A Crevice on the Crane Beach: Finite-Degree Predicates Michaël Cadilhac Charles Paperman WSI, Universität Tübingen WSI, Universität Tübingen Email: michael@cadilhac.name Email: charles.paperman@gmail.com Abstract—First-order logic (FO) over words is shown to expressible in FO[ARB], where ARB denotes all possible be equiexpressive with FO equipped with a restricted set of numerical predicates (expressing numerical properties of the 7 numericalpredicates,namelytheorder,abinarypredicateMSB0, positions in a word). Further, as all regular neutral letter 01 TanhdeCthreanfienBitee-adcehgrPereopperretdyic(aCtBesP:)F,iOnt[rAodRuB]ce=dFmOor[≤et,hManSBa0d,eFcaINde]. languages of FO[ARB] are star-free [10], i.e., in FO[≤], the 2 ago, is true of a logic if all the expressible languages admitting Crane Beach Conjecture asked: n a neutral letter are regular. Although it is known that FO[ARB] Are all neutral letter languages of FO[ARB] in FO[≤]? does not have the CBP, it is shown here that the (strong form Notethatthisechoestheaboveintuitiononuniformity,since a J ofthe)CBPholdsforbothFO[≤,FIN]andFO[≤,MSB0].Thus the numerical predicates correspond precisely to the allowed 2 sFtOill[≤ex,pFrIeNss]aexwhiidbeitsvaariefotyrmofolafnlgoucaalgiteys,awnhdilethbeeiCngBPo,naensdimcpalne power to compute the circuit for a given input length [11]. The 1 predicate away from the expressive power of FO[ARB]. The intuition on the logic side is even more compelling: if a letter counting ability of FO[≤,FIN] is studied as an application. can be introduced anywhere without impacting membership, ] O thentheonlymeaningfulrelationthatcanrelatepositionsisthe I. INTRODUCTION linear order. However, first-order logic can “count” up to logn L Ajtai [1] and Furst, Saxe, and Sipser [2] showed some 30 (see, e.g., [12]), meaning that even within a word with neutral . s yearsagothatParity,thelanguageofwordsover{0,1}having letters, FO[ARB] can assert some property on the number of c an even number of 1, is not computable by families of shallow nonneutral letters. This is, in essence, why nonregular neutral [ circuits, namely AC0 circuits. Since then, a wealth of precise letter languages can be expressed in FO[ARB]. 3 expressiveness properties of AC0 has been derived from this In the recent years, a great deal of efforts was put into v sole result [3], [4]. Naturally aiming at a better understanding studying the Crane Beach Property in different logics, i.e., 3 7 ofthecorereasonsbehindthislowerbound,acontinuouseffort whether the definable neutral letter languages are regular. 6 has been made to provide alternative proofs of Parity∈/ AC0. Krebs and Sreejith [13], building on the work of Roy and 2 However, this has been a rather fruitless endeavor, with the Straubing [14], show that all first-order logics with monoidal 0 notable exception of the early works of Razborov [5] and quantifiers and + as the sole numerical predicate have the . 1 Smolenski [6] that develop a less combinatorial approach with Crane Beach Property. Lautemann et al. [15] show Crane 0 an algebraic flavor. For instance, Koucký et al. [7] foray into Beach Properties for classes of bounded-width branching pro- 7 descriptive complexity and use model-theoretic tools to obtain grams, with an algebraic approach relying on communication 1 Parity ∈/ AC0, but assert that “contrary to [their] original complexity. Some expressiveness results were also derived : v hope, [their] Ehrenfeucht-Fraïssé game arguments are not from Crane Beach Properties, for instance Lee [16] shows that i X simpler than classical lower bounds.” More recent promising FO[+] is strictly included in FO[≤,×] by proving that only approaches, especially the topological ones of [8], [9], have the former has the Crane Beach Property. Notably, all these r a yet to yield strong lower bounds. logics are quite far from full FO[ARB], and in that sense, fail A different take originated from a conjecture of Lautemann to identify the part of the arbitrary numerical predicates