Verbrugge, Rineke

English

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

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Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

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   University of GroningenZero-one laws with respect to models of provability logic and two Grzegorczyk logicsVerbrugge, RinekeIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Verbrugge, R. (2018). Zero-one laws with respect to models of provability logic and two Grzegorczyk logics.115-120. Paper presented at Advances in Modal Logic, Bern, Switzerland.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 13-11-2019Zero-one laws with respect to models ofprovability logic and two Grzegorczyk logicsRineke Verbrugge 1Institute of Artificial IntelligenceFaculty of Science and EngineeringUniversity of GroningenPO Box 407, 9700 AK Groningen, The NetherlandsKeywords: Provability logic, Grzegorczyk logics, zero-one laws1 IntroductionIn the late 1960s, Glebskii and colleagues proved that first-order logic withoutfunction symbols satisfies a zero-one law: every formula is either almost alwaystrue or almost always false in finite models [6]. More formally, let L be alanguage of first-order logic and let An(L) be the set of all L-models withuniverse {1, . . . , n}. Now let µn(σ) be the fraction of members of An(L) inwhich σ is true:µn(σ) =|M ∈ An(L) : M |= σ || An(L) |Then for every σ ∈ L, limn→∞ µn(σ) = 1 or limn→∞ µn(ϕ) = 0. This was alsoproved later but independently by Fagin [5]; Carnap had already proved thezero-one law for first-order languages with only unary predicate symbols [3].The above zero-one laws and other limit laws have found applications indatabase theory and AI.In this article, we are interested in zero-one laws forsome modal logics that impose structural restrictions on their models; all threelogics that we are interested in are sound and complete with respect to finitepartial orders, with different extra restrictions per logic. The zero-one lawfor first-order logic also holds when restricted to partial orders, both reflexiveand irreflexive ones [4]. The proof uses a surprising combinatorial result byKleitman and Rothschild [9] on which we will also rely for our results.1.1 Kleitman and Rothschild’s result on finite partial ordersKleitman and Rothschild proved that with asymptotic probability 1, finitepartial orders have a very special structure: There are no chains u < v < w < zof more than three objects and the structure can be divided into three levels:1 L.C.Verbrugge@rug.nl2 Zero-one laws with respect to models of provability logic and two Grzegorczyk logics• L1, the set of minimal elements;• L2, the set of elements immediately succeeding elements in L1;• L3, the set of elements immediately succeeding elements in L2.Moreover, in partial orders of size n, the sizes of these sets tend to n4 for bothL1 and L3 while the size of L2 tends ton2 . As n increases, each elementin L1 has as immediate successors asymptotically half of the elements of L2and each element in L2 has as immediate successors asymptotically half of theelements of L3 [9]. Kleitman and Rothschild’s theorem holds both for reflexive(non-strict) and for irreflexive (strict) partial orders.1.2 Zero-one laws for modal logicsLet Φ = {p1, . . . , pk} be a finite set of propositional atoms 2 and let L(Φ) bethe modal language over Φ, inductively defined as the smallest set closed under:(i) If p ∈ Φ, then p ∈ L(Φ).