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Σ 2 ∃ es 1 1 ic e: al ∗ ) 2 ∗ ∗ predecessors asymptotically half of the elements of L2 [19]. • (ϕ) = ∀y(Rxy → ϕ [y/x]). Kleitman and Rothschild’s 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- 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 in each with its first-order part in one of the Bernays-Sch¨onfinkel or world a value Vw(p), which can be either 0 or 1. The Ackermann classes (see [21]). truth definition is as usual in , 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 worlds W = {1,...,n}. We take νn,Φ to be the uniform of four zero-one laws with respect to the classes of finite distribution on Mn,Φ. Let νn,Φ(ϕ) be the measure frames corresponding to K, T, S4 and S5. [21, Theorem in Mn,Φ of the set of models in which ϕ is valid. 5.1 and Theorem 5.15]: Either limn→∞ µn,Φ(ϕ) = 0 or Let Fn,Φ be the set of finite frames with set of worlds limn→∞ µn,Φ(ϕ)=1. W = {1,...,n}. We take µn,Φ to be the uniform probability They proposed four axiomatizations for the sets of formulas distribution on Fn. Let µn,Φ(ϕ) be the measure in Fn of the that would be almost always valid in the corresponding four set of frames in which ϕ is valid. classes of frames [21]. However, almost 10 years later, Le Bars surprisingly proved them wrong with respect to the Halpern and Kapron proved that every formula ϕ in modal zero-one law for K-frames [26]. By proving that the formula language L(Φ) is either valid in almost all models (“almost q ∧ ¬p ∧ ((p ∨ q) → ¬♦(p ∨ q)) ∧ ♦p does not have an surely true”) or not valid in almost all models (“almost surely asymptotic probability, he showed that in fact no zero-one law false”) [21, Corollary 4.2]: holds with respect to all finite frames. Doubt had already been

Either lim νn,Φ(ϕ)=0 or lim νn,Φ(ϕ)=1. cast on the zero-one law for frame validity by Goranko and n→∞ n→∞ Kapron, who proved that the formula ¬(p ↔ ¬♦p) fails in In fact, this zero-one law for models already follows from the the countably infinite random frame, while it is almost surely zero-one law for first-order logic [1], [3] by Van Benthem’s valid in K-frames [5]. (See also [27, Section 9.5]).4 Currently, translation method [22], [23]. As reminder, let ∗ be given by: the problem of axiomatizing the modal logic of almost sure ∗ 5 • pi = Pi(x) for atomic sentences pi ∈ Φ; frame validities for finite K-frames appears to be open. ∗ ∗ • (¬ϕ) = ¬ϕ ; As a reaction to Le Bars’ counter-example, Halpern and ∗ ∗ ∗ • (ϕ∧ψ) = (ϕ ∧ψ ) (similar for other binary operators); Kapron [29] published an erratum, in which they showed

