over a single-letter alphabet, has the independence property if for all $n>0$ there are vectors $\vec{a_{0}},\vec{a_{1}},\ldots,\vec{a_{n-1}}$, each of $\mathbb{N}^{k}$, for which for any $M\subseteq[n]$, there is a vector $\vec{b_{M}}\in\mathbb{N}^{\ell}$ such that:333Note that we evaluate a formula over an infinite domain; this is well defined in our case since we only use unvaried predicates and the letter predicates are irrelevant.

[TABLE]

The logic $\text{FO}[\mathcal{N}]$ has the independence property if it contains such a $\varphi$.

Intuitively, a logic has the independence property iff it can encode arbitrary sets. Barrington et al. [10], relying on a deep result of Baldwin and Benedikt [21], show that:

Theorem 3** ([10, Corollary 4.13]).**

If a logic does not have the independence property, then it has the strong Crane Beach Property.

We note that this powerful tool cannot show that the logic we consider exhibits the Crane Beach Property:

Proposition 1**.**

\text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]* has the independence property.*

Proof.

Let $n>0$, and define $a_{i}=2^{n}+2^{i}$ for $0\leq i<n$. Now for $M\subseteq[n]$, let $b_{M}=2^{n}+\sum_{i\in M}2^{i}$. It holds that $i\in M$ iff the binary AND of $a_{i}$ and $b_{M}$ is $a_{i}$. Consider this latter binary predicate; its behavior on two arguments that do not share the same MSB is irrelevant, and we can thus decide that such inputs are rejected. Thanks to this, we obtain a finite-degree predicate. Consequently, the formula that consists of this single predicate has the independence property. ∎

VI \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] has the Crane Beach Property

VI-A Communication complexity

We will show the Crane Beach Property of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] by a communication complexity argument. This approach is mostly unrelated to the use of communication complexity of [15, 22]; in particular, we are concerned with two-party protocols with a split of the input in two contiguous parts, as opposed to worst-case partitioning of the input among multiple players. We rely on a characterization of [23] of the class of languages expressible in monadic second-order with varied monadic numerical predicates. Writing this class \text{MSO}[\leq,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}}], they state in particular the following:

Proposition 2** ([23, Theorem 2.2]).**

Let $L\subseteq A^{*}$ and define, for all $p\in\mathbb{N}$, the equivalence relation $\sim_{p}$ over $A^{*}$ as: $u\sim_{p}v$ iff for all $w\in A^{p}$, $u\cdot w\in L\Leftrightarrow v\cdot w\in L$. If there is a $N\in\mathbb{N}$ such that for all $p\in\mathbb{N}$, $\sim_{p}$ has at most $N$ equivalence classes, then L\in\mbox{\text{MSO}[\leq,\text{$\mathcal{M}$\kern-0.50003pt\scalebox{0.6}{$\mathcal{ON}$}}]}.

Lemma 1**.**

Let $L\subseteq A^{*}$. Suppose there are functions $f_{\text{Alice}}\colon A^{*}\times\mathbb{N}\times\{0,1\}^{*}\to\{0,1\}$ and $f_{\text{Bob}}\colon A^{*}\times\mathbb{N}\times\{0,1\}^{*}$ and a constant $K\in\mathbb{N}$ such that for any $u,v\in A^{*}$, the sequence, for $1\leq i\leq K$:

\begin{array}[]{@{}l@{\,}lll}\text{\textbullet}\;\;a_{i}=f_{\text{Alice}}&(u,&|u\cdot v|,&b_{1}b_{2}\cdots b_{i-1})\\ \text{\textbullet}\;\;b_{i}=f_{\text{Bob}}&(v,&|u\cdot v|,&a_{1}a_{2}\cdots a_{i});\end{array}**

is such that $b_{K}=1$ iff $u\cdot v\in L$. Then L\in\mbox{\text{MSO}[\leq,\text{$\mathcal{M}$\kern-0.50003pt\scalebox{0.6}{$\mathcal{ON}$}}]}.

Proof.

We adapt the (folklore) proof that $L$ is regular iff such functions exist where $f_{\text{Alice}}$ and $f_{\text{Bob}}$ do not use their second parameter.

