Zero-one laws for provability logic: Axiomatizing validity in almost all models and almost all frames Rineke Verbrugge Department of Artificial Intelligence, University of Groningen, e-mail [email protected] Abstract—It has been shown in the late 1960s that each formula [5] for nice historical overviews of zero-one laws). Later, of first-order logic without constants and function symbols obeys Kaufmann showed that monadic existential second-order logic a zero-one law: As the number of elements of finite models does not satisfy a zero-one law [8]. Kolaitis and Vardi have increases, every formula holds either in almost all or in almost no made the border more precise by showing that a zero-one law models of that size. Therefore, many properties of models, such 1 as having an even number of elements, cannot be expressed in holds for the fragment of existential second-order logic (Σ1) the language of first-order logic. For modal logics, limit behavior in which the first-order part of the formula belongs to the for models and frames may differ. Halpern and Kapron proved Bernays-Sch¨onfinkel class (∃∗∀∗ prefix) or the Ackermann zero-one laws for classes of models corresponding to the modal class (∃∗∀∃∗ prefix) [9], [10]; however, no zero-one law logics K, T, S4, and S5. They also proposed zero-one laws for 2 ∗ the corresponding classes of frames, but their zero-one law for holds for any other class, for example, the G¨odel class (∀ ∃ K-frames has since been disproved. prefix) [11]. Kolaitis and Vardi proved that a zero-one law does ω In this paper, we prove zero-one laws for provability logic hold for the infinitary finite-variable logic L∞ω, which implies with respect to both model and frame validity. Moreover, we that a zero-one law also holds for LFP(FO), the extension of axiomatize validity in almost all irreflexive transitive finite models first-order logic with a least fixed-point operator [12]. and in almost all irreflexive transitive finite frames, leading to two different axiom systems. In the proofs, we use a combinatorial The above zero-one laws and other limit laws have found ap- result by Kleitman and Rothschild about the structure of almost plications in database theory [13], [14], [15] and algebra [16]. all finite partial orders. On the way, we also show that a previous In AI, there has been great interest in asymptotic conditional result by Halpern and Kapron about the axiomatization of almost probabilities and their relation to default reasoning and degrees sure frame validity for S4 is not correct. Finally, we consider the complexity of deciding whether a given formula is almost surely of belief [17], [14]. valid in the relevant finite models and frames. In this article, we focus on zero-one laws for a modal logic that imposes structural restrictions on its models, namely, I. INTRODUCTION provability logic, which is sound and complete with respect In the late 1960s, Glebskii and colleagues proved that first- to finite strict (irreflexive) partial orders [18]. order logic without function symbols satisfies a zero-one law, The zero-one law for first-order logic also holds when that is, every formula is either almost always true or almost restricted to partial orders, both reflexive and irreflexive ones, always false in finite models [1]. More formally, let L be a as proved by Compton [4]. To prove this, he used a surprising language of first-order logic and let An(L) be the set of all combinatorial result by Kleitman and Rothschild [19] on labelled L-models with universe {1,...,n}. Now let µn(σ) which we will also rely for our results. Let us give a summary. be the fraction of members of An(L) in which σ is true, i.e., |{M ∈ An(L): M |= σ} | A. Kleitman and Rothschild’s result on finite partial orders arXiv:2102.05947v2 [cs.LO] 24 May 2021 µn(σ)= | An(L) | Kleitman and Rothschild proved that with asymptotic prob- Then for every σ ∈ L, limn→∞ µn(σ) = 1 or 1 ability 1, finite partial orders have a very special structure: limn→∞ µn(ϕ)=0. This was also proved later but independently by Fagin [3]; There are no chains u<v<w<z of more than three Carnap had already proved the zero-one law for first-order objects and the structure can be divided into three levels: languages with only unary predicate symbols [6] (see [7], • L1, the set of minimal elements; • L2, the set of elements immediately succeeding elements 1 The distinction