(ii) If A ∈ L(Φ) and B ∈ L(Φ), then also ¬A ∈ L(Φ), 2A ∈ L(Φ), 3(ϕ) ∈L(Φ), (A ∧B) ∈ L(Φ), (A ∨B) ∈ L(Φ), and (A→ B) ∈ L(Φ).Let Mn,Φ be the set of finite Kripke models over Φ with set of worlds W ={1, . . . , n}. We take νn,Φ to be the uniform probability distribution on Mn,Φ.Let νn,Φ(ϕ) be the measure in Mn,Φ of the set of Kripke models in which ϕ isvalid. Halpern and Kapron proved that every formula ϕ in L(Φ) is either validin almost all models or not valid in almost all models [8, Corollary 4.2]:Either limn→∞ νn,Φ(ϕ) = 0 or limn→∞ νn,Φ(ϕ) = 1.By the Kleitman-Rothschild theorem, this modal zero-one law can also berestricted to finite models on reflexive or irreflexive partial orders, so that theexistence of zero-one laws for finite models of provability logic and Grzegorczyklogic immediately follow. However, one would like to prove a stronger resultand axiomatize the set formulas ϕ for which limn→∞ νn,Φ(ϕ) = 1.The result about GL was proved in my 1995 LMPS presentation [12], butthe proof was not published before. The 0-1 laws for Grz and WGrz are new.2 Provability logic and two of its cousinsHere follow brief reminders about provability logic GL, Grzegorczyk logic Grz,and weak Grzegorczyk logic wGrz.2.1 Provability LogicThe most widely used provability logic is called GL after Go¨del and Lo¨b. Asaxioms, it contains all axiom schemes from K and the extra scheme GL:All (instances of) propositional tautologies (A1)(ϕ→ ψ) → (ϕ→ ψ) (A2)(ϕ→ ϕ) → ϕ (GL)2 In the rest of this paper, we take Φ to be finite, although the results can be extended toenumerably infinite Φ by the methods described in [8].Verbrugge 3The rules of inference of GL are modus ponens and necessitation (if GL ` ϕ,then GL ` ϕ). Note that GL ` ϕ → ϕ, as first proved by De Jonghand Sambin [1,13], but that the reflexivity axiom ϕ → ϕ does not follow.Indeed, Segerberg proved in 1971 that provability logic is sound and completewith respect to all finite, transitive, irreflexive frames [11].2.2 Grzegorczyk logicGrzegorczyk Logic Grz, first introduced in [7], has the same axiom schemesand inference rules as GL, except that axiom GL is replaced by Grz:2(2(ϕ→ 2ϕ) → ϕ) → ϕ (Grz)Grz is sound and complete with respect to the class of all finite transitive,reflexive and anti-symmetric frames [1, Chapter 12].2.3 Weak Grzegorczyk logicWeak Grzegorczyk Logic Grz has the same axiom schemes and inference rulesas GL, except that axiom GL is replaced by wGrz, in which 2+ψ := 2ψ ∧ ψ:2+(2(ϕ→ 2ϕ) → ϕ) → ϕ (wGrz)wGrz is sound and complete with respect to the class of all finite transitive,anti-symmetric frames (which need be neither irreflexive nor reflexive) [10].3 Zero-one laws over relevant classes of finite models3.1 GL: 0-1 law in finite irreflexive transitive modelsWe provide an axiomatization for almost sure model validity with respect to therelevant finite models corresponding to GL, namely the irreflexive transitiveones. The axiom system AXΦ,MGL has the same axioms and rules as GL plus:222⊥ (T3)3> → 3A (C1)33> → 3(B ∧3C) (C2)In the axiom schemes C1 and C2, the formulas A, B and C all stand forconsistent conjunctions of literals over Φ. 3 Note that AXΦ,MGL is not a normalmodal logic, because one cannot substitute just any formula for A,B,C. 4Definition 3.1 Define MΦGL = (W,R, V ), the canonical asymptotic Kripkemodel over Φ, with W,R, V as follows (see Fig. 1):W = {bv,mv, uv | v a propositional valuation on Φ};R = {〈bv,mv′〉 | v, v′ propositional valuations on Φ} ∪{〈mv, uv′〉 | v, v′ propositional valuations on Φ} ∪{〈dv, uv′〉 | v, v′ propositional valuations on Φ}; andfor all pi ∈ Φ, the valuation V is defined by Vbv (pi) = 1 iff v(pi) = 1; Vmv (pi) =1 iff v(pi) = 1; Vuv (pi) = 1 iff v(pi) = 1.3 C1 and C2 have been inspired by Carnap’s consistency axiom: 3ϕ for any ϕ that is aconsistent propositional formula [2], and used by Halpern and Kapron [8] for axiomatizingalmost sure model validities for K-models.4 For example, substituting ⊥ for A in C1 would make C1 equivalent to 2⊥.4 Zero-one laws with respect to models of provability logic and two Grzegorczyk logicsbv1p1, p2bv2p1bv3p2bv4mv1p1, p2 mv2p1 mv3 p2 mv4uv1p1, p2uv2p1uv3p2uv4Fig. 1. The canonical asymptotic Kripke model MΦGL = (W,R, V ) for Φ = {p1, p2},defined in Definition 3.1. The accessibility relation is the transitive closure of the onedrawn in the picture. The figure shows only propositional atoms true at each world.The zero-one law for model validity now follows:Theorem 3.2 For every formula ϕ ∈ L(Φ), the following are equivalent:(i) MΦGL |= ϕ;(ii) AXΦ,MGL ` ϕ;(iii) limn→∞ νn,Φ(ϕ) = 1;(iv) limn→∞ νn,Φ(ϕ) 6= 0.3.2 Grz: 0-1 law in finite