Let $p\in\mathbb{N}$. For any $u\in A^{*}$, let $c(u)$ be the set of pairs $(a_{1}a_{2}\cdots a_{K},b_{1}b_{2}\cdots b_{K-1})$ such that for all $1\leq i\leq K$, it holds that $a_{i}=f_{\text{Alice}}(u,|u|+p,b_{1}b_{2}\cdots b_{i-1})$. Define the equivalence relation $\equiv$ by letting $u\equiv v$ iff $c(u)=c(v)$; it clearly has a finite number $N=N(K)$ of equivalence classes. Moreover, if $u\equiv v$ and $w\in A^{p}$, then $(u,w)$ and $(v,w)$ define the same sequences of $a_{i}$’s and $b_{i}$’s, by a simple induction. Hence $u\cdot w\in L\Leftrightarrow v\cdot w\in L$. This shows that $\equiv$ refines $\sim_{p}$, implying, by Proposition 2, that L\in\mbox{\text{MSO}[\leq,\text{$\mathcal{M}$\kern-0.50003pt\scalebox{0.6}{$\mathcal{ON}$}}]}. ∎

We shall adopt the classical communication complexity view here, and consider $f_{\text{Alice}}$ and $f_{\text{Bob}}$ as two players, Alice and Bob, that alternate exchanging a bounded number of bits in order to decide if the concatenation of their respective inputs is in $L$. To show that $L$ is in \text{MSO}[\leq,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}}], the protocol between Alice and Bob should end in a constant number of rounds. We will then rely on the fact that:

Theorem 4** ([23, Theorem 4.6]).**

\text{MSO}[\leq,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}}]* has the Crane Beach Property.*

VI-B A toy example: \mbox{\text{FO}[<]}\subseteq\mbox{\text{MSO}[\leq,\text{$\mathcal{M}$\kern-0.50003pt\scalebox{0.6}{$\mathcal{ON}$}}]}

We will demonstrate how the communication complexity approach will be used with a toy example. Doing so, the requirements for this protocol to work will be emphasized, and they will be enforced when showing the Crane Beach Property of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] in Section VI-C.

Let us consider the following formula over $A=\{a,b,c\}$:

[TABLE]

depicted as a tree in Figure 1.a. The formula $\varphi$ asserts that the all the letters after the last $a$ are $b$’s. In this example, Alice will receive $u=aa$, and Bob $v=bb$. Naturally, $\varphi$ over words of length $4$ is equivalent to the formula where $\exists x$ is replaced by $\bigvee_{x=0}^{3}$, and $\forall y$ is replaced by $\bigwedge_{y=0}^{3}$; our approach will be to split this rewriting between Alice and Bob.

Consider the variable $x$. To check the validity of the formula over a $u\cdot v$, the variable should range over the positions of both players. In other words, the formula is true if there is a position $x$ of Alice verifying $(\forall y)[\psi]$ or a position $x$ of Bob verifying it—likewise for the universal quantifier. We thus “split” the quantifiers by enforcing the domain to be either Alice’s ($\forall^{\textsf{A}},\exists^{\textsf{A}}$) or Bob’s ($\forall^{\textsf{B}},\exists^{\textsf{B}}$), obtaining Figure 1.b.

Alice will now expand her quantifiers to range over her word; she will thus replace, e.g., $(\forall^{\textsf{A}}y)[\psi]$ by $\bigwedge_{y=0}^{1}\psi$. Crucially, at the leaves of the formula, it is known which variables were quantified by each player, and if they are Alice’s, their values. Consider for instance a leaf where Alice substituted $y$ with a numerical value. The letter predicate $\mathbf{b}(y)$ can thus be replaced by its truth value. More importantly, the predicate $x<y$ can also be evaluated: Either Alice quantified $x$, and it has a numerical value, or she did not, and we know for sure that $x<y$ does not hold, since $x$ will be quantified by Bob. Applied to our example, we obtain the tree of Figure 1.c.

The resulting formulas at the leaves are thus free from the variables quantified by Alice. Moreover, for each internal node of the tree, its children represent subformulas of bounded quantifier depth, and there are thus a finite number of possible nonequivalent subformulas. Once only one subformula per equivalence class is kept, the resulting tree is of bounded depth and each node has a bounded number of children. Hence the size of this tree is bounded by a value that only depends on $\varphi$. Alice can thus communicate this tree to Bob. In our example, simplifying the tree, we obtain the formula:

[TABLE]

Finally, Bob can actually quantify his variables, resulting in a formula with no quantified variable, that he can evaluate, concluding the protocol.