that and Thérien, investigated by Barrington et al. [10]: the Crane fit the intuition that they are rendered useless by the presence Beach Conjecture. They noticed that the letter 0 acts as a of a neutral letter. neutral letter in Parity, i.e., 0 can be added or removed from In the present paper, we identify a large class of predicates, any word without affecting its membership to the language. If the finite-degree predicates, and a predicate MSB such that 0 a circuit family recognizes a language with a neutral letter, it any numerical predicate can be first-order defined using them seems convincing that the circuits for two given input sizes and the order; in symbols, FO[≤,MSB0,FIN] = FO[ARB]. should look very similar, that is: the circuit family must be We show that, strikingly, both FO[≤,MSB0] and FO[≤,FIN] highly uniform. It was thus conjectured that all neutral letter have the Crane Beach Property, this latter statement being our languages in AC0 were regular, and this was disproved in [10]. main result. Hence showing that some nonregular neutral letter This however sparked an interest in the study of neutral languageisnotexpressibleinFO[ARB]couldbedonebyshow- letter languages, in particular from the descriptive complexity ing that MSB0 may be removed from any FO[≤,MSB0,FIN] view. Indeed, AC0 circuits recognize precisely the languages formula expressing it. TheprooffortheCraneBeachPropertyofFO[≤,FIN]relies the length n of u is denoted |u|. We write ε for the empty on a communication complexity argument different from that word and A≤k for words of length ≤k. of [15]. It is also unrelated to the database collapse techniques B. Logic on words of[10](succinctlyput,nologicwiththeCraneBeachProperty For an alphabet A, let σ be the vocabulary {a|a∈A} of has the so-called independence property, i.e., can encode A unary letter predicates. A (finite) word u=u u ···u ∈ arbitrary large sets). We will show that in fact FO[≤,FIN] 0 1 n−1 A∗ is naturally associated with the structure over σ with doeshavetheindependenceproperty.Thisprovides,tothebest A universe [n] and with a interpreted as the set of positions i of our knowledge, the first example of a logic that exhibits such that u =a, for any a∈A. A numerical predicate is a both the independence and the Crane Beach properties. i k-aryrelationsymboltogetherwithaninterpretationin[n]k for TheaforementionedcountingpropertyofFO[ARB]ledtothe each possible universe size n. Given a formula ϕ that relies on conjecture [10], [16] that a logic has the Crane Beach Property some numerical predicates and a word u, we write u|=ϕ to if and only if it cannot count beyond a constant. To the best mean that ϕ is true of the σ -structure for u augmented with of our knowledge, neither of the directions is known; we show A the interpretations of the numerical predicates for the universe however that FO[≤,FIN] can only count up to a constant, by of size |u|. A formula ϕ thus defines or expresses the language showing that it cannot even express very restricted forms of {u∈A∗ |u|=ϕ}. the addition. This adds evidence to the “if” direction of the conjecture. C. Classes of formulas Structure of the paper. In Section II, we introduce the WeletARBbethesetofallnumericalpredicates.Givenaset required notions, although some familiarity with language N ⊆ARB, we write FO[N] for the set of first-order formulas theory and logic on words is assumed (see, e.g., [4]). In built using the symbols from N ∪σ , for any alphabet A. A Section III, we show that FO[≤,MSB0,FIN] = FO[ARB]. In Similarly, MSO[N] denotes monadic second-order formulas Section IV, we present a simple proof, relying on a much builtwiththosesymbols.WefurtherdefinethequantifiersMaj harder result from [10], that FO[≤,MSB0] has the Crane and ∃≡, for i∈N, that will only be used in discussions: Beach Property. The failing of the aforementioned collapse i • u|=(Majx)[ϕ(x)] iff there is strict majority of positions technique for FO[≤,FIN] is shown in Section V. We tackle i∈[|u|] such