2Interestingly, it was recently found experimentally that for smaller n there 4We will show in this paper that for irreflexive partial orders, almost-sure are strong oscillations, while the behavior stabilizes only around n = 45 [20]. frame validity in the finite does transfer to validity in the corresponding 3In the rest of this paper in the parts on almost sure model validity, we take countable random Kleitman-Rothschild frame, and that the validities are quite Φ to be finite, although the results can be extended to enumerably infinite Φ different from those for almost all K frames (see Section V). by the methods described in [21], [17]. 5For up to 2006: see [27]; for more recently: [28] . exactly where their erstwhile of [21, Theorem 5.1] had p1,p2 w2 w4 p1, ¬p2 gone wrong. It may be that the problem they point out also invalidates their similar proof of the zero-one law with respect T to finite reflexive frames, corresponding to [21, Theorem ¬p1,p2 w1 w3 ¬p1, ¬p2 5.15 a]. However, with respect to frame validity for T-frames, as far as we know, no counterexample to a zero-one law has yet been published and Le Bars’ counterexample cannot easily be adapted to reflexive frames; therefore, the situation remains w0 p1, ¬p2 6 unsettled for T. Fig. 1. Counter-model showing that the formula χ :=(p1 ∧ ♦(¬p1 ∧ ♦p1 ∧ (p1 → p2))) → ((¬p1 ∧ ♦p1) → ♦(p1 → p2)) does not hold in w0 D. Halpern and Kapron’s axiomatization for almost sure of this three-layer model. The relation in the model is the reflexive transitive frame validities for S4 fails closure of the one represented by the arrows. Unfortunately, Halpern and Kapron’s proof of the 0-1 law for reflexive, transitive frames and the axiomatization w1Rw3 and w2Rw3. This confluence follows from Compton’s of the almost sure frame validities for reflexive, transitive extension axiom (b) [4] (similar to (b) in Proposition 4 of frames [21, Theorem 5.16] turn out to be incorrect as well, the current paper). Therefore by M, w1 |= (p1 → p2), also 7 ′ as follows. Halpern and Kapron introduce the axiom DEP2 M, w3 |= (p1 → p2), so M, w2 |= ♦(p1 → p2). Therefore and they axiomatize almost-sure frame validities in reflexive M, w |= ((¬p1 ∧ ♦p1) → ♦(p1 → p2)). Now, because transitive frames by S4+DEP2′ [21, Theorem 5.16], where w ∈ W was arbitrary, we have M |= χ. ′ DEP2 is:¬(p1 ∧ ♦(¬p1 ∧ ♦(p1 ∧ ♦¬p1))). Therefore, the axiom system given in [21, Theorem 5.16] is The axiom DEP2′ precludes R-chains tRuRvRw of more than not complete with respect to almost-sure frame validities for three different states. finite reflexive transitive orders. Fortunately, it seems possible to mend the situation and still obtain an axiom system that Proposition 1. Suppose Φ= {p1,p2}. Now take the following is sound and complete with respect to almost sure S4 frame sentence χ: validity, by adding extra characterizing the umbrella- χ := (p1 ∧ ♦(¬p1 ∧ ♦p1 ∧ (p1 → p2))) → and diamond properties that we will also use for the provability ((¬p1 ∧ ♦p1) → ♦(p1 → p2)) logic GL in Section V; the S4 version is future work. ′ Then S4+DEP2 6⊢ χ but limn→∞ µn,Φ(χ)=1 E. Almost sure model validity does not coincide with almost Proof. It is easy to see that S4+DEP2′ 6⊢ χ by taking the sure frame validity five-point reflexive transitive frame of Figure 1, where Interestingly, whereas for full classes of frames, ‘validity ♦ ♦  M, w0 |= p1 ∧ (¬p1 ∧ p1 ∧ (p1 → p2)) in all finite models’ coincides with ‘validity in all finite but M, w3 6|= (¬p1 ∧ ♦p1) → ♦(p1 → p2), so frames’ of the class, this is not the case for ‘almost sure M, w0 6|= ((¬p1 ∧ ♦p1) → ♦(p1 → p2)). validity’. In particular, for both the class of reflexive transitive frames ( 4) and the class of reflexive transitive symmetric Now we sketch a proof that χ is valid in almost all reflexive S frames ( 5), there are many more formulas that are ‘valid Kleitman-Rothschild frames.8 So let M = (W, R, V ) be an S almost all almost all arbitrary large enough (with appropriate extension axioms in finite models’ than ‘valid in finite frames’ of the appropriate kinds. Our work has been greatly holding) Kleitman-Rothschild frame (W, R) together with an inspired by Halpern and Kapron’s paper [21] and we also arbitrary valuation V . Let w be arbitrary in W and suppose use some of the previous results that they applied, notably M, w |= p1 ∧ ♦(¬p1 ∧ ♦p1 ∧ (p1 → p2)). Then there is a the above-mentioned combinatorial result by Kleitman and w1 ∈ W with wRw1 and M, w1 |= ¬p1 ∧ ♦p1 ∧ (p1 → p2). Rothschild about finite partial orders. We want to show that M, w |= ((¬p1 ∧ ♦p1) → ♦(p1 → p2)). To do this, suppose w2 is arbitrary in W with wRw2 The rest of this paper is structured as follows. In Section II, and M, w2 |= ¬p1 ∧ ♦p1. The above facts imply that both we give a brief reminder of the axiom system and w1 and w2 are in the middle layer and w is in the bottom of provability logic. In the central Sections III, IV and V, layer. Then almost surely, there is a w3 in the top layer with we show why provability logic obeys zero-one laws both 6Joe Halpern and Bruce Kapron (personal communication) and Jean-Marie with respect to its models and with respect to its frames. We Le Bars (personal communication) confirmed the current non-settledness of provide two axiom systems characterizing the formulas that are the problem for T. almost always valid in the relevant models, respectively almost 7The author of this paper discovered the counter-example after a colleague had pointed out that the author’s earlier attempt at a proof of the 0-1 always valid in the relevant frames. When discussing almost law for frames of provability logic, inspired by Halpern and Kapron’s [21] sure frame validity, we will investigate both the almost sure axiomatiation, contained a serious gap. validity in finite irreflexive transitive frames and validity in the 8Halpern and Kapron [21, Theorem 4.14] proved that almost surely, every reflexive transitive relation is in fact a partial order, so the Kleitman-Rothschild countable random Kleitman-Rothschild frame, and show that result also holds for finite frames with reflexive transitive relations. there is transfer between them. Section VI provides a sketch of the complexity of the problems of almost sure 3) if ϕ ∈ Θ and (ϕ → ψ) ∈ Θ then ψ ∈ Θ; model and almost sure frame validity for provability logic. 4) if Θ ⊢S ϕ then ϕ ∈ Θ. Finally, Section VII presents a conclusion and some questions for future work. III. VALIDITY IN ALMOST ALL FINITE IRREFLEXIVE The result on models in Section III was proved 26 years TRANSITIVE MODELS Φ,M ago, and presented in [30], [31], but the proofs have not been The axiom system AXGL has the same axioms and rules published before in an archival venue. The results about almost as GL (see Section II) plus the following axioms: sure frame validities for GL are new, as well as the counter-  example against the axiomatization by Halpern and Kapron of ⊥ (T3) almost sure S4 frame validities.9 ♦⊤ → ♦A (C1) ♦♦⊤ → ♦(B ∧ ♦C) (C2) II. PROVABILITY LOGIC In this section, a brief reminder is provided about the In the axiom schemes C1 and C2, the formulas A, B and C protagonist of this paper: the provability logic GL, named all stand for consistent conjunctions of literals over Φ. after G¨odel and L¨ob. As axioms, it contains all axiom schemes These axiom schemes have been inspired by Carnap’s from K and the extra scheme GL. Here follows the full set of consistency axiom: ♦ϕ for any ϕ that is a consistent propo- axiom schemes of GL: sitional formula [35], which has been used by Halpern and Kapron [21] for axiomatizing almost sure model validities for All (instances of) propositional tautologies (A1) K-models.    Φ,M (ϕ → ψ) → ( ϕ → ψ) (A2) Note that AXGL is not a normal modal logic, because one (ϕ → ϕ) → ϕ (GL) cannot substitute just any formula for A, B, C; for example, substituting p1 ∧ ¬p1 for A in C1 would make that formula The rules of are and necessitation: equivalent to ¬♦⊤, which is clearly undesired. However, even GL GL GL Φ,M if ⊢ ϕ → ψ and ⊢ ϕ, then ⊢ ϕ. though AXGL is not closed under uniform , it is if GL ⊢ ϕ, then GL ⊢ ϕ. still a propositional theory, in the sense that it is closed under Note that the transitivity axiom ϕ → ϕ follows from modus ponens. GL, which was first proved by De Jongh and Sambin [32], Example 1. For Φ = {p1,p2}, the axiom scheme C1 boils [33], but that the reflexivity axiom ϕ → ϕ does not follow. down to the following four axioms: Indeed, Segerberg proved in 1971 that provability logic is sound and complete with respect to all transitive, converse ♦⊤ → ♦(p1 ∧ p2) (1) well-founded frames (i.e., for each non-empty set X, there ♦⊤ → ♦(p1 ∧ ¬p2) (2) is an R-greatest element of X; or equivalently: there is no ♦⊤ → ♦(¬p1 ∧ p2) (3) infinitely ascending sequence x1Rx2Rx3Rx4,...). Segerberg also proved with respect to all finite, transitive, ♦⊤ → ♦(¬p1 ∧ ¬p2) (4) irreflexive frames [18]. The latter and completeness The axiom scheme C2 covers 16 axioms, corresponding to result will be relevant for our purposes. For more information the 24 possible choices of positive or negative literals, as on provability logic, see, for example, [34], [32], [33]. captured by the following scheme, where “[¬]” is shorthand In the next three sections, we provide axiomatizations, first for a negation being present or absent at the current location: for almost sure model validity and then for almost sure frame validity, with respect to the relevant finite frames correspond- ♦♦⊤ → ♦([¬]p1 ∧ [¬]p2 ∧ ♦([¬]p1 ∧ [¬]p2)) ing to GL, namely the irreflexive transitive ones. For the proofs of the zero-one laws for almost sure model The following definition of the canonical asymptotic model and frame validity, we will need completeness proofs of the over a finite set of propositional atoms Φ is based on the relevant axiomatic theories – let us refer to such a theory by set of propositional valuations on Φ, namely, the functions v S for the moment – with respect to almost sure model validity from the set of propositional atoms Φ to the set of truth values and with respect to almost sure frame validity. Here we will {0, 1}. As worlds, we introduce for each such valuation v three use Lindenbaum’s lemma and maximal S-consistent sets of new distinct objects, for mnemonic called uv (Upper), formulas. For such sets, the following useful properties hold, mv (Middle), and bv (Bottom); see Figure 2. as usual [18], [25]: Φ Definition 1. Define MGL = (W, R, V ), the canonical asymp- Proposition 2. Let Θ be a maximal S-consistent set of for- totic model over Φ, with W, R, V as follows: mulas in L(Φ). Then for each pair of formulas ϕ, ψ ∈ L(Φ): W = {bv | v a propositional valuation on Φ} ∪ {m | v a propositional valuation on Φ} ∪ 1) ϕ ∈ Θ iff ¬ ϕ 6∈ Θ; v {uv | v a propositional valuation on Φ} 2) (ϕ ∧ ψ) ∈ Θ ⇔ ϕ ∈ Θ and ψ ∈ Θ; ′ R = {hbv,mv′ i | v, v propositional valuations on Φ} ∪ ′ 9Due to the length restriction, this paper includes proof sketches of the main {hmv,uv′ i | v, v propositional valuations on Φ} ∪ ′ results. Full proofs are to be included in an extended version for a journal. {hbv,uv′ i | v, v propositional valuations on Φ}; and for all pi ∈ Φ and all propositional valuations v on Φ, form, determined by which of the four conjunctions the modal valuation V is defined by: of relevant literals [¬]p1 ∧ [¬]p2 is an element. 10 Vbv (pi)= Vmv (pi)= Vuv (pi)= v(pi). Because ♦♦⊤ ∈ Γ, by axiom C2 and Proposition 2, all these four maximal consistent sets contain the 16 For the proof of the zero-one law for model validity, we will Φ M formulas ♦([¬]p1 ∧ [¬]p2 ∧ ♦([¬]p1 ∧ [¬]p2)). By need a completeness proof of AX , with respect to almost GL the definition of R′, this means that all four worlds sure model validity, including use of Lindenbaum’s lemma Φ M in this bottom level B will have direct access to all and Proposition 2, applied to AX , . The zero-one law for GL the four worlds in middle level M as well as access model validity follows directly from the following theorem: in two steps to all four worlds in upper level U. Theorem 1. For every formula ϕ ∈ L(Φ), the following are equivalent: ′ Φ,M Note that R is transitive because AXGL extends GL, so for Φ 1) M |= ϕ; all maximal consistent sets Γ and all formulas ψ ∈ L(Φ), we GLΦ M , ′ 2) AXGL ⊢ ϕ; have that ψ → ψ ∈ Γ. Also R is irreflexive. Because 3) limn→∞ νn,Φ(ϕ)=1; each world contains either ⊥ and ¬⊥ (for U), or ⊥ and 4) limn→∞ νn,Φ(ϕ) 6=0. ¬⊥ (for M), or ⊥ and ¬⊥ (for B), by definition ′ Proof. We show a circle of implications. Let ϕ ∈ L(Φ). of R , none of the worlds has a relation to itself.