Takeaway. This protocol relies on the fact that predicates that involve variables from both Alice and Bob can be evaluated by Alice alone. This enables Alice to remove “her” variables before sending the partially evaluated tree to Bob, who can quantify the remainder of the variables.

VI-C The case of \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]

Theorem 5**.**

\text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]* has the strong Crane Beach Property.*

Proof.

Let $\varphi$ be a formula over an alphabet $A$ in $\text{FO}[\leq,\mathcal{N}]$, for some finite subset $\mathcal{N}$ of $\mathcal{F}$

$\mathcal{IN}$

, and suppose $\varphi$ expresses a language $L$ that admits a neutral letter $e$. We show that L\in\mbox{\text{MSO}[\leq,\text{$\mathcal{M}$\kern-0.50003pt\scalebox{0.6}{$\mathcal{ON}$}}]} using Lemma 1. This concludes the proof since by Theorem 4, $L$ is a neutral letter regular language in \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}], and it thus belongs to $\text{FO}[\leq]$ (see [10]; this is essentially a consequence of {\text{{Parity}}}\notin{\text{AC{}^{0}}}).

Let us write $u\in A^{*}$ for Alice’s word, and $v$ for Bob’s. Both players will compute a value $N>0$ that depends solely on $\varphi$ and $|u\cdot v|$, and the protocol will then decide whether $u\cdot e^{N}\cdot v\in L$, which is equivalent to $u\cdot v\in L$ by hypothesis. We suppose that a large enough $N$ has been picked for the protocol to work, and delay to the end of the proof its computation.

We will henceforth suppose that $\varphi$ is given in prenex normal form and that all variables are quantified only once:

[TABLE]

with $\psi$ quantifier-free and $Q_{i}\in\{\forall,\exists\}$. We again see formulas as trees with leaves containing quantifier-free formulas.

Rather than splitting the domain $[|u\cdot e^{N}\cdot v|]$ at a precise position, and tasking Alice to quantify over the first half and Bob over the second half, we will rely on a third group, that is “far enough” from both Alice’s and Bob’s words. The core of this proof is to formalize this notion. Let us first introduce the tools that will enable this formalization: one set of definitions, and two facts that will be used later on.

Definition 4**.**

Let $C$ be the set of pairs of integers $(p_{1},p_{2})$ that appear in a same tuple of a relation in $\mathcal{N}$. Define the link graph $G=(\mathbb{N},E)$ as the undirected graph defined by $(p_{1},p_{2})\in E$ iff $p_{1}=p_{2}$ or there are integers $p_{1}^{\prime}\leq\{p_{1},p_{2}\}\leq p_{2}^{\prime}$ such that $(p_{1}^{\prime},p_{2}^{\prime})\in C$. For $p\in\mathbb{N}$, $\mathcal{L}(p)$ (resp. $\mathcal{R}(p)$) is the greatest $q<p$ (resp. smallest $q>p$) which is not a neighbor of $p$ in $G$. Equivalently, $\mathcal{L}(p)$ is the smallest neighbor of $p$ minus 1, and $\mathcal{R}(p)$ is the greatest neighbor of $p$ plus 1.

Note that $\mathcal{L}$ and $\mathcal{R}$ are well defined since each vertex of $G$ has a finite number of neighbors. This directly implies that:

Fact 1**.**

The functions $\mathcal{L}$ and $\mathcal{R}$ are nondecreasing and unbounded. Moreover, for any $p\in\mathbb{N}$, $\mathcal{L}(p)<p<\mathcal{R}(p)$.

Writing $\mathcal{R}^{n}$ for the function $\mathcal{R}$ composed $n$ times with itself, and similarly for $\mathcal{L}$, we have:

Fact 2**.**

For any position $p$ and $n>m\geq 0$:

• •

$\mathcal{L}^{m}(\mathcal{R}^{n}(p))\geq\mathcal{R}(p)$;

• •

$\mathcal{R}^{m}(\mathcal{L}^{n}(p))\leq\mathcal{L}(p)$.

Proof.