that (cid:104)u,x:=i(cid:105)|=ϕ; tSheectCiornanVeIB,eaafctherPtrhoepenretyceosfsaFrOy[≤to,oFlsINh]a,voeurbemenainderveseuloltp,eidn. • u |= (∃≡i )[ϕ(x)] iff the number of positions i ∈ [|u|] verifying (cid:104)u,x:=i(cid:105)|=ϕ is a multiple of i. Finally, in Section VII, we focus on the counting inabilities of We will write MAJ[N] and FO+MAJ[N] with the obvious FO[≤,FIN]. meanings. Further, FO+MOD[N] allows all the quantifiers ∃≡ Previous works. Finite-degree predicates were introduced i in FO[N] formulas. by the second author in [17], in the context of two-variable logics. Therein, it is shown that the two-variable fragment of D. On numerical predicates FO[≤,FIN] has the Crane Beach Property, and, even stronger, The most ubiquitous numerical predicate here will be the that the neutral letter languages expressible with k quantifier binary order predicate ≤. The predicate that zeroes the most alternations can be expressed without the finite-degree pred- significant bit (MSB) of a number will also be important: icates with the same amount of quantifier alternations. The (m,n)∈MSB iffn=m−2(cid:98)logm(cid:99).Notethatbothpredicates 0 techniques used in [17] are specific to two-variable logics, do not depend on the universe size, and we single out this relying heavily on the fact that each quantification depends on concept: a single previously quantified variable. We thus stress that the communication complexity argument developed in Section VI Definition 1. A k-ary numerical predicate P is unvaried if is unrelated to [17]. there is a set E ⊆ Nk such that the interpretation of P on ThefactthattwosetsofpredicatescanbothverifytheCrane universes of size n is E ∩[n]k. In this case, we identify P Beach Property while their union does not has already been with the set E. It is varied otherwise.1 We write ARBu for the witnessed in [10]. Indeed, letting MON be the set of monoidal set of unvaried numerical predicates. numerical predicates, the Property holds for both FO[≤,+] Naturally, any varied predicate can be converted to an and FO[≤,MON] but fails for FO[≤,+,MON], although this unvaried one by turning the universe length into an argument latter class is less expressive than FO[ARB] (this can be shown andquantifyingthemaximumposition;thisimpliesinparticular using the same proof as [7, Proposition 5]). that FO[ARB] = FO[ARBu]. This is however not entirely innocuous, as will be discussed in Section VII. II. PRELIMINARIES We will rely on the following class of unvaried predicates, A. Generalities generalizing a definition of [17] (see also the older notion of We write N = {0,1,2,...} for the set of nonnegative “finite formula” [18]): numbers.Forn∈N,welet[n]={0,1,...,n−1}.Afunction f: N→N is nondecreasing if m>n implies f(m)≥f(n). 1Therelevanceofthisconcepthasbeennotedinpreviousworks(e.g., [10]), butwasleftunnamed.Thesecondauthorusedin[17]theterms(non)uniform, An alphabet A is a finite set of letters (symbols), and we an unfortunate coinage in this context. We prefer here the less conflicting write A∗ for the set of finite words. For u = u0u1···un−1, terms(un)varied. 2 Definition2. AnunvariedpredicateP ⊆Nk isoffinitedegree2 The work zone has two salient properties: 1. Checking that if for all n∈N, n appears in a finite number of tuples in P. a number k ∈[(cid:96)] belongs to it amounts to checking that k has We write FIN for the class of such predicates. exactly one greater power of two; in particular, two work-zone positions share the same MSB; 2. Any number in [(cid:96)] outside Note that this does not imply that there is a N that bounds the work zone can be obtained by replacing the MSB of a the number of appearance for all n’s. Some examples: number in the work zone with some other bits (0, 10, and 11, • MSB0 is not a finite-degree predicate, as, e.g., (2n,0)∈ for the first, third, and fourth zone, respectively); we call this MSB for any n, hence 0 appears infinitely often; 0 a translation to a zone, e.g., in our example above, 10101 is • Any