between labelled and unlabelled probabilities was intro- in L1; duced by Compton [2]. The unlabelled count function counts the number of isomorphism types of size n, while the labelled count function counts • L3, the set of elements immediately succeeding elements the number of labelled structures of size n, that is, the number of relevant in L2. structures on the universe {1,...,n}. It has been proved both for the general Moreover, the ratio of the expected size of L1 to n and of the zero-one law and for partial orders that in the limit, the distinction between 1 labelled and unlabelled probabilities does not make a difference for zero-one expected size of L3 to n are both 4 , while the ratio of the 1 laws [3], [2], [4]. Per finite size n, labelled probabilities are easier to work expected size of L2 to n is 2 . As n increases, each element with than unlabelled ones [5], so we will use them in the rest of the article. in L1 has as immediate successors asymptotically half of the 978-1-6654-4895-6/21/$31.00 ©2021 IEEE elements of L2 and each element in L3 has as immediate 2 ∗ ∗ predecessors asymptotically half of the elements of L2 [19]. • (ϕ) = ∀y(Rxy → ϕ [y/x]). Kleitman and Rothschild’s theorem holds both for reflexive Van Benthem mapped each model M = (W, R, V ) to a (non-strict) and for irreflexive (strict) partial orders. classical model M ∗ with as objects the worlds in W and B. Zero-one laws for modal logics: Almost sure model validity the obvious binary relation R, while for each atom pi ∈ Φ, Pi = {w ∈ W | M, w |= pi} = {w ∈ W | Vw(pi)=1}. In order to describe the known results about zero-one laws Van Benthem then proved that for all ϕ ∈ L(Φ), M |= ϕ iff for modal logics with respect to the relevant classes of models M ∗ |= ∀x ϕ∗ [23]. Halpern and Kapron [24], [21] showed that and frames, we first give reminders of some well-known a zero-one law for modal models immediately follows by Van definitions and results. 3 Benthem’s result and the zero-one law for first-order logic. Let Φ = {p1,...,pk} be a finite set of propositional atoms By Compton’s above-mentioned result that the zero-one and let L(Φ) be the modal language over Φ, inductively law for first-order logic holds when restricted to the partial defined as the smallest set closed under: orders [4], this modal zero-one law can also be restricted 1) If p ∈ Φ, then p ∈ L(Φ). to finite models on reflexive or irreflexive partial orders, so 2) If A ∈ L(Φ) and B ∈ L(Φ), then also ¬A ∈ L(Φ), that a zero-one law for finite models of provability logic A ∈ L(Φ), ♦(ϕ) ∈ L(Φ), (A∧B) ∈ L(Φ), (A∨B) ∈ immediately follows. However, one would like to prove a L(Φ), and (A → B) ∈ L(Φ). stronger result and axiomatize the set of formulas ϕ for which A Kripke frame (henceforth: frame) is a pair F = (W, R) limn→∞ νn,Φ(ϕ)=1. Also, Van Benthem’s result does not where W is a non-empty set of worlds and R is a binary allow proving zero-one laws for classes of frames instead of ∗ ∗ accessibility relation. A Kripke model (henceforth: model) models: We have F |= ϕ iff F |= ∀P1 ... ∀Pn∀xϕ , but the 1 M = (W, R, V ) consists of a frame (W, R) and a valuation latter formula is not necessarily a negation of a formula in Σ1 function V that assigns to each atomic proposition in each with its first-order part in one of the Bernays-Sch¨onfinkel or world a truth value Vw(p), which can be either 0 or 1. The Ackermann classes (see [21]). truth definition is as usual in modal logic, including the clause: Halpern and Kapron [24], [21] aimed to fill in the above- M, w |= ϕ if and only if mentioned gaps for the modal logics K, T, S4 and S5 (see [25] for all w′ such that wRw′,M,w′ |= ϕ. for definitions). They proved zero-one laws for the relevant classes of finite models for these logics. For all four, they A formula ϕ is valid in model M = (W, R, V ) (notation axiomatized the classes of sentences that are almost surely M |= ϕ) iff for all w ∈ W , M, w |= ϕ. true in the relevant finite models. A formula ϕ is valid in frame F = (W, R) (notation F |= ϕ) iff for all valuations V , ϕ is valid in the model (W, R, V ). C. The quest for zero-one laws for frame validity Let Mn,Φ be the set of finite models over Φ with set of Halpern and Kapron’s paper also contains descriptions worlds W = {1,...,n}. We take νn,Φ to be the uniform of four zero-one laws with respect to the classes of finite probability distribution on Mn,Φ.