This is easily shown by induction; we prove the first item, the second being similar. For $n=1$, this is clear. Let $n>1$. If $m=0$, this is immediate from Fact 1, let thus $m>0$. We have that:

[TABLE]

with $p^{\prime}=\mathcal{R}(p)$. By induction hypothesis and the fact that $\mathcal{L}$ is nondecreasing, it holds that:

[TABLE]

Let $p^{\prime\prime}=\mathcal{R}(p^{\prime})$. By definition of $\mathcal{L}$, $(q+1,p^{\prime\prime})$ is an edge in $G$. Now by definition of $G$, if $q<p^{\prime}$, then $(p^{\prime},p^{\prime\prime})$ should also be an edge in $G$, which contradicts the definition of $p^{\prime\prime}$. Hence $q\geq p^{\prime}$, showing the property. ∎

Let us now suppose we have two large positions $|u|\ll\ell_{0}\ll r_{0}\ll|v|$, the requirements on which will be made clear shortly. Let us deem a position $p$ to be Alicic if $p\leq\ell_{0}$, Bobic if $p\geq r_{0}$, and Neutral otherwise; we call this the type of the position. We wish to ensure that two positions of two different types cannot be linked in $G$, so that they cannot appear in a tuple of a predicate in $\mathcal{N}$. This surely is not the case if the typing of positions does not reflect previously typed positions, e.g., $\ell_{0}-1$ is Alicic, but $\ell_{0}$ is Neutral, and their distance may not be large enough to ensure that they do not form an edge in $G$. Thus the boundaries of the zones, $\ell_{0}$ and $r_{0}$, will be moving with each new typing. Formally, let $T=\{{\text{Alice}},{\text{Neutral}},{\text{Bob}}\}$ be an alphabet, and define the function $\mathsf{bounds}\colon T^{\leq k}\to[|u\cdot e^{N}\cdot v|]^{2}$ by:

[TABLE]

Assumption**.**

We henceforth assume that if $(\ell,r)=\mathsf{bounds}(h)$ for some word $h\in T^{\leq k}$, then $|u|<\ell<r<|u|+N$. This will have to be guaranteed by carefully picking $N$, $\ell_{0}$ and $r_{0}$.

The type of a position $p$ under type history $t_{1}t_{2}\cdots t_{i}\in T^{*}$ is computed by first taking $(\ell,r)=\mathsf{bounds}(t_{1}t_{2}\cdots t_{i})$, and reasoning as before: it is Alicic if $p\leq\ell$, Bobic if $p\geq r$, and Neutral otherwise. This is well defined since $\ell<r$ by our Assumption. The crucial property here is as follows:

Fact 3**.**

Let $p_{1},p_{2},\ldots,p_{k}$ be positions, and inductively define the type $t_{i}$ of $p_{i}$ as its type under type history $t_{1}t_{2}\cdots t_{i-1}$.

Two positions with different types do not form an edge in $G$; 2. 2.

All Alicic positions are strictly smaller than the Neutral ones, which are strictly smaller than the Bobic ones; 3. 3.

All Neutral positions are labeled with the neutral letter.

Proof.

(Points 1 and 2.) Suppose $p_{i}$ is Alicic and $p_{j}$ is Neutral, with $i<j$. Let $(\ell,r)=\mathsf{bounds}(t_{1}t_{2}\cdots t_{i-1})$, we thus have that $p_{i}$ is maximally $\ell$. Let $(\ell^{\prime},r^{\prime})=\mathsf{bounds}(t_{1}t_{2}\cdots t_{j-1})$, then $p_{j}$ is minimally $\ell^{\prime}+1$. By definition, once the types of $p_{1},p_{2},\ldots,p_{i}$ are fixed, the smallest $\ell^{\prime}$ that can be obtained with the types $t_{>i}$ is by having all positions $p_{t}$, with $i<t<j$, Neutral. In that case, an easy computation shows that $\ell^{\prime}$ would be:

[TABLE]

That is, $\mathcal{L}$ is composed with itself $m$ times with:

[TABLE]

Hence $\ell^{\prime}$ is at most $\mathcal{L}^{m}(\mathcal{R}^{n}(\ell))$ with $m<n$, and by Fact 2, $\ell^{\prime}\geq\mathcal{R}(\ell)$. Hence $(p_{i},p_{j})$ is not an edge in $G$, and $p_{i}<p_{j}$.