unvaried monadic numerical predicate is of finite the translation of 01101 to the third zone. degree, this implies in particular that any language over a More formally, we can define a formula work(x) which is unary alphabet is expressed by a FO[≤,FIN] formula; true iff x belongs to the work zone, by expressing that there • The graph of any nondecreasing unbounded function is exactly one power of two strictly greater than x, using the f: N→N defines a finite-degree predicate, since f−1(n) monadic predicate true on powers of two. Moreover, we can is a finite set for all n; define formulas trans(i)(x,y), 1≤i≤4, which are true if x • Theorder,sum,andmultiplicationarenotoffinitedegree; is in the work zone and y is its translation to the i-th zone; let • One can usually “translate” unvaried predicates to make us treat the case i=3, the others being similar. The formula themfinitedegree;forinstance,thepredicatetrueof(x,y) trans(3)(x,y) is true if y is obtained by replacing the MSB if y−x0,anddefinea =2n+2i for0≤i 0 there are vectors is such that bK =1 iff u·v ∈L. Then L∈MSO[≤,MON]. #– #– # – a ,a ,...,a , each of Nk, for which for any M ⊆ [n], Proof. We adapt the (folklore) proof that L is regular iff such 0 1 n−1 # – there is a vector b ∈N(cid:96) such that:3 functions exist where f and f do not use their second M Alice Bob #– #– #– # – parameter. (cid:104)N,x :=a , y :=b (cid:105)|=ϕ iff i∈M . i M Let p ∈ N. For any u ∈ A∗, let c(u) be the set of pairs The logic FO[N] has the independence property if it contains (a1a2···aK,b1b2···bK−1) such that for all 1 ≤ i ≤ K, such a ϕ. it holds that ai = fAlice(u,|u|+p,b1b2···bi−1). Define the equivalence relation ≡ by letting u ≡ v iff c(u) = c(v); it Intuitively, a logic has the independence property iff it can clearly has a finite number N =N(K) of equivalence classes. encode arbitrary sets. Barrington et al. [10], relying on a deep Moreover, if u≡v and w ∈Ap, then (u,w) and (v,w) define result of Baldwin and Benedikt [21], show that: the same sequences of a ’s and b ’s, by a simple induction. i i Theorem 3 ([10, Corollary 4.13]). If a logic does not have Hence u·w ∈L⇔v·w ∈L. This shows that ≡ refines ∼p, the independence property, then it has the strong Crane Beach implying, by Proposition 2, that L∈MSO[≤,MON]. Property. Weshalladopttheclassicalcommunicationcomplexityview We note that this powerful tool cannot show that the logic here, and consider f and f as two players, Alice and Alice Bob we consider exhibits the Crane Beach Property: Bob, that alternate exchanging a bounded number of bits in order to decide if the concatenation of their respective inputs Proposition 1. FO[≤,FIN] has the independence property. is in L. To show that L is in MSO[≤,MON], the protocol 3Notethatweevaluateaformulaoveraninfinitedomain;thisiswelldefined between Alice and Bob should end in a constant number of inourcasesinceweonlyuseunvariedpredicatesandtheletterpredicatesare rounds. We will then rely on the fact that: irrelevant. 4 ∨ a) b) ∃Ax ∃Bx ∃x ∧ ∧ ∀y ∀Ay ∀By ∀Ay ∀By ψ ψ ψ ψ ψ Quantified by Alice ∨ c) (cid:87) ∃Bx x=0 x=1 ∧ ∧ ∧ (cid:86) (cid:86) ∀By ∀By (cid:86) ∀By y=1 y=0 y=1 y=0 (cid:62)∧ (cid:62)∧ y=1 (cid:62)∧ y=0 a(x)∧ (cid:62)→⊥ ⊥→⊥ (cid:62)→b(y) ⊥→⊥ (cid:62)∧ (cid:62)∧ (cid:62)∧ a(x)∧ ⊥→⊥ (cid:62)→b(y) ⊥→⊥ ψ ⊥→⊥ Fig.1. TheformulaϕasitgetsevaluatedbyAliceandBob. Theorem4([23,Theorem4.6]). MSO[≤,MON]hastheCrane Alicewillnowexpandherquantifierstorangeoverherword; Beach Property. she will thus replace, e.g., (∀Ay)[ψ] by (cid:86)1 ψ. Crucially, at y=0 the leaves of the formula, it is known which variables were B. A toy example: FO[<]⊆MSO[≤,MON] quantified by each player, and if they are Alice’s, their values. We will demonstrate how the communication complexity Consider for instance a leaf where Alice substituted y with a approach will be used with a toy example. Doing so, the numericalvalue.Theletterpredicateb(y)canthusbereplaced requirements for this protocol to work will be emphasized, and by its truth value. More