1 ⇒ 2 The next step is to prove by induction that a truth lemma AXΦ,M holds: For all ψ in the language L(Φ) and for all maximal By contraposition. Suppose that GL 6⊢ ϕ, then ¬ϕ is Φ,M Φ,M AXGL -consistent sets Γ, the following holds: AXGL -consistent. By Lindenbaum’s lemma, we can extend Φ,M Φ {¬ϕ} to a maximal AXGL -consistent set ∆ over Φ. We use MCGL, wΓ |= ψ iff ψ ∈ Γ. a standard canonical model construction; here, we illustrate ′ how that works for the finite set Φ= {p1,p2}, but the method For atoms, this follows by the definition of V . The steps 11 works for any finite Φ= {p1,...,pk}. It will turn out that for the propositional connectives are as usual, using the the model we define is isomorphic to the model of Definition 1. properties of maximal consistent sets (see Proposition 2). Φ ′ ′ ′ Let us define the model MCGL = (W , R , V ): ′ Φ,M AXΦ,M • W = {wΓ | Γ is maximal AXGL -consistent, For the -step, let Γ be a maximal GL -consistent set based on Φ}. and let us suppose as induction hypothesis that for some ′ ′ AXΦ,M • R = {hwΓ1 , wΓ2 i | wΓ1 , wΓ2 ∈ W and arbitrary formula χ, for all maximal GL -consistent sets Φ for all ψ ∈ Γ1, it holds that ψ ∈ Γ2} Π, MCGL, wΠ |= χ iff χ ∈ Π. We want to show that ′ ′ Φ  iff  . • For each wΓ ∈ W : VwΓ (p)=1 iff p ∈ Γ MCGL, wΓ |= χ χ ∈ Γ Because the worlds of this model correspond to the maximal For the direction from right to left, suppose that χ ∈ Γ, Φ,M ′ ′ ′ AXGL -consistent sets, all worlds wΓ ∈ W can be distin- then by definition of R , for all Π with wΓR wΠ, we have χ ∈ Φ guished into three kinds, exhaustively and without overlap: Π, so by induction hypothesis, MCGL, wΠ |= χ. Therefore, by Φ  U ⊥ ∈ Γ; there are exactly four maximal consistent the truth definition, MCGL, wΓ |= χ. sets Γ of this form, determined by which of the four For the direction from left to right, let us use contraposition  conjunctions of relevant literals [¬]p1 ∧ [¬]p2 is an and suppose that χ 6∈ Γ. Now we will show that the set AXΦ,M element. These comprise the upper level U of the {ξ | ξ ∈ Γ}∪{¬χ} is GL -consistent. For otherwise, model. there would be some ξ1,...,ξn for which ξ1,..., ξn ∈ Γ such that ξ1,...,ξn ⊢ Φ,M χ, so by necessitation, M ¬⊥ ∈ Γ and ⊥ ∈ Γ; there are exactly four AXGL A2, and propositional logic, ξ1,..., ξn ⊢ Φ,M χ, maximal consistent sets Γ of this form, determined AXGL by which of the four conjunctions of relevant literals therefore by maximal consistency of Γ and Proposition 2(iv), [¬]p1 ∧[¬]p2 is an element. By axiom C1 and Propo- also χ ∈ Γ, contradicting our assumption. Therefore, sition 2, all these four maximal consistent sets con- by Lindenbaum’s lemma there is a maximal consistent set ′ tain the four formulas of the form ♦([¬]p1 ∧ [¬]p2); Π ⊇{ξ | ξ ∈ Γ}∪{¬χ}. It is clear by definition of R that ′ ′ Φ by definition of R and using the fact that ⊥ ∈ Γ, wΓR wΠ, and by induction hypothesis, MCGL, wΠ |= ¬χ, i.e., Φ Φ this means that all the four worlds in this middle level MCGL, wΠ 6|= χ, so by the truth definition, MCGL, wΓ 6|= χ. M will have access to all the four worlds in the upper This finishes the inductive proof of the truth lemma. level U. B ¬⊥ ∈ Γ and ¬⊥ ∈ Γ and ⊥ ∈ Γ; there Finally, from the truth lemma and the fact stated at the are exactly four maximal consistent sets Γ of this beginning of the proof of 1 ⇒ 2 that ¬ϕ ∈ ∆, we have that Φ MCGL, w∆ 6|= ϕ, so we have found our counter-model. 10If Φ were enumerably infinite, the definition could be adapted so that precisely those propositional valuations are used that make only finitely many propositional atoms true, see also [21]. With its three layers (Upper, Middle, and Bottom) of four 11For adapting to the enumerably infinite case, see [21, Theorem 4.15]. worlds each, corresponding to each consistent conjunction p1,p2 p1, ¬p2 ¬p1,p2 ¬p1, ¬p2