The other cases are similar. For instance, if $p_{i}$ is Neutral and $p_{j}$ Bobic, with $i<j$, then, with the same notation as above, $\ell^{\prime}$ can be at most $\mathcal{L}^{m}(\mathcal{R}^{n}(\ell))$, and by Fact 2, $\ell^{\prime}\geq\mathcal{L}(\ell)$.

(Point 3.) This is a direct consequence of the Assumption. Consider $(\ell,r)=\mathsf{bounds}({\text{Neutral}}^{k})$; this provides the minimal $\ell$ and maximal $r$ between which a position can be labeled Neutral. By the Assumption, $|u|<\ell<r<|u|+N$, hence a Neutral position has a neutral letter.

∎

We are now ready to present the protocol. First, we rewrite quantifiers using Alicic/Neutral/Bobic annotated quantifiers:

• •

$(\forall x)[\rho]\leadsto(\forall^{\textsf{A}}x)[\rho]\;\land\;(\forall^{\textsf{N}}x)[\rho]\;\land\;(\forall^{\textsf{B}}x)[\rho]$,

• •

$(\exists x)[\rho]\leadsto(\exists^{\textsf{A}}x)[\rho]\;\lor\;(\forall^{\textsf{N}}x)[\rho]\;\lor\;(\exists^{\textsf{B}}x)[\rho]$.

Let us further equip each node with the type history of the variables quantified before it; that is, each node holds a string $t_{1}t_{2}\cdots t_{n}\in T^{\leq k}$ where $t_{i}$ is the annotation of the $i$-th quantifier from the root to the node, excluding the node itself.

Now if we were given the entire word $u\cdot e^{N}\cdot v$, a way to evaluate the formula that respects the semantic of “Alicic”, “Neutral”, and “Bobic” is as follows:

This is precisely the algorithm that Alice and Bob will execute. First, Alice will quantify her variables according to the $\mathsf{bounds}$ of the type history of each node, as in Algorithm 1. At the leaves, she will thus obtain the formula $\psi$, and have a set of quantified Alicic variables. She can then evaluate $\psi$ partially: if an atomic formula only relies on Alicic variables, she can compute its value. If an atomic formula uses a mix of Alicic and non-Alicic variables, then she can also evaluate it: if the formula is a numerical predicate, then by Fact 3.1, it will be valued false; if the formula is of the form $x<y$, then it is true iff $x$ is Alicic, by Fact 3.2. Alice now simplifies her tree: logically equivalent leaves with the same parent are merged, and inductively, each internal node keeps only a single occurrence per formula appearing as a child. We remark that the semantic of the tree is preserved. This results in a tree whose size depends solely on $\varphi$, and the values of $N$, $\ell_{0}$, and $r_{0}$, and Alice can thus send it to Bob.

Bob will now expand the remaining quantifiers (Neutral and Bobic), respecting the bounds of the type history, as in Algorithm 1. He can then evaluate all the leaves, since, by Fact 3.3, the only letter predicate true of a Neutral position is that of the neutral letter. This concludes the protocol, which clearly produces the same result as Algorithm 1.

What are $N$, $\ell_{0}$, $r_{0}$? We check that Alice and Bob can agree on these values without communication. The requirements were made explicit in our Assumption. The values computed by the function $\mathsf{bounds}$ are obtained by applying $\mathcal{L}$ and $\mathcal{R}$ on $\ell_{0}$ and $r_{0}$ at most $n=\sum_{i=0}^{k-1}2^{i}$ times. From Fact 1, it is clear that any $(\ell,r)=\mathsf{bounds}(h)$, for $h\in T^{\leq k}$, verifies:

• •

$\ell_{\text{min}}=\mathcal{L}^{n}(\ell_{0})\leq\ell\leq\mathcal{R}^{n}(\ell_{0})=\ell_{\text{max}}$;

• •

$r_{\text{min}}=\mathcal{L}^{n}(r_{0})\leq r\leq\mathcal{R}^{n}(r_{0})=r_{\text{max}}$.