importantly, the predicate x < y they will be enforced when showing the Crane Beach Property can also be evaluated: Either Alice quantified x, and it has a of FO[≤,FIN] in Section VI-C. numerical value, or she did not, and we know for sure that Let us consider the following formula over A={a,b,c}: x1.Ifm=0,thisisimmediatefromFact1,letthusm>0. involve variables from both Alice and Bob can be evaluated We have that: by Alice alone. This enables Alice to remove “her” variables Lm(Rn(p))=L(Lm−1(Rn−1(p(cid:48)))) , before sending the partially evaluated tree to Bob, who can quantify the remainder of the variables. with p(cid:48) =R(p). By induction hypothesis and the fact that L is nondecreasing, it holds that: C. The case of FO[≤,FIN] Theorem 5. FO[≤,FIN]hasthestrongCraneBeachProperty. Lm(Rn(p))≥L(R(p(cid:48)))=q . Proof. Let ϕ be a formula over an alphabet A in FO[≤,N], Let p(cid:48)(cid:48) = R(p(cid:48)). By definition of L, (q+1,p(cid:48)(cid:48)) is an edge for some finite subset N of FIN, and suppose ϕ expresses in G. Now by definition of G, if q 0 that depends solely We wish to ensure that two positions of two different types on ϕ and |u·v|, and the protocol will then decide whether cannot be linked in G, so that they cannot appear in a tuple of u·eN·v ∈L,whichisequivalenttou·v ∈Lbyhypothesis.We a predicate in N. This surely is not the case if the typing of supposethatalargeenoughN hasbeenpickedfortheprotocol positionsdoesnotreflectpreviouslytypedpositions,e.g.,(cid:96) −1 to work, and delay to the end of the proof its computation. 0 is Alicic, but (cid:96) is Neutral, and their distance may not be large Wewillhenceforthsupposethatϕisgiveninprenexnormal 0 enough to ensure that they do not form an edge in G. Thus form and that all variables are quantified only once: the boundaries of the zones, (cid:96) and r , will be moving with 0 0 ϕ≡(Q x )(Q x )···(Q x )[ψ] , eachnewtyping.Formally,letT ={Alice,Neutral,Bob}bean 1 1 2 2 k k alphabet,anddefinethefunctionbounds: T≤k →[|u·eN·v|]2 with ψ quantifier-free and Q ∈{∀,∃}. We again see formulas i by: as trees with leaves containing quantifier-free formulas. Rather than splitting the domain [|u·eN ·v|] at a precise bounds(ε)=(l ,r ) 0 0 position, and tasking Alice to quantify over the first half and bounds(t t ···t )= 1 2 i Bob over the second half, we will rely on a third group, that is  (Rn((cid:96)),r) if t =Alice “far enough” from both Alice’s and Bob’s words. The core of  i this proof is to formalize this notion. Let us first introduce the (Ln((cid:96)),Rn(r)) if ti =Neutral tools that will enable this formalization: one set of definitions, ((cid:96),Ln(r)) if t =Bob i and two facts that will be used later on. with ((cid:96),r)=bounds(t t ···t ) and n=2k−i. 1 2 i−1 Definition 4. LetC bethesetofpairsofintegers(p ,p )that 1 2 Assumption. Wehenceforthassumethatif((cid:96),r)=bounds(h) appearinasametupleofarelationinN.Definethelinkgraph for some word h ∈ T≤k, then |u| < (cid:96) < r < |u|+N. This G=(N,E) as the undirected graph defined by (p ,p )∈E 1 2 will have to be guaranteed by carefully picking N, (cid:96) and r . iff p =p or there are integers p(cid:48) ≤{p ,p }≤p(cid:48) such that 0 0 1 2 1 1 2 2 (p(cid:48),p(cid:48)) ∈ C. For p ∈ N, L(p) (resp. R(p)) is the greatest The type of a position p under type history t t ···t ∈T∗ 1 2 1 2 i q

p) which is not a neighbor of p in is computed by first taking ((cid:96),r) = bounds(t t ···t ), and 1 2 i G. Equivalently, L(p) is the smallest neighbor of p minus 1, reasoning as before: it is Alicic if p≤(cid:96), Bobic if p≥r, and and R(p) is the greatest neighbor of p plus 1. Neutral otherwise. This is well defined since (cid:96) < r by our Assumption. The crucial property here is as follows: Note that L and R are well defined since each vertex of G has a finite number of neighbors. This directly implies that: Fact 3. Let p ,p ,...,p be positions, and inductively define 1 2 k the type t of p as its type under type history t t ···t . Fact 1. The functions