uv1 uv2 uv3 uv4

p1,p2 mv1 p1, ¬p2 mv2 mv3 ¬p1,p2 mv4 ¬p1, ¬p2

bv1 bv2 bv3 bv4 p1,p2 p1, ¬p2 ¬p1,p2 ¬p1, ¬p2

Φ Fig. 2. The canonical asymptotic model MGL = (W, R, V ) for Φ= {p1,p2}, defined in Definition 1. The accessibility relation is the transitive closure of the relation given by the arrows drawn in the picture. The four relevant valuations are v1, v2, v3, v4, given by v1(p1)= v1(p2) = 1; v2(p1) = 1, v2(p2) = 0; v3(p1) = 0, v3(p2) = 1; v4(p1)= v4(p2) = 0. of literals, and with each world corresponding to maximal C1 is valid in almost all Kleitman-Rothschild models, i.e., consistent sets containing axioms C1 and C2 and therefore limn→∞ νn,Φ(♦⊤ → ♦A)=1. being related to precisely all those worlds in the layers above, Φ the model MCGL that we construct in the completeness proof Similarly, we sketch a proof that axiom C2, namely ♦♦⊤ → Φ above is isomorphic to the canonical asymptotic model MGL ♦(B ∧ ♦C) where B, C are consistent conjunctions of literals of Definition 1; for Φ= {p1,p2}, see Figure 2. over Φ, is valid in almost all irreflexive transitive Kleitman- Rothschild models (W, R, V ) of depth 3 with an arbitrary V . 2 ⇒ 3 Let Φ= {p1,...,pk}. Again, let us consider a state s in such a Φ,M Suppose that AXGL ⊢ ϕ. We will show that the axioms model of n elements where ♦♦⊤ holds, then s is in the bottom Φ,M of AXGL hold in almost all irreflexive transitive Kleitman- of the three layers; therefore, the model being of Kleitman- Rothschild models of depth 3 (see Subsection I-A). First, Rothschild type, s has as direct successors approximately half it is immediate that GL is sound with respect to all finite of the states in the middle layer, which contains asymptotically 1 irreflexive transitive converse well-founded models, that at least 2 of the model’s states. So s has asymptotically at least 1 axiom ⊥ is sound with respect to those of depth 3, 4 · n direct successors. and that almost sure model validity is closed under MP and The probability that a given state t is a direct successor of Necessitation. It remains to show the almost sure model s with the right valuation to make B true is therefore at least 1 1 1 validity of axiom schemes C1 and C2 over finite irreflexive 4 · 2k = 2k+2 . Similarly, given such a t, the probability that models of the Kleitman-Rothschild variety. a given state t′ in the top layer is a direct successor of t in 1 1 1 which C holds is asymptotically at least 2k+2 · 2k+3 = 22k+5 We will now sketch a proof that the ‘Carnap-like’ axiom C1, Therefore, the probability that for the given s there are no namely ♦⊤ → ♦A where A is a consistent conjunction t,t′ with sRtRt′ with B true at t and C true at t′ is at most 1 n of literals over Φ, is valid in almost all irreflexive (1 − 22k+5 ) . Summing up, the probability that there is at transitive models (W, R, V ) of depth 3 of the Kleitman- least one s in a Kleitman-Rothschild model not having any Rothschild variety with an arbitrary V . Let us suppose that pair of successors sRtRt′ with B true at t and C true at t′ is k 1 n 1 n Φ = {p1,...,pk}, so there are 2 possible valuations. Let at most n · (1 − 22k+5 ) . Again, limn→∞ n · (1 − 22k+5 ) =0, us consider a state s in such a model of n elements where so C2 holds in almost all Kleitman-Rothschild models, i.e. ♦⊤ holds; then, being a Kleitman-Rothschild model, s has limn→∞ νn,Φ(♦♦⊤ → ♦(B ∧ ♦C))=1. as direct successors approximately half of the states in the directly higher layer, which contains asymptotically at least 3 ⇒ 4 1 1 4 of the model’s states. So s has asymptotically at least 8 · n Obvious, because 0 6=1. direct successors. The probability that a given state t is a direct successor of s with the right valuation to make A true 4 ⇒ 1 1 1 1 is therefore at least 8 · 2k = 2k+3 . Thus, the probability that By contraposition. Suppose as before that Φ= {p1,...,pk}. Φ s does not have any direct successors in which A holds is at Now suppose that the canonical asymptotic model MGL 6|= ϕ 1 n Φ most (1 − 2k+3 ) . Therefore, the probability that there is at for some ϕ ∈ L(Φ), say, MGL,s 6|= ϕ, for some s ∈ W . We least one s in a Kleitman-Rothschild model not having any claim that almost surely for a sufficiently large finite Kleitman- 1 n ′ ′ ′ ′ direct successors satisfying A is at most n · (1 − 2k+3 ) . By Rothschild type model M = (W , R , V ) of three layers with 1 n ′ Φ a standard calculus result, limn→∞ n · (1 − 2k+3 ) = 0, so V random, there is a bisimulation relation Z from MGL to M ′. We now sketch how to define the bisimulation Z. ∀x,y,z((x

12 Definition 2 (Extension axioms). The theory Tas-irr includes Notice that by extension axiom (c) and transitivity, in almost the axioms for strict partial orders, namely, ∀x¬(x < x) and all Kleitman-Rothschild models, all points in the bottom layer L1 are connected to all points in the top layer L3. 12Here, the subscript as-irr stands for “almost sure - irreflexive”. Now that we have shown that the extension axioms hold in 5) If ♦ψ ∈ Λ and ψ itself is not of the form ♦ξ or ¬ξ, FKR as well as in almost all finite strict partial orders, we then ♦♦ψ ∈ Λ, and also ¬ψ, ¬ψ ∈ Λ. have enough background to be able to prove the modal zero- 6) If ψ ∈ Λ and ψ itself is not of the form ξ or ¬♦ξ, one law with respect to the class of finite irreflexive transitive then ψ ∈ Λ, and also ♦¬ψ, ♦♦¬ψ ∈ Λ. frames corresponding to provability logic. Note that Λ is a finite set of formulas, of size polynomial in V. VALIDITY IN ALMOST ALL FINITE IRREFLEXIVE the length of the formula ϕ from which it is built. TRANSITIVE FRAMES Definition 4. Let Λ be a closure as defined above and let Φ F Take Φ = {p1,...,pk} or Φ = {pi | i ∈ N}. The axiom AX , Φ,F ∆, ∆1, ∆2 be maximal GL -consistent sets. We define: system AXGL corresponding to validity in almost all finite Λ • ∆ := ∆ ∩ Λ; frames of provability logic has the same axioms and rules as • ∆1 ≺ ∆2 iff for all χ ∈ ∆1, we have χ ∈ ∆2; GL, plus the following axiom schemas, for all k ∈ N, where Λ Λ • ∆1 ≺ ∆2 iff ∆1 ≺ ∆2. all ϕi ∈ L(Φ): Theorem 2. For every formula ϕ ∈ L(Φ), the following are ⊥ (T3) equivalent: ♦♦ ♦ ♦   ♦ ♦ Φ,F ⊤ ∧ ^ ( ⊤ ∧ ϕi) → ( ⊤ → (^ ϕi)) 1) AXGL ⊢ ϕ; i≤k i≤k 2) FKR |= ϕ; (DIAMOND-k) 3) limn→∞ µn,Φ(ϕ)=1; ♦♦⊤ ∧ ^ ♦(⊥ ∧ ϕi) → ♦(^ ♦ϕi) (UMBRELLA-k) 4) limn→∞ µn,Φ(ϕ) 6=0. i≤k i≤k Proof. We show a circle of implications. Let ϕ ∈ L(Φ). Here, UMBRELLA-0 is the formula ♦♦⊤ ∧ ♦(⊥ ∧ ϕ0) → ♦♦ϕ0, which represents the property that direct successors of 1 ⇒ 2 Φ,F bottom layer worlds are never endpoints but have at least one Suppose AXGL ⊢ ϕ. Because finite irreflexive Kleitman- successor in the top layer. Rothschild frames are finite strict partial orders that have no The formula DIAMOND-0 has been inspired by the well- chains of length greater than 3, the axioms and of GL known axiom ♦ϕ → ♦ϕ that characterizes confluence, + ⊥ hold in all Kleitman-Rothschild frames, therefore also known as the diamond property: for all x,y,z, if xRy they are valid in FKR. and xRz, then there is a w such that yRw and zRw. So we only need to check the validity of the DIAMOND-k AXΦ,M Note that in contrast to the theory GL introduced in and UMBRELLA-k axioms in FKR for all k ≥ 0. Φ,F Section III, the axiom system AXGL gives a normal modal logic, closed under uniform substitution. DIAMOND-k-1: Fix k ≥ 1, take sentences ϕi ∈ L(Φ) for Φ,F Also notice that AXGL is given by an infinite set of ♦♦ ♦ ♦  i =1,...,k − 1 and let ϕ = ⊤ ∧ Vi≤k−1 ( ⊤ ∧ ϕi) → axioms. It turns out that if we base our logic on an infinite (♦⊤ → ♦( 1 ϕi)). By Proposition 3, we know that N N Vi≤k− set of atoms Φ = {pi | i ∈ }, then for each k ∈ , each of the extension axioms of the form (b) holds in FKR. DIAMOND-k+1 and UMBRELLA-k+1 are strictly stronger We want to show that ϕ is valid in FKR. than DIAMOND-k and UMBRELLA-k, respectively. So we To this end, let V be any valuation on the set of labelled have two infinite sets of axioms that both strictly increase in states W of FKR and let M = (FKR, V ). Now take strength, thus by a classical result of Tarski, the modal theory an arbitrary b ∈ W and suppose that M,b |= ♦♦⊤ ∧ Φ,F generated by AXGL is not finitely axiomatizable. ♦ ♦  Vi≤k−1 ( ⊤ ∧ ϕi). Then b is in the bottom layer L1 and For the proof of the zero-one law for frame validity, we there are worlds x0,...,xk−1 (not necessarily distinct) in the AXΦ,F will again need a completeness proof, this time of GL middle layer L2 such that for all i ≤ k − 1, we have b < xi with respect to almost sure frame validity, including use of and M, xi |= ϕi. Now take any xk in L2 with b < xk. Then, AXΦ,F Lindenbaum’s lemma and finitely many maximal GL - by the extension axiom (b), there is an element z in the upper consistent sets of formulas, each intersected with a finite set layer L3 such that Vi≤k xi < z. Now for that z, we have of relevant formulas Λ. ♦ that M,z |= Vi≤k−1 ϕi. But then M, xk |= (Vi≤k−1 ϕi), Below, we will define the closure of a sentence ϕ ∈ L(Φ). so because xk is an arbitrary direct successor of b, we have We may view this closure as the set of formulas that are  ♦ ♦ M,b |= ( ⊤ → (Vi≤k−1 ϕi)). To conclude, relevant for making a (finite) countermodel against ϕ. M,b |= ♦♦⊤∧ ^ ♦(♦⊤∧ϕi) → (♦⊤ → ♦( ^ ϕi)), Definition 3 (Closure of a formula). The closure of ϕ with 1 1 Φ,F Φ,F i≤k− i≤k− respect to AXGL is the minimal set Λ of AXGL -formulas such that: so because b and V were arbitrary, we have 1) ϕ ∈ Λ. FKR |= ♦♦⊤∧ ^ ♦(♦⊤∧ϕi) → (♦⊤ → ♦( ^ ϕi)), 2) ⊥ ∈ Λ. i≤k−1 i≤k−1 3) If ψ ∈ Λ and χ is a subformula of ψ, then χ ∈ Λ. as desired. 4) If ψ ∈ Λ and ψ itself is not a negation, then ¬ψ ∈ Λ. UMBRELLA-k-1: Fix k ≥ 1, take sentences ϕi ∈ L(Φ) for Kleitman-Rothschild frames of any large enough size to show ♦♦ ♦  i =1,...,k − 1 and let ϕ = ⊤ ∧ Vi≤k−1 ( ⊥ ∧ ϕi) → that limn→∞ µn,Φ(ϕ)=0 (Step 4 ⇒ 1 (b)). ♦ ♦ (Vi≤k−1 ϕi). By Proposition 3, we know that each of the extension axioms of the form (c) holds in FKR. We want to Step 4 ⇒ 1 (a) show that ϕ is valid in FKR. By the Lindenbaum Lemma, we can extend {¬ϕ} to a Φ,F Λ To this end, let V be any valuation on the set of labelled maximal AXGL -consistent set Ψ. Now define Ψ := Ψ ∩ Λ, states W of FKR and let M = (FKR, V ). Now take where Λ is as in Definition 3, and introduce a world sΨ an arbitrary b ∈ W and suppose that M,b |= ♦♦⊤ ∧ corresponding to ΨΛ. ♦  Vi≤k−1 ( ⊥∧ϕi). Then b is in the bottom layer L1 and there We distinguish three cases for the step-by-step construction: are accessible worlds x0,...,xk−1 (not necessarily distinct) U (upper layer), M (middle layer), and B (bottom layer). in the upper layer L3 such that for all i ≤ k − 1, we have Λ b < xi and M, xi |= ϕi. By the extension axiom (c) from Case U, with ⊥ ∈ Ψ : Definition 2, there is an element z in the middle layer L2 In this case we are done: a one-point counter-model such that b