Hence we pick $\ell_{0}=\mathcal{R}^{n+1}(|u|)$, ensuring, by Fact 2, that $\ell_{\text{min}}>|u|$. Next, we pick $r_{0}$ to be $\mathcal{R}^{n+1}(\ell_{\text{max}})$, ensuring that $r_{\text{min}}>\ell_{\text{max}}$ by the same Fact 2. Finally, we pick $N=\mathcal{R}^{n+1}(r_{0})$, ensuring, by Fact 1, that $N>r_{\text{max}}$, so that in particular, $r_{\text{max}}<|u|+N$. We then indeed obtain that $|u|<\ell<r<|u|+N$, as required. Note that these computations depend solely on $\varphi$ and the lengths of $u$ and $v$. ∎

Remark*.*

It should be noted that the crux of this proof is that a relation $R(x,y)$ with $x$ Alicic and $y$ Neutral or Bobic can be readily evaluated by Alice. If $R$ were monadic, then it could not mix two positions of different types, hence Alice could still remove all of her variables at the end of her evaluation. The rest of the protocol will be similar, with Bob quantifying the remaining positions. This shows that \text{FO}[\leq,\text{\mathcal{M}\kern-0.50003pt\scalebox{0.6}{\mathcal{ON}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] also has the Crane Beach Property.

VII On counting

A compelling notion of computational power, for a logic, is the extent to which it is able to precisely evaluate the number of positions that verify a formula. This is formalized with the following standard definition:

Definition 5**.**

For a nondecreasing function $f(n)\leq n$, a logic is said to count up to $f(n)$ if there is a formula $\varphi(c)$ in this logic such that for all $n$ and $w\in\{0,1\}^{n}$:

[TABLE]

It is known from [10] that if a logic can count up to $\log(\log(\cdots(\log n)))$, for some number of iterations of $\log$, then the logic does not have the Crane Beach Property. It has also been conjectured [10, 16] that a logic has the Crane Beach Property iff it cannot count beyond a constant. It is not known whether there exists a set of predicates $\mathcal{N}$ such that $\text{FO}[\mathcal{N}]$ can count beyond a constant but not up to $\log n$.

We define a much weaker ability:

Definition 6**.**

For a nondecreasing function $f(n)\leq n$, a logic is said to sum through $f(n)$ if there is a formula $\varphi(a,b,c)$ in this logic such that for all $n$ and $w\in\{0,1\}^{n}$:

[TABLE]

This is in general even weaker than being able to sum “up to” $f(n)$, that is, having a formula expressing that $a=b+c$ and $c\leq f(n)$. Naturally, counting and summing are related:

Proposition 3**.**

Let $\mathcal{N}$ be a set of unvaried numerical predicates. If $\text{FO}[\leq,\mathcal{N}]$ can count up to $f(n)$, it can sum through $f(n)$.

Proof.

Letting $\varphi(c)$ be the formula that counts up to $f(n)$, we modify it into $\varphi^{\prime}(a,b,c)$ by changing the letter predicates to consider that there is a $1$ in position $p$ iff $b\leq p<a$. This expresses that $a=b+c$ provided that $c\leq f(n)$.

Next, the graph $F$ of $f$ is obtained as follows. First, modify $\varphi(c)$ into $\varphi^{\prime}(c,c^{\prime})$, by restricting all quantifications to $c$ and replacing the letter predicates to have $1$’s in all positions below $c^{\prime}$. Second, $(c,c^{\prime})\in F$ iff $c^{\prime}$ is maximal among those that verify $\varphi^{\prime}(c,c^{\prime})$. This relies on the fact that $\mathcal{N}$ consists solely of unvaried predicates.

The logic can then sum through $f(n)$ by:

[TABLE]

Remark*.*

Proposition 3 depends crucially on the fact that the predicates are unvaried to show that the graph of the summing function is expressible. Writing $\mathcal{S}$ for the set of varied monadic predicates $S=(S_{n})_{n\geq 0}$ with $|S_{n}|=1$ for all $n$, it is easily shown that $\text{FO}[\leq,+,\times,\mathcal{S}]$ can count up to any function $\leq\log n$. However, we conjecture that there are functions whose graphs are not expressible in this logic.

Proposition 4**.**

\text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]* cannot sum through beyond a constant.*

Proof.