L and R are nondecreasing and i i 1 2 i−1 unbounded. Moreover, for any p∈N, L(p)m≥0: • Lm(Rn(p))≥R(p); Proof. (Points 1 and 2.) Suppose pi is Alicic and pj is • Rm(Ln(p))≤L(p). Neutral, with i < j. Let ((cid:96),r) = bounds(t1t2···ti−1), 6 we thus have that p is maximally (cid:96). Let ((cid:96)(cid:48),r(cid:48)) = This is precisely the algorithm that Alice and Bob will i bounds(t t ···t ),thenp isminimally(cid:96)(cid:48)+1.Bydefinition, execute. First, Alice will quantify her variables according to 1 2 j−1 j once the types of p ,p ,...,p are fixed, the smallest (cid:96)(cid:48) that the boundsof thetype historyof eachnode, asin Algorithm1. 1 2 i can be obtained with the types t is by having all positions At the leaves, she will thus obtain the formula ψ, and have >i p , with i |u|. Next, we pick r to be Rn+1((cid:96) ), ensuring Now if we were given the entire word u·eN ·v, a way min 0 max that r > (cid:96) by the same Fact 2. Finally, we pick N = min max to evaluate the formula that respects the semantic of “Alicic”, Rn+1(r ), ensuring, by Fact 1, that N > r , so that in 0 max “Neutral”, and “Bobic” is as follows: particular,r <|u|+N.Wethenindeedobtainthat|u|<(cid:96)< max r <|u|+N, as required. Note that these computations depend Algorithm 1 Formula Evaluation solely on ϕ and the lengths of u and v. of Theorem 5. 1: foreach quantifier node ∀Ax or ∃Ax do 2: ((cid:96),r):=bounds(type history at node) 3: if node is ∀Ax then Remark. It should be noted that the crux of this proof is that a 4: Replace node with (cid:86)(cid:96) relation R(x,y) with x Alicic and y Neutral or Bobic can be x=0 readily evaluated by Alice. If R were monadic, then it could (cid:87) 5: Similarly with ∃ becoming notmixtwopositionsofdifferenttypes,henceAlicecouldstill 6: end remove all of her variables at the end of her evaluation. The 7: Evaluate the part of the leaves than can be evaluated rest of the protocol will be similar, with Bob quantifying the 8: foreach quantifier node do remaining positions. This shows that FO[≤,MON,FIN] also 9: ((cid:96),r):=bounds(type history at node) has the Crane Beach Property. 10: if node is ∀Nx then 11: Replace node with (cid:86)r−1 x=(cid:96)+1 12: else if node is ∀Bx then VII. ONCOUNTING 13: Replace node with (cid:86)|ueNv| x=r (cid:87) A compelling notion of computational power, for a logic, is 14: Similarly with ∃ becoming the extent to which it is able to precisely evaluate the number 15: end of positions that verify a formula. This is formalized with the 16: Finish evaluating the tree following standard definition: 7 Definition 5. For a nondecreasing function f(n)≤n, a logic logn, hence n appears a finite number of time as (n,y) in is said to count up to f(n) if there is a formula ϕ(c) in this Bit(cid:48). Suppose (x,n)∈Bit(cid:48), then n−f(x)>0, but for x large logic such that for all n and w ∈{0,1}n: enough, f(x)>n, hence there can only be a finite number of pairs (x,n) in Bit(cid:48). w |=ϕ(c) ⇔ c≤f(n)∧c=number of 1’s in w . Now Bit can be defined in FO[≤,FIN] using ϕ, since It is known from [10] that if a logic can count up to Bit(x,y) holds iff (∃z)[ϕ(z,y,x)∧Bit(cid:48)(x,z)], a contradiction log(log(···(logn))), for some number of iterations of log, concluding the proof. then the logic does not have the Crane Beach Property. It has Corollary 1. FO[≤,FIN] cannot count beyond a constant. also been conjectured [10], [16] that a logic has the Crane Beach Property iff it cannot count beyond a constant. It is not VIII. CONCLUSION known whether there exists a set of predicates N such that We showed that FO[≤,FIN] is one simple predicate away FO[N] can count beyond a constant but not up to logn. from expressing all of FO[ARB], and that it exhibits the Crane We define a much weaker ability: Beach Property. This logic is thus really on the brink of a Definition 6. For a nondecreasing function f(n)≤n, a logic crevice on the Crane Beach, and exemplifies a diverse set of is said to sum through f(n) if there is a formula ϕ(a,b,c) in