Vardi [9, Lemma 4], we can prove that if every finite {χj, χj | j ∈{1,...,k}} ⊢AXΦ,F (¬ψi → ¬ψi). ∗ GL of Tas−irr ∪ {∃x¬ϕ (P1,...,Pn)} is satisfiable over the extended vocabulary with 1 , then by compactness, Because we have ⊢ Φ,F χj → χj for all j =1,...,k P ,...,Pn AXGL ∗ Tas−irr ∪ {∃x¬ϕ (P1,...,Pn)} has a countable model over and ⊢ Φ,F (¬ψi → ¬ψi) → ¬ψi, we can conclude: AXGL the extended vocabulary. The reduct of this model to the {χ | χ ∈ Ψ}⊢ Φ,F ¬ψi. old language is still a countable model of Tas−irr, and is AXGL therefore isomorphic to FKR (by Proposition 3). But then ∗ Using Proposition 2(4) and the fact that ¬ψi ∈ Λ, this leads FKR |= ∃P1,...,Pn∃x¬ϕ , a . Λ Λ to ¬ψi ∈ Ψ , contradicting our assumption that ♦ψi ∈ Ψ . Also note that because ⊥ ∈ Ψ, by definition, ⊥ ∈ ∆i. 3 ⇒ 4 Φ F We can now extend ∆ to a maximal AX , -consistent Obvious, because 0 6=1. i GL set Ψi by the Lindenbaum Lemma, and we define for each Λ i ∈ {1,...,n} the set Ψ := Ψ ∩ Λ and a world sΨ 4 ⇒ 1 i i i AXΦ,F corresponding to it. We have for all i ∈ {1,...,n} that By contraposition. Let ϕ ∈ L(Φ) and suppose that GL 6⊢ Λ Λ Λ Φ,F Ψ ≺ Ψi as well as ψi, ¬ψi ∈ Ψi . ϕ. Then ¬ϕ is AXGL -consistent. We will do a completeness proof by the finite step-by-step method (see, for example, [37], We have now finished creating a two-layer counter-model [38]), but based on infinite maximal consistent sets, each M = (W, R, V ), which has: of which is intersected with the same finite set of relevant ϕ • formulas Λ, so that the constructed counter-model remains W = {sΨ,sΨ1 ,...,sΨn }; finite (see [39], [40, footnote 3]). • R = {hsΨ,sΨi i | i ∈{1,...,n}}; Λ In the following, we are first going to construct a model • For each p ∈ Φ and sΓ ∈ W : VsΓ (p)=1 iff p ∈ Γ . Mϕ = (W, R, V ) that will contain a world where ¬ϕ holds A truth lemma can be proved as in Case B below (but easier). (Step 4 ⇒ 1 (a)). Then we will embed this model into Λ Φ,F Case B, with ⊥ 6∈ Ψ : Therefore, by maximal AXGL -consistency of Ψ, we have by Λ In this case, we also look at all formulas of the form ♦ψ ∈ Ψ . Proposition 2 that ♦(⊥ ∧ ξi) ∈ Ψ for each i ∈ {1,...,k}. We first divide this into two sets, as follows: We also have ♦♦⊤ ∈ Ψ. UMBRELLA-k now gives us ♦ Λ 1) The set of -formulas in Ψ for which we have that Ψ ⊢AXΦ,F ♦♦⊤ ∧ ♦(⊥ ∧ ξi) → Λ Λ GL ^ ♦ξk+1,..., ♦ξl ∈ Ψ but ♦♦ξk+1,..., ♦♦ξl 6∈ Ψ for i=1,...,k Λ 13 some l ∈ N, so ¬ξk+1,..., ¬ξl ∈ Ψ . Λ ♦ ♦ 2) The set of ♦♦-formulas with ♦♦ξ1,..., ♦♦ξk ∈ Ψ . ( ^ ξi) Note that for these formulas, we also have i=1,...,k Λ Φ F ♦ξ1,..., ♦ξ ∈ Ψ , because GL ⊢ ♦♦ξ → ♦ξ . , k i i We conclude from maximal AXGL -consistency of Ψ and We will treat these pairs ♦♦ξi, ♦ξi for i = 1,...,k at ♦ ♦ Proposition 2(4) that (Vi=1,...,k ξi) ∈ Ψ. the same go. This means that we can construct one direct successor of Λ Note that (1) and (2) lead to disjoint sets which together Ψ containing all the ♦ξi for i ∈{1,...,k}. To this end, let exhaust the ♦-formulas in ΨΛ. Altogether, that set now ∆1 := {χ,χ | χ ∈ Ψ}∪{♦ξ1,..., ♦ξk} contains {♦ξ1,..., ♦ξk, ♦♦ξ1,..., ♦♦ξk, ♦ξk+1,..., ♦ξl}. Φ,F Claim: ∆1 is AXGL -consistent. For if not, we would have:

Let us first check the formulas of type (1): ♦ξk+1,..., ♦ξl ∈ Λ Λ {χ,χ | χ ∈ Ψ}⊢AXΦ,F ¬( ^ ♦ξi) Ψ , but ¬ξk+1,..., ¬ξl ∈ Ψ . We can now show by GL i=1,...,k similar reasoning as in Case M that for each i ∈{k+1,...,l}, Φ,F But then by the same reasoning as we used before (“boxing ∆i = {χ,χ | χ ∈ Ψ}∪{ξi, ¬ξi} is AXGL -consistent, Φ,F both sides” and using GL ⊢ χ → χ) we conclude that so we can extend them to maximal AXGL -consistent sets Ψi and define Λ with Λ Λ, and corresponding Ψi := Ψi ∩ Λ Ψ ≺ Ψi {χ | χ ∈ Ψ}⊢ Φ,F ¬( ♦ξi). AXGL ^ worlds sΨi for all i ∈{k +1,...,l}. i=1,...,k We now claim that for all i ∈ {k +1,...,l}, the world ♦ ♦ This contradicts (Vi=1,...,k ξi) ∈ Ψ, which we showed sΨi is not in the top layer of the model with root sΨ. To Φ,F above. Now that we know ∆1 to be AXGL -consistent, we can derive a contradiction, suppose that it is in the top layer, so Φ,F Λ extend it by the Lindenbaum Lemma to a maximal AXGL - ⊥ ∈ Ψi . Then also ⊥ ∧ ξi ∈ Ψi, so because Ψ ≺ Ψi, we ♦  consistent set, which we call Ψ1 ⊇ ∆1. Let sΨ1 be the world have ( ⊥ ∧ ξi) ∈ Ψ. By UMBRELLA-0, we know that Λ Λ Λ corresponding to Ψ1 , with Ψ ≺ Ψ1 . ⊢AXΦ,F ♦♦⊤ ∧ ♦(⊥ ∧ ξi) → ♦♦ξi. Now we can use the same method as in Case M to GL Λ find the required direct successors of Ψ1 . Namely, for all Φ,F Also having ♦♦⊤ ∈ Ψ, we can now use Proposition 2(4) to i ∈ {1,...,k} we find maximal AXGL -consistent sets Ξi ♦♦ ♦♦ Λ conclude that ξi ∈ Ψ. Therefore, because ξi ∈ Λ, we and let sΞi be the worlds corresponding to the Ξi , with ♦♦ Λ Λ Λ also have ξi ∈ Ψ , contradicting our starting assumption Ψ1 ≺ Ξi and ξi ∈ Ξi. Λ that ♦ξi is a type (1) formula. We conclude that ⊥ 6∈ Ψi , Λ therefore, sΨi is in the middle layer. We have now handled making direct successors of Ψ for all

Let us now look for each of these sΨi with i in k +1,...,l, the formulas of type (1) and type (2). We can then finish off the which direct successors in the top layer they require. Any step-by-step construction for Case B by populating the upper Λ formulas of the form ♦χ ∈ Ψi have to be among the formulas layer U using one appropriate restriction to Λ of a maximal Λ Λ ♦ξ1,..., ♦ξk of type (2), for which ♦♦ξ1,..., ♦♦ξk ∈ Ψ . consistent set Ξ0, as follows. We note that ¬ξi ∈ Ψi for i Λ Suppose ♦ξj ∈ Ψi for some j in 1,...,k and i in k+1,...,l. in k+1,...,l, and that ⊥ ∈ Ψ1 . Let us take the following Then we can show (just like in Case M) that there is a maximal instance of the DIAMOND-(l-k) axiom scheme: Λ consistent set Xi,j with Ψi ≺ Xi,j and ξj , ⊥ ∈ Xi,j . The Λ ♦♦⊤ ∧ ^ ♦(♦⊤ ∧ ¬ξi) → world in the top layer corresponding to Xi,j will be called Λ i∈{k+1,...,l} sXi,j . Because Xi,j is finite, we can describe it by ⊥ and a finite conjunction of literals, which we represent as χi,j . For (♦⊤ → ♦( ^ ¬ξi)) ease of in the next step, let us define: i∈{k+1,...,l} Λ Now we have ♦♦⊤ ∈ Ψ . Because Ψ ≺ Ψi and ♦⊤∧¬ξi ∈ A := {hi, ji|♦ξj ∈Ψi with i ∈k +1,...,l and j ∈1,...,k} Ψi for all i in k +1,...,l, we derive that Λ For the formulas of type (2), we have ♦♦ξi ∈ Ψ . Moreover, ^ ♦(♦⊤ ∧ ¬ξi) ∈ Ψ. we have for each i ∈{1,...,k}: i∈{k+1,...,l}

GL + ⊥⊢ ♦♦ξi → ♦(⊥ ∧ ξi). Now by one more application of Proposition 2(4), we have

