Zero-one laws for provability logic: Axiomatizing validity in almost all models and almost all frames
aa r X i v : . [ c s . L O ] F e b Zero-one laws for provability logic: Axiomatizingvalidity in almost all models and almost all frames
Rineke VerbruggeDepartment of Artificial Intelligence, University of Groningen, e-mail [email protected]
Abstract —It has been shown in the late 1960s that each formulaof first-order logic without constants and function symbols obeysa zero-one law: As the number of elements of finite modelsincreases, every formula holds either in almost all or in almostno models of that size. Therefore, many properties of models,such as having an even number of elements, cannot be expressedin the language of first-order logic. Halpern and Kapron provedzero-one laws for classes of models corresponding to the modallogics K, T, S4, and S5.In this paper, we prove zero-one laws for provability logicwith respect to both model and frame validity. Moreover, weaxiomatize validity in almost all relevant finite models and inalmost all relevant finite frames, leading to two different axiomsystems. In the proofs, we use a combinatorial result by Kleitmanand Rothschild about the structure of almost all finite partialorders. On the way, we also show that a previous result byHalpern and Kapron about the axiomatization of almost sureframe validity for S4 is not correct. Finally, we consider thecomplexity of deciding whether a given formula is almost surelyvalid in the relevant finite models and frames.
I. I
NTRODUCTION
In the late 1960s, Glebskii and colleagues proved that first-order logic without function symbols satisfies a zero-one law,that is, every formula is either almost always true or almostalways false in finite models [1]. More formally, let L be alanguage of first-order logic and let A n ( L ) be the set of all labelled L -models with universe { , . . . , n } . Now let µ n ( σ ) be the fraction of members of A n ( L ) in which σ is true, thatis, µ n ( σ ) = | M ∈ A n ( L ) : M | = σ || A n ( L ) | Then for every σ ∈ L , lim n →∞ µ n ( σ ) = 1 or lim n →∞ µ n ( ϕ ) = 0 . This was also proved later but independently by Fagin [3];Carnap had already proved the zero-one law for first-orderlanguages with only unary predicate symbols [6] (see [7],[5] for nice historical overviews of zero-one laws). Later, The distinction between labelled and unlabelled probabilities was intro-duced by Compton [2]. The unlabelled count function counts the numberof isomorphism types of size n , while the labelled count function countsthe number of labelled structures of size n , that is, the number of relevantstructures on the universe { , . . . , n } . It has been proved both for the generalzero-one law and for partial orders that in the limit, the distinction betweenlabelled and unlabelled probabilities does not make a difference for zero-one laws. This is because almost all relevant structures (in our case partialorders) are automorphism-rigid in the sense that their only automorphism isthe identity [3], [2], [4]. Per finite size n , labelled probabilities are easier towork with than unlabelled ones [5], so we will use them in the rest of thearticle. Kaufmann showed that monadic existential second-order logicdoes not satisfy a zero-one law [8]. Kolaitis and Vardi havemade the border more precise by showing that a zero-one lawholds for the fragment of existential second-order logic ( Σ )in which the first-order part of the formula belongs to theBernays-Sch¨onfinkel class ( ∃ ∗ ∀ ∗ prefix) or the Ackermannclass ( ∃ ∗ ∀∃ ∗ prefix) [9], [10]; however, no zero-one lawholds for any other class, for example, the G¨odel class ( ∀ ∃ ∗ prefix) [11]. Blass, Gurevich and Kozen have proved that azero-one law does hold for LFP(FO), the extension of first-order logic with a least fixed-point operator [12].The above zero-one laws and other limit laws have foundapplications in database theory [13], [14] and algebra [15].In AI, there has been great interest in asymptotic conditionalprobabilities and their relation to default reasoning and degreesof belief [16], [17], [14].In this article, we focus on zero-one laws for a modallogic that imposes structural restrictions on its models, namely,provability logic, which is sound and complete with respectto finite strict (irreflexive) partial orders [18].The zero-one law for first-order logic also holds whenrestricted to partial orders, both reflexive and irreflexive ones,as proved by Compton [4]. To prove this, he used a surprisingcombinatorial result by Kleitman and Rothschild [19] onwhich we will also rely for our results. Let us give a shortsummary. A. Kleitman and Rothschild’s result on finite partial orders
Kleitman and Rothschild proved that with asymptotic prob-ability 1, finite partial orders have a very special structure:There are no chains u < v < w < z of more than threeobjects and the structure can be divided into three levels: • L , the set of minimal elements; • L , the set of elements immediately succeeding elementsin L ; • L , the set of elements immediately succeeding elementsin L .Moreover, in partial orders of size n , the sizes of these setstend to n for both L and L while the size of the middlelayer L tends to n . As n increases, each element in L hasas immediate successors asymptotically half of the elementsof L and each element in L has as immediate predecessorssymptotically half of the elements of L [19]. Kleitman andRothschild’s theorem holds both for reflexive (non-strict) andfor irreflexive (strict) partial orders. In addition, Halpern andKapron [21],[22, Theorem 4.14] proved that almost surely,every reflexive transitive order is in fact a partial order, sothe above result also holds for finite frames with reflexivetransitive relations.