behaviors that fit the intuition that neutral letters should render this logic such that for all n and w ∈{0,1}n: numerical predicates essentially useless. We emphasize some future research directions: w |=ϕ(a,b,c) ⇔ a=b+f(c) . • As a consequence of our results, one can show that a This is in general even weaker than being able to sum “up nonregular neutral letter language L is not in AC0 as to” f(n), that is, having a formula expressing that a=b+c follows.AssumeL∈AC0 foracontradiction,andletϕ∈ and c≤f(n). Naturally, counting and summing are related: FO[≤,MSB0,FIN] be a formula expressing it. Suppose that one can show that ϕ can be rewritten without the Proposition3. LetN beasetofunvariednumericalpredicates. If FO[≤,N] can count up to f(n), it can sum through f(n). predicate MSB0, then L ∈ FO[≤,FIN], and thus L is regular, a contradiction. We hope to be able to apply this Proof. Letting ϕ(c) be the formula that counts up to f(n), we strategy in the future. modify it into ϕ(cid:48)(a,b,c) by changing the letter predicates to • As noted in [14] and [10] and studied in particular consider that there is a 1 in position p iff b ≤ p < a. This in [13], the interest in circuit complexity calls for the expresses that a=b+c provided that c≤f(n). study of logics with more sophisticated quantifiers, no- Next, the graph F of f is obtained as follows. First, modify tably modular quantifiers and, more generally, monoidal ϕ(c) into ϕ(cid:48)(c,c(cid:48)), by restricting all quantifications to c and quantifiers. Hence the natural question here is whether replacing the letter predicates to have 1’s in all positions below FO+MOD[≤,FIN] has the Crane Beach Property. c(cid:48). Second, (c,c(cid:48)) ∈ F iff c(cid:48) is maximal among those that • As asked in [10], can we dispense from our implicit verify ϕ(cid:48)(c,c(cid:48)). This relies on the fact that N consists solely reliance on the lower bound Parity∈/ AC0? In the cases of unvaried predicates. of [10], and as noted by the authors, this would be The logic can then sum through f(n) by: very difficult, as their results imply the lower bound. ψ(a,b,c)≡(∃c(cid:48))[F(c,c(cid:48))∧a=b+c(cid:48)] . Here, the strong Crane Beach Property for FO[≤,FIN] does not directly imply the lower bound. To show that Remark. Proposition 3 depends crucially on the fact that the Parity ∈/ AC0, one could additionally prove that all the predicates are unvaried to show that the graph of the summing regular, neutral letter languages of FO[≤,MSB0,FIN] are functionisexpressible.WritingS forthesetofvariedmonadic in FO[≤,FIN]—we know that this statement holds, but predicates S = (S ) with |S | = 1 for all n, it is easily only thanks to Parity∈/ AC0. n n≥0 n shown that FO[≤,+,×,S] can count up to any function ≤ • ArewereallyonthebrinkoffallingofftheCraneBeach? logn. However, we conjecture that there are functions whose That is, are there unvaried predicates that cannot be graphs are not expressible in this logic. expressed in FO[≤,FIN] but can still be added to the logic while preserving the Crane Beach Property? We Proposition 4. FO[≤,FIN] cannot sum through beyond a noted that all varied monadic predicates can be added constant. safely,butalreadyverysimplepredicatesfalsifytheCrane Proof. Suppose for a contradiction that FO[≤,FIN] can sum Beach Property. For instance, with F the graph of the 2- throughanondecreasingunboundedfunctionf usingaformula adic valuation, FO[≤,F] is as expressive as FO[≤,+,×] ϕ(a,b,c). Let Bit be the binary predicate true of (x,y) if the (see [24, Theorem 3]), which does not have the Crane y-th bit of x is 1. We define a translated version as: Beach Property [10]. • Numerical predicates correspond in a precise sense [11] Bit(cid:48) ={(x,y)|(x,y−f(x))∈Bit} . to the computing power allowed to construct circuit We show that Bit(cid:48) is of finite degree. Let n∈N, and suppose families for a language. Is there a natural way to present (n,y) ∈ Bit(cid:48). 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