reflexive transitive anti-symmetricmodelsDefine axiom system AXΦ,MGrz as Grz plus the following axioms:¬(ϕ ∧3(¬ϕ ∧ ψ ∧3(¬ψ ∧ χ ∧3¬χ))) (D3)(ϕ ∧3¬ϕ) → 3A (C3)(ϕ ∧3(¬ϕ ∧ ψ ∧3¬ψ) → 3(B ∧3C) (C4)In the axiom schemes above, ϕ, ψ, χ stand for any formulas in L(Φ), whileA,B and C stand for consistent conjunctions of literals over Φ. 5Definition 3.3 Define the canonical asymptotic Kripke model MΦGrz =(W,R, V ), where:W = {bv,mv, uv | v a propositional valuation on Φ};R = {〈w,w〉 | w ∈W} ∪{〈bv,mv′〉 | v, v′ propositional valuations on Φ} ∪{〈mv, uv′〉 | v, v′ propositional valuations on Φ} ∪{〈dv, uv′〉 | v, v′ propositional valuations on Φ}; andVbv (p) = 1 iff v(p) = 1; Vmv (p) = 1 iff v(p) = 1; Vuv (p) = 1 iff v(p) = 1.5 The axioms D3, C3 and C4 have been inspired by the axioms proposed in [8, Theorem4.16] for the almost sure validities in finite S4 models.Verbrugge 5Note that MΦGrz is just the reflexive closure of MΦGL (Definition 3.1).Theorem 3.4 For every ϕ ∈ L(Φ), the following are equivalent: (i) MΦGrz |=ϕ; (ii) AXΦ,MGrz ` ϕ; (iii) limn→∞ νn,Φ(ϕ) = 1; (iv) limn→∞ νn,Φ(ϕ) 6= 0.3.3 wGRz: 0-1 law in finite transitive anti-symmetric modelsDefine the axiom system AXΦ,MwGrz as wGrz plus axioms D3, C3 and C4.Definition 3.5 The canonical asymptotic Kripke model MΦwGrz is a com-bination of the irreflexive transitive MΦGL and the reflexive transitive anti-symmetric MΦGrz (Def. 3.1 and 3.3), having a reflexive and irreflexive copyof each valuation-related world in each layer; it is transitive and antisymmetricand has direct accessibility from all states in the bottom layer to all states inthe middle layer and all states in the middle layer to all states in the top layer.Theorem 3.6 For every ϕ ∈ L(Φ), the following are equivalent: (i) MΦwGrz |=ϕ; (ii) AXΦ,MwGrz ` ϕ; (iii) limn→∞ νn(ϕ) = 1; (iv) limn→∞ νn(ϕ) 6= 0.ConclusionWe have formulated zero-one laws for provability logic, Grzegorczyk logic andweak Grzegorczyk logic, with respect to model validity. On the way, we haveaxiomatized validity in almost all relevant finite models, leading to three axiomsystems. 6 Many questions are left open for future research, most notably, thoseabout almost sure frame validity.References[1] Boolos, G., “The Logic of Provability,” Cambridge University Press, Cambridge, 1993.[2] Carnap, R., “Meaning and Necessity,” University of Chicago Press, Chicago, IL, 1947.[3] Carnap, R., “Logical Foundations of Probability,” University of Chicago Press, 1950.[4] Compton, K. J., The computational complexity of asymptotic problems I: Partial orders,Information and Computation 78 (1988), pp. 108–123.[5] Fagin, R., Probabilities on finite models 1, Journal of Symbolic Logic 41 (1976), pp. 50–58.[6] Glebskii, Y. V., D. I. Kogan, M. Liogon’kii and V. Talanov, Range and degree ofrealizability of formulas in the restricted predicate calculus, Cybernetics and SystemsAnalysis 5 (1969), pp. 142–154.[7] Grzegorczyk, A., Some relational systems and the associated topological spaces,Fundamenta Mathematicae 60 (1967), pp. 223–231.[8] Halpern, J. Y. and B. Kapron, Zero-one laws for modal logic, Annals of Pure and AppliedLogic 69 (1994), pp. 157–193.[9] Kleitman, D. J. and B. L. Rothschild, Asymptotic enumeration of partial orders on afinite set, Transactions of the American Mathematical Society 205 (1975), pp. 205–220.[10] Litak, T., The non-reflexive counterpart of Grz, Bulletin of the Section of Logic Univ.Lo´dz 36 (2007), pp. 195–208.[11] Segerberg, K., “An Essay in Classical Modal Logic,” Ph.D. thesis, Uppsala Univ. (1971).[12] Verbrugge, R., Zero-one laws for provability logic (abstract), in: Proc. of the Tenth Int.Congress of Logic, Methodology and Philosophy of Science (LMPS95), 1995, pp. 79–80.[13] Verbrugge, R., Provability logic, in: E. N. Zalta, editor, The Stanford Encyclopedia ofPhilosophy (Fall 2017 Edition), Stanford University, 2017 .6 Proofs can be found in the full paper.

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Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

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