B. Zero-one laws for modal logics: Almost sure model validity
In order to describe the known results about zero-one lawsfor modal logics with respect to the relevant classes of modelsand frames, we first give reminders of some well-knowndefinitions and results.Let
Φ = { p , . . . , p k } be a finite set of propositional atoms and let L (Φ) be the modal language over Φ , inductivelydefined as the smallest set closed under:1) If p ∈ Φ , then p ∈ L (Φ) .2) If A ∈ L (Φ) and B ∈ L (Φ) , then also ¬ A ∈ L (Φ) , (cid:3) A ∈ L (Φ) , ♦ ( ϕ ) ∈ L (Φ) , ( A ∧ B ) ∈ L (Φ) , ( A ∨ B ) ∈ L (Φ) , and ( A → B ) ∈ L (Φ) .A frame is a pair F = ( W, R ) where W is a non-empty setof worlds and R is a binary accessibility relation. A model M = ( W, R, V ) consists of a frame ( W, R ) and a valuationfunction V that assigns to each atomic proposition in eachworld a truth value V w ( p ) , which can be either 0 or 1. Thetruth definition is as usual in modal logic, including the clause: M, w | = (cid:3) ϕ if and only iffor all w ′ such that wRw ′ , M, w ′ | = ϕ. A formula ϕ is valid in model M = ( W, R, V ) (notation M | = ϕ ) iff for all w ∈ W , M, w | = ϕ .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 ) .Let M n, Φ be the set of finite Kripke models over Φ with setof worlds W = { , . . . , n } . We take ν n, Φ to be the uniformprobability distribution on M n, Φ . Let ν n, Φ ( ϕ ) be the measurein M n, Φ of the set of Kripke models in which ϕ is valid.Let F n, Φ be the set of finite Kripke frames with set of worlds W = { , . . . , n } . We take µ n, Φ to be the uniform probabilitydistribution on F n . Let µ n, Φ ( ϕ ) be the measure in F n of theset of Kripke frames in which ϕ is valid.Halpern and Kapron proved that every formula ϕ in modallanguage L (Φ) is either valid in almost all models (“almostsurely true”) or not valid in almost all models (“almost surelyfalse”) [22, Corollary 4.2]:Either lim n →∞ ν n, Φ ( ϕ ) = 0 or lim n →∞ ν n, Φ ( ϕ ) = 1 . Interestingly, it was recently found experimentally that for smaller n thereare strong oscillations, while the behavior appears to stabilize only around n = 45 [20]. In the rest of this paper in the parts on almost sure model validity, we take Φ to be finite, although the results can be extended to enumerably infinite Φ by the methods described in [22], [17]. In fact, this zero-one law for models already follows from thezero-one law for first-order logic [1], [3] by Van Benthem’stranslation method [23], [24]. As reminder, let ∗ be given by: • p ∗ i = P i ( x ) for atomic sentences p i ∈ Φ ; • ( ¬ ϕ ) ∗ = ¬ ϕ ∗ ; • ( ϕ ∧ ψ ) ∗ = ( ϕ ∗ ∧ ψ ∗ ) (and similarly for the other binaryconnectives); • ( (cid:3) ϕ ) ∗ = ∀ y ( Rxy → ϕ ∗ [ y/x ]) .Van Benthem mapped each Kripke model M = ( W, R, V ) toa classical model M ∗ with as objects the worlds in W andthe obvious binary relation R , while for each atom p i ∈ Φ , P i = { w ∈ W | M, w | = p i } = { w ∈ W | V w ( p i ) = 1 } .Van Benthem then proved that for all ϕ ∈ L (Φ) , M | = ϕ iff M ∗ | = ∀ x ϕ ∗ [24]. Halpern and Kapron [21], [22] showed thata zero-one law for modal models immediately follows by VanBenthem’s result and the zero-one law for first-order logic.By Compton’s above-mentioned result that the zero-onelaw for first-order logic holds when restricted to the partialorders [4], this modal zero-one law can also be restrictedto finite models on reflexive or irreflexive partial orders, sothat a zero-one law for finite models of provability logicimmediately follows. However, one would like to prove astronger result and axiomatize the set of formulas ϕ for which lim n →∞ ν n, Φ ( ϕ ) = 1 . Also, Van Benthem’s result does notallow proving zero-one laws for classes of frames instead ofmodels: We have F | = ϕ iff F ∗ | = ∀ P . . . ∀ P n ∀ xϕ ∗ , but thelatter formula is not necessarily a (negation of) a formula in Σ with its first-order part in one of the Bernays-Sch¨onfinkelor Ackermann classes (see [22]).Halpern and Kapron [21], [22] aimed to fill in the above-mentioned gaps for the modal logics K , T , S4 and S5 . Theyproved zero-one laws for the relevant classes of finite modelsfor these logics. For all four, they axiomatized the classesof sentences that are almost surely true in the relevant finitemodels. C. The quest for zero-one laws for frame validity
Halpern and Kapron’s paper also contains descriptionsof four zero-one laws with respect to the classes of finiteframes corresponding to K , T , S4 and S5 . [22, Theorem5.1 and Theorem 5.15]: Either lim n →∞ µ n, Φ ( ϕ ) = 0 or lim n →∞ µ n, Φ ( ϕ ) = 1 .They proposed four axiomatizations for the sets of formulasthat would be almost always valid in the corresponding fourclasses of frames [22]. However, almost 10 years later, LeBars surprisingly proved them wrong with respect to the zero-one law for K -frames [25]. By proving that the formula q ∧ ¬ p ∧ (cid:3)(cid:3) (( p ∨ q ) → ¬ ♦ ( p ∨ q )) ∧ (cid:3)♦ p does not have an asymptotic probability, he showed that in fact no zero-one law holds with respect to all finite Kripke frames.Doubt had already been cast on the zero-one law for framevalidity by Goranko and Kapron, who proved that the formula ¬ (cid:3)(cid:3) ( p ↔ ¬ ♦ p ) fails in the countably infinite random frame,while it is almost surely valid in K -frames [5]. (See also26, Section 9.5]). Currently, the problem of axiomatizing themodal logic of almost sure frame validities for finite K -framesappears to be open. As a reaction to Le Bars’ counter-example, Halpern andKapron [28] published an erratum, in which they showedexactly where their erstwhile proof of [22, Theorem 5.1] hadgone wrong. It may be that the problem they point out alsoinvalidates their similar proof of the zero-one law with respectto finite reflexive frames, corresponding to T [22, Theorem5.15 a]. However, with respect to frame validity for T -frames,as far as we know, no counterexample to a zero-one law hasyet been published and Le Bars’ counterexample cannot easilybe adapted to reflexive frames; therefore, the situation remainsunsettled for T . D. Halpern and Kapron’s axiomatization for almost sureframe validities for S4 fails
Unfortunately, Halpern and Kapron’s proof of the 0-1 lawfor reflexive, transitive frames and the axiomatization of thealmost sure frame validities for reflexive, transitive frames[22, Theorem 5.16] turn out to be incorrect as well, asfollows. Halpern and Kapron introduce the axiom DEP2 ′ and they axiomatize almost-sure frame validities in reflexivetransitive frames by S4 +DEP2 ′ [22, Theorem 5.16], whereDEP2 ′ is: ¬ ( p ∧ ♦ ( ¬ p ∧ ♦ ( p ∧ ♦ ¬ p ))) . The axiom DEP2 ′ precludes R -chains tRuRvRw of morethan three different states. Proposition 1.
Suppose
Φ = { p , p } . Now take the followingsentence χ : χ := ( p ∧ ♦ ( ¬ p ∧ p ∧ (cid:3) ( p → p ))) → (cid:3) (( ¬ p ∧ ♦ p ) → ♦ p ) Then S4 +DEP2 ′ χ but lim n →∞ µ n, Φ ( χ ) = 1 Proof.
It is easy to see that S4 +DEP2 ′ χ by taking thefive-point reflexive transitive frame of Figure 1, where M, w | = ( p ∧ ♦ ( ¬ p ∧ p ∧ (cid:3) ( p → p )) but M, w = ( ¬ p ∧ ♦ p ) → ♦ p ) , so M, w = (cid:3) (( ¬ p ∧ ♦ p ) → ♦ p ) . However, χ is true in almost all reflexive Kleitman-Rothschild frames: If a world in the bottom layer has twosuccessors in the middle layer, then there is a world in the We will show in this paper that for partial orders, almost-sure framevalidity in the finite does transfer to validity in the corresponding countablerandom Kleitman-Rothschild frame, and that the validities are quite differentfrom those for almost all K frames (see Section V). For up to 2006: see [26]; for more recently: [27] . Joe Halpern and Bruce Kapron (personal communication) and Jean-MarieLe Bars (personal communication) confirmed the current non-settledness ofthe problem for T . The author of this paper discovered the counter-example after a colleaguehad pointed out that the author’s earlier attempt at a proof of the 0-1 lawfor provability logic, inspired by Halpern and Kapron’s [22] axiomatiation,contained a serious gap. w p , ¬ p w ¬ p , p w ¬ p , ¬ p w p , p w p , ¬ p Fig. 1. Counter-model showing that the formula χ , namely, ( p ∧ ♦ ( ¬ p ∧ p ∧ (cid:3) ( p → p ))) → (cid:3) (( ¬ p ∧ ♦ p ) → ♦ p ) does not hold in w of this three-layer model. The relationin the model is the reflexive transitive closure of the relationrepresented by the arrows. top layer to which both of these middle worlds have access(the diamond property); this is because each extension axiomfrom Compton’s theory T as [4] holds in almost all finitereflexive transitive frames (similar to Proposition 4 of thecurrent paper).Therefore, the axiom system given in [22, Theorem 5.16]is not complete with respect to almost-sure frame validitiesfor finite reflexive transitive orders.Fortunately, there is a way to mend the situation and stillobtain an axiom system that is sound and complete withrespect to almost sure S GL in Section V. E. Almost sure model validity does not coincide with almostsure frame validity
Interestingly, whereas for full classes of frames, ‘validityin all finite models’ coincides with ‘validity in all finiteframes’ of the class, this is not the case for ‘almost surevalidity’. In particular, for both the class of reflexive transitiveframes ( S
4) and the class of reflexive transitive symmetricframes ( S almost all finite models’ than ‘valid in almost all finiteframes’ of the appropriate kinds. Our work has been greatlyinspired by Halpern and Kapron’s paper [22] and we alsouse some of the previous results that they applied, notablythe above-mentioned combinatorial result by Kleitman andRothschild about finite partial orders.The rest of this paper is structured as follows. In Section II,we give a brief reminder of the axiom system and semanticsof provability logic. In the central Sections III, IV and V,we show why provability logic obeys zero-one laws both withrespect to its models and with respect to its frames. We providetwo axiom systems characterizing the formulas that are almostalways valid in the relevant models, respectively almost alwaysvalid in the relevant frames. When discussing almost sureframe validity, we will investigate both the almost sure validityin finite frames and validity in the countable random frame,and show that there is transfer between them. Section VIprovides a sketch of the complexity of the decidability prob-lems of almost sure model and almost sure frame validity forrovability logic. Finally, Section VII presents a conclusionand some questions for future work.The result on models in Section III was proved 25 yearsago, and presented in the 1995 LMPS presentation [29], butthe proofs have not been published before in an archival venue.The results about almost sure frame validities for GL are new,as well as the counter-example against the axiomatization byHalpern and Kapron of almost sure S ROVABILITY LOGIC
In this section, a brief reminder is provided about theprotagonist of this paper: the provability logic GL , namedafter G¨odel and L¨ob. As axioms, it contains all axiom schemesfrom K and the extra scheme GL. Here follows the full set ofaxiom schemes of GL :All (instances of) propositional tautologies (A1) (cid:3) ( ϕ → ψ ) → ( (cid:3) ϕ → (cid:3) ψ ) (A2) (cid:3) ( (cid:3) ϕ → ϕ ) → (cid:3) ϕ (GL)The rules of inference are modus ponens and necessitation:if GL ⊢ ϕ → ψ and GL ⊢ ϕ , then GL ⊢ ϕ .if GL ⊢ ϕ , then GL ⊢ (cid:3) ϕ .Note that GL ⊢ (cid:3) ϕ → (cid:3)(cid:3) ϕ , which was first proved byDe Jongh and Sambin [30], [31], but that the reflexivityaxiom (cid:3) ϕ → ϕ does not follow. Indeed, Segerberg provedin 1971 that provability logic is sound and complete withrespect to all transitive, converse well-founded frames (i.e.,for each non-empty set X , there is an R-greatest elementof X ; or equivalently: there is no infinitely ascending se-quence x Rx Rx Rx , . . . ). Segerberg also proved complete-ness with respect to all finite, transitive, irreflexive frames [18].The latter soundness and completeness result will be relevantfor our purposes. For more information on provability logic,see, for example, [32], [30], [31].In the next three sections, we provide axiomatizations, firstfor almost sure model validity and then for almost sure framevalidity, with respect to the relevant finite frames correspond-ing to GL , namely the irreflexive transitive ones.For the proofs of the zero-one laws for almost sure modeland frame validity, we will need completeness proofs of therelevant axiomatic theories – let us refer to such a theory by S for the moment – with respect to almost sure model validityand with respect to almost sure frame validity. Here we willuse Lindenbaum’s lemma and maximal S -consistent sets offormulas. For such sets, the following useful properties hold,as usual [18], [33]: Proposition 2.
Let Θ be a maximal S -consistent set of for-mulas in L (Φ) . Then for each pair of formulas ϕ, ψ ∈ L (Φ) : ϕ ∈ Θ iff ¬ ϕ Θ ; ( ϕ ∧ ψ ) ∈ Θ ⇔ ϕ ∈ Θ and ψ ∈ Θ ; if ϕ ∈