Suppose for a contradiction that \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] can sum through a nondecreasing unbounded function $f$ using a formula $\varphi(a,b,c)$. Let Bit be the binary predicate true of $(x,y)$ if the $y$-th bit of $x$ is $1$. We define a translated version as:

[TABLE]

We show that ${\text{Bit}}^{\prime}$ is of finite degree. Let $n\in\mathbb{N}$, and suppose $(n,y)\in{\text{Bit}}^{\prime}$. This implies in particular that $0<y-f(n)<\log n$, hence $n$ appears a finite number of time as $(n,y)$ in ${\text{Bit}}^{\prime}$. Suppose $(x,n)\in{\text{Bit}}^{\prime}$, then $n-f(x)>0$, but for $x$ large enough, $f(x)>n$, hence there can only be a finite number of pairs $(x,n)$ in ${\text{Bit}}^{\prime}$.

Now Bit can be defined in \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] using $\varphi$, since ${\text{Bit}}(x,y)$ holds iff $(\exists z)[\varphi(z,y,x)\land{\text{Bit}}^{\prime}(x,z)]$, a contradiction concluding the proof. ∎

Corollary 1**.**

\text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]* cannot count beyond a constant.*

VIII Conclusion

We showed that \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] is one simple predicate away from expressing all of \text{FO}[\text{\mathcal{A}\kern-0.20004pt\scalebox{0.6}{\mathcal{RB}}}], and that it exhibits the Crane Beach Property. This logic is thus really on the brink of a crevice on the Crane Beach, and exemplifies a diverse set of behaviors that fit the intuition that neutral letters should render numerical predicates essentially useless. We emphasize some future research directions:

• •

As a consequence of our results, one can show that a nonregular neutral letter language $L$ is not in AC0 as follows. Assume L\in{\text{AC{}^{0}}} for a contradiction, and let \varphi\in\mbox{\text{FO}[\leq,{\text{MSB${}_{0}$}},\text{$\mathcal{F}$\kern-1.00006pt\scalebox{0.6}{$\mathcal{IN}$}}]} be a formula expressing it. Suppose that one can show that $\varphi$ can be rewritten without the predicate MSB0, then L\in\mbox{\text{FO}[\leq,\text{$\mathcal{F}$\kern-1.00006pt\scalebox{0.6}{$\mathcal{IN}$}}]}, and thus $L$ is regular, a contradiction. We hope to be able to apply this strategy in the future.

• •

As noted in [14] and [10] and studied in particular in [13], the interest in circuit complexity calls for the study of logics with more sophisticated quantifiers, notably modular quantifiers and, more generally, monoidal quantifiers. Hence the natural question here is whether \text{FO+MOD}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] has the Crane Beach Property.

• •

As asked in [10], can we dispense from our implicit reliance on the lower bound {\text{{Parity}}}\notin{\text{AC{}^{0}}}? In the cases of [10], and as noted by the authors, this would be very difficult, as their results imply the lower bound. Here, the strong Crane Beach Property for \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] does not directly imply the lower bound. To show that {\text{{Parity}}}\notin{\text{AC{}^{0}}}, one could additionally prove that all the regular, neutral letter languages of \text{FO}[\leq,{\text{MSB{}_{0}}},\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] are in \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]—we know that this statement holds, but only thanks to {\text{{Parity}}}\notin{\text{AC{}^{0}}}.

• •

Are we really on the brink of falling off the Crane Beach? That is, are there unvaried predicates that cannot be expressed in \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}] but can still be added to the logic while preserving the Crane Beach Property? We noted that all varied monadic predicates can be added safely, but already very simple predicates falsify the Crane Beach Property. For instance, with $F$ the graph of the 2-adic valuation, $\text{FO}[\leq,F]$ is as expressive as $\text{FO}[\leq,+,\times]$ (see [24, Theorem 3]), which does not have the Crane Beach Property [10].

• •

Numerical predicates correspond in a precise sense [11] to the computing power allowed to construct circuit families for a language. Is there a natural way to present \text{FO}[\leq,\text{\mathcal{F}\kern-1.00006pt\scalebox{0.6}{\mathcal{IN}}}]-uniform circuits?

Acknowledgment

The authors would like to thank Thomas Colcombet, Arnaud Durand, Andreas Krebs, and Pierre McKenzie for enlightening discussions, and Michael Blondin for his careful proofreading.