(♦⊤ → ♦( ^ ¬ξi)) ∈ Ψ. 13  The formulas of the form ¬ξj are in Λ because of Def. 3, clause 5. i∈{k+1,...,l} Λ 14 Because Ψ ≺ Ψj and ♦⊤ ∈ Ψj for all j in 1, k +1,...,l, we we have ♦¬χ ∈ Γ . Then in the step-by-step conclude that construction, in Case M or Case B, we have constructed Φ,F a maximal AXGL -consistent set Ξ with Γ ≺ Ξ and ♦⊤ → ♦( ¬ξi) ∈ Ψj for all j ∈{1, k +1,...,l}. Λ ^ sΓRsΞ, with ¬χ ∈ Ξ, thus ¬χ ∈ Ξ . Now by the i∈{k+1,...,l} induction hypothesis, we have Mϕ,sΞ 6|= χ, so by the Λ truth definition, Mϕ,sΓ 6|= χ. Now we can find one world sΞ0 corresponding to Ξ0 such Λ Λ that for all j in 1, k + 1,...,l, we have Ψj ≺ Ξ0 . And Λ moreover, ¬ξi ∈ Ξ0 for all i in k +1,...,l. Finally, from the truth lemma and the fact above that Λ ¬ϕ ∈ Ψ , we have Mϕ,sΨ 6|= ϕ, so we have found our We have now finished creating our finite counter-model Mϕ = counter-model. (W, R, V ), which has: Step 4 ⇒ 1 (b) • W = {sΨ}∪{sΨ1 ,sΨk+1 ,...,sΨl }∪ Now we need to show that limn→∞ µn,Φ(ϕ)=0. {sXi,j | hi, ji ∈ A}∪{sΞi | i ∈{1,...,k}}∪{sΞ0 } • R is the transitive closure of: We claim that almost surely for a sufficiently large {hsΨ,sΨi i | i ∈{1, k +1,...,l}}∪ finite Kleitman-Rothschild type frame F ′ = (W ′, R′) {hsΨi ,sXi,j i | hi, ji ∈ A}∪ {hsΨ ,sΞ i | i ∈{1,...,k}}∪ of three layers, there is a bisimulation relation Z from 1 i ′ Mϕ = (W, R, V ) defined in the (a) part of this step to F , {hsΨi ,sΞ0 i | i ∈{1, k +1,...,l}}; Λ such that the is a generated subframe F ′′ = (W ′′, R′′) • For each p ∈ Φ and sΓ ∈ W : VsΓ (p)=1 iff p ∈ Γ . ′ ′′ ′ ′′ ′ Now we can relatively easily prove a truth lemma, restricted of F , with R = R ∩ W . We will define a valuation V on ′ and define ′′ as the restriction of ′ to ′′ and to formulas from Λ, as follows. F V V W such that for all w ∈ W, w′′ ∈ W ′′: If wZw′′, then w and w′′ have the same valuation. Then, once the bisimulation Z Truth Lemma ′′ ′′ ′′ is given, suppose that sΨZs for some s ∈ W . By the For all ψ in Λ and all worlds sΓ in W : Λ bisimulation theorem [23], we have that for all ψ ∈ L(Φ), Mϕ,sΓ |= ψ iff ψ ∈ Γ . ′′ ′′ ′′ ′′ Mϕ,sΨ |= ψ ⇔ M ,s |= ψ, in particular, M ,s 6|= ϕ. Because M ′′ is a generated submodel of M ′ = (W ′, R′, V ′), Proof By induction on the construction of the formula. For ′ ′′ Λ we also have M ,s 6|= ϕ. Conclusion: limn→∞ µn,Φ(ϕ)=0. atoms p ∈ Λ, the fact that Mϕ,sΓ |= p iff p ∈ Γ follows by the definition of V . We now sketch how to define the above-claimed bisimulation Z from M to a generated subframe of such a sufficiently Induction Hypothesis: Suppose for some arbitrary χ, ξ ∈ Λ, ϕ large Kleitman-Rothschild frame F ′ = (W ′, R′). There are we have that for all worlds s∆ in W : Λ Λ three cases, corresponding to Case U, Case M, and Case M ,s∆ |= χ iff χ ∈ ∆ and M ,s∆ |= ξ iff ξ ∈ ∆ . ϕ ϕ B of the step-by-step construction of the counter-model M in Step 4 ⇒ 1 (a). One by one, we will show that Inductive step: ϕ the constructed counter-model can almost surely be mapped • Negation: Suppose ¬χ ∈ Λ. Now by the truth definition, by a bisimulation to a generated subframe of a Kleitman- Mϕ,sΓ |= ¬χ iff Mϕ,sΓ 6|= χ. By the induction Rothschild frame, as the number of nodes grows large enough. hypothesis, the latter is equivalent to χ 6∈ ΓΛ. But this in Λ turn is equivalent by Proposition 2(1) to ¬χ ∈ Γ . Case U • Conjunction: Suppose χ∧ξ ∈ Λ. Now by the truth defini- The one-point counter-model Mϕ = (W, R, V ) against ϕ, tion, Mϕ,sΓ |= χ ∧ ξ iff Mϕ,sΓ |= χ and Mϕ,sΓ |= χ. with W = {sΨ}, can be turned into a counter-model on every By the induction hypothesis, the latter is equivalent to ′ Λ Λ three-layer Kleitman-Rothschild frame F as follows. Take a χ ∈ Γ and ξ ∈ Γ , which by Proposition 2(2) is ′ Λ world u in the top layer of F , define Z by sΨZu and take equivalent to χ ∧ ξ ∈ Γ . a valuation on L(Φ) that corresponds on that world u with • Box: Suppose χ ∈ Λ. We know by the induction Λ the valuation of world sΨ in Mϕ. Then Z is a bisimulation hypothesis that for all sets ∆ in W , Mϕ,s∆ |= χ from M to a model on a one-point generated subframe Λ  ϕ iff χ ∈ ∆ . We want to show that Mϕ,sΓ |= χ iff of F ′. This world provides a counterexample showing F ′ 6|= ϕ. χ ∈ ΓΛ.  Λ For one direction, suppose that χ ∈ Γ , then by Case M definition of R, for all s∆ with sΓRs∆, we have Γ ≺ ∆ Λ The two-layer model Mϕ defined in Case M of part (a) of so χ ∈ ∆ , so by induction hypothesis, for all these this step almost surely has as a bisimilar image a generated s∆, Mϕ,s∆ |= χ. Therefore by the truth definition, subframe of a large enough Kleitman-Rothschild frame Mϕ,sΓ |= χ.  For the other direction, suppose that χ ∈ Λ but 14or, if χ is of the form ¬χ1, then ♦χ1 ∈ Γ, with ♦χ1 logically equivalent Λ χ 6∈ Γ . Then (by Definition 3 and Proposition 2(4)), to ♦¬χ; in that case we further with ♦χ1. ′ ′ ′ ′ ′′ ′′ ′′ ′′ F = (W , R , V ), as follows. Take a world m in the middle • Back: Suppose sΓZs and s R v . Then case by case, ′ layer of the Kleitman-Rothschild frame F with sufficiently one can show that there is an s∆ ∈ W such that sΓRs∆ ′′ many (at least n) successors in the top layer, say, u1,...,um and s∆Zv . with m ≥ n. Take F ′′ = (W ′′, R′′, V ′′) to be the generated upward-closed frame of F ′ with root m. Define a mapping Hereby we have sketched a proof that on almost all large enough Kleitman Rothschild frame frames, ϕ is not valid. Z by sΨZm, sΨi Zui for all i

• sΨi Zmi for i ∈{k +1,...,l}; it is always valid. For comparison, remember that for first-

• sXi,j Zui,j for all hi, ji ∈ A; order logic the difference between validity and almost sure

• sΞi Zvi for all i ∈{1,...,k}; validity is a lot starker still: Grandjean [41] proved that the ′ • sΞ0 Zv for all other v (not of the form ui,j ) with miR v decidability problem of almost sure validity in the finite is only for i ∈{k +1,...,l}; PSPACE-complete, while the validity problem on all structures

• sΞ0 Zw0; is undecidable [42], [43] and the validity problem on all finite

• sΨ1 Zm for all m in L2 other than mk+1,...,ml; structures is not even recursively enumerable [44]. • Divide the (many) w ∈ L3 such that for all i ∈ ′ VII. CONCLUSIONANDFUTUREWORK {k +1,...,l} not miR w, randomly into a partition of about equal-sized , one by one corresponding to We have proved zero-one laws for provability logic with respect to both model and frame validity. On the way, we have Z- images of each of sΞ0 , sΞi for each i ∈{1,...,k}. Now define the valuation V ′′ on F ′′ such that for all p ∈ Φ axiomatized validity in almost all relevant finite models and ′′ ′′ ′′ ′′ in almost all relevant finite frames, leading to two different and all sΓ ∈ W and s ∈ W , if sΓZs , then V ′′ (p)=1 iff s axiom systems. If the polynomial hierarchy does not collapse, VsΓ (p)=1. Finally, one can check that Z also satisfies the two other conditions for bisimulations: the two problems of ‘almost sure model/frame validity’ are ′′ less complex than ‘validity in all models/frames’. • Forth: Suppose sΓZs and sΓRs∆. Then case by case, Among finite frames in general, partial orders are pretty one can show that there is a v′′ ∈ W ′′ such that s′′R′′v′′ ′′ rare – using Fagin’s extension axioms, it is easy to show that and s∆Zv ; almost all finite frames are not partial orders. Therefore, results 15This is possible because there are at least k · (l − k)+ k elements of L3. about almost sure frame validities in the finite do not transfer between frames in general and strict partial orders. Indeed, [17] A. J. Grove, J. Y. Halpern, and D. Koller, “Asymptotic conditional the logic of frame validities on finite irreflexive partial orders probabilities: The unary case,” SIAM Journal on Computing, vol. 25, no. 1, pp. 1–51, 1996. studied here is quite different from the modal logic of the [18] K. Segerberg, “An essay in classical modal logic,” Ph.D. dissertation, validities in almost all finite frames [5], [28]. One of the most Uppsala University, 1971. interesting results in [5] is that frame validity does not transfer [19] D. J. Kleitman and B. L. Rothschild, “Asymptotic enumeration of partial orders on a finite set,” Transactions of the American Mathematical from almost all finite K-frames to the countable random frame, Society, vol. 205, pp. 205–220, 1975. although it does transfer in the other direction. In contrast, we [20] J. Henson, D. Rideout, R. D. Sorkin, and S. Surya, “Onset of the have shown that for irreflexive transitive frames, validity does asymptotic regime for (uniformly random) finite orders,” Experimental Mathematics, vol. 26, no. 3, pp. 253–266, 2017. transfer in both directions between almost all finite frames and [21] J. Y. Halpern and B. Kapron, “Zero-one laws for modal logic,” Annals the countable random irreflexive Kleitman-Rothschild frame. of Pure and Applied Logic, vol. 69, no. 2-3, pp. 157–193, 1994. [22] J. v. Benthem, “Modal correspondence theory,” Ph.D. dissertation, A. Future work University of Amsterdam, 1976. [23] ——, Modal Logic and . Naples: Bibliopolis, 1983. Currently, we are proving similar 0-1 laws for logics of [24] J. Y. Halpern and B. M. Kapron, “Zero-one laws for modal logic,” in reflexive transitive frames, such as S4 and Grzegorczyk logic, Proceedings 7th Annual IEEE Symposium on Logic in Computer Science axiomatizing both almost sure model validity and almost (LICS), 1992, pp. 369–380. [25] B. F. Chellas, Modal Logic: An Introduction. Cambridge: Cambridge sure frame validity. It turns out that Halpern and Kapron’s University Press, 1980. claim that there is a 0-1 law for S4 frame validity can still [26] J.-M. Le Bars, “The 0-1 law fails for frame satisfiability of propositional be salvaged, albeit with a different, stronger axiom system, modal logic,” in Proceedings 17th Annual IEEE Symposium on Logic in Computer Science (LICS), 2002, pp. 225–234. containing two infinite series of umbrella and diamond axioms [27] V. Goranko and M. Otto, “ of modal logic,” in Handbook similar to the ones in the current paper. Furthermore, it appears of Modal Logic, P. Blackburn, J. van Benthem, and F. Wolter, Eds. that one can do the same for logics of transitive frames that Elsevier, 2006, pp. 249–329. [28] V. Goranko, “The modal logic of almost sure frame validities in the may be neither reflexive nor irreflexive, such as K4 and weak finite,” in Proceedings of the 13th International Conference on Advances Grzegorczyk logic. in Modal Logic (AiML), N. Olivetti and R. Verbrugge, Eds. College Publications, 2020, pp. 249–268. [29] J. Y. Halpern and B. M. Kapron, “Erratum to “zero-one laws for modal [1] Y. V. Glebskii, D. I. Kogan, M. Liogon’kii, and V. Talanov, “Range and logic” [Ann. Pure Appl. Logic 69 (1994) 157–193],” Annals of Pure degree of realizability of formulas in the restricted predicate calculus,” and Applied Logic, vol. 121, no. 2-3, pp. 281–283, 2003. Cybernetics and Systems Analysis, vol. 5, no. 2, pp. 142–154, 1969. [30] R. Verbrugge, “Zero-one laws for provability logic (abstract),” in Pro- [2] K. J. Compton, “A logical approach to asymptotic combinatorics I. first ceedings of the Tenth International Congress of Logic, Methodology and order properties,” Advances in Mathematics, vol. 65, pp. 65–96, 1987. Philosophy of Science (LMPS), 1995, pp. 79–80. [3] R. Fagin, “Probabilities on finite models 1,” Journal of Symbolic Logic, [31] ——, “Zero-one laws with respect to models of provability logic and two vol. 41, no. 1, pp. 50–58, 1976. Grzegorczyk logics,” in Advances in Modal Logic 2018: Accepted Short [4] K. J. Compton, “The computational complexity of asymptotic problems Papers, G. D’Agostino and G. Bezhanishvilii, Eds., 2018, pp. 115–120. I: Partial orders,” Information and Computation, vol. 78, no. 2, pp. 108– [32] G. Boolos, The Logic of Provability. Cambridge: Cambridge University 123, 1988. Press, 1993. [5] V. Goranko and B. Kapron, “The modal logic of the countable random [33] R. Verbrugge, “Provability logic,” in The Stanford Encyclopedia of frame,” Archive for , vol. 42, no. 3, pp. 221–243, Philosophy (Fall 2017 Edition), E. N. Zalta, Ed. Stanford University, 2003. 2017. [6] R. Carnap, Logical Foundations of Probability. University of Chicago [34] C. Smory´nski, Self-Reference and Modal Logic. Berlin: Springer Press, 1950. Verlag, 1985. [7] K. J. Compton, “0-1 laws in logic and combinatorics,” in Proceedings [35] R. Carnap, Meaning and Necessity. Chicago, IL: University of Chicago 1987 NATO Advanced Study Institute Algorithms and Order, I. Rival, Press, 1947. Ed. Dordrecht: Reidel, 1988, pp. 353–383. [36] H. Gaifman, “Concerning measures in first order calculi,” Israel Journal [8] M. Kaufmann, “A counterexample to the 0-1 law for existential monadic of Mathematics, vol. 2, no. 1, pp. 1–18, 1964. second-order logic,” in CLI Internal Note 32. Comp. Logic Inc., 1987. [37] J. P. Burgess, “Basic tense logic,” in Handbook of , [9] P. Kolaitis and M. Vardi, “The decision problem for the probabilities of F. Guenthner and D. Gabbay, Eds. Springer, 1984, pp. 89–133. higher-order properties,” in Proceedings 19th Annual ACM Symposium [38] D. de Jongh, F. Veltman, and R. Verbrugge, “Completeness by con- on the Theory of Computing (STOC), 1987, pp. 425–435. struction for tense logics of linear time,” in Liber Amicorum for Dick [10] P. G. Kolaitis and M. Y. Vardi, “0–1 laws and decision problems for de Jongh, L. Afanasiev and M. Marx, Eds. Amsterdam: Institute of fragments of second-order logic,” Information and Computation, vol. 87, Logic, Language and Computation, 2004. no. 1-2, pp. 302–338, 1990. [39] R. Verbrugge, “Does Solovay’s completeness theorem extend to bounded [11] L. Pacholski and W. Szwast, “The 0-1 law fails for the class of existential arithmetic?” Master’s thesis, University of Amsterdam, 1988. second order G¨odel sentences with equality,” in Proceedings 30th Annual [40] J. J. Joosten, J. M. Rovira, L. Mikec, and M. Vukovi´c, “An overview Symposium on Foundations of Computer Science (FOCS). IEEE, 1989, of generalised Veltman semantics,” arXiv preprint arXiv:2007.04722, pp. 160–163. 2020. [12] P. G. Kolaitis and M. Y. Vardi, “Infinitary logics and 0–1 laws,” [41] E. Grandjean, “Complexity of the first-order theory of almost all finite Information and Computation, vol. 98, no. 2, pp. 258–294, 1992. structures,” Information and Control, vol. 57, pp. 180–204, 1983. [13] Y. Gurevich, “Zero-one laws,” in Current Trends In Theoretical Com- [42] A. Church, “A note on the ,” The Journal of puter Science. World Scientific, 1993, pp. 293–309. Symbolic Logic, vol. 1, no. 1, pp. 40–41, 1936. [14] J. Y. Halpern, “From statistical knowledge bases to degrees of belief: An [43] A. Turing, “On computable numbers, with an application to the Entschei- overview,” in Proc. 25th ACM SIGMOD-SIGACT-SIGART Symposium dungsproblem,” Proceedings of the London Mathematical Society, vol. 2, on Principles of Database Systems. ACM, 2006, pp. 110–113. no. 1, pp. 230–265, 1937. [15] L. Libkin, Elements of . Berlin: Springer, 2013. [44] B. A. Trakhtenbrot, “Impossibility of an algorithm for the decision [16] F. A. Zaid, A. Dawar, E. Gr¨adel, and W. Pakusa, “Definability of problem in finite classes,” Doklady Akademii Nauk SSSR, vol. 70, no. 4, summation problems for abelian groups and semigroups,” in Proceedings pp. 569–572, 1950, (in Russian). 32nd Annual IEEE Symposium on Logic in Computer Science (LICS). IEEE, 2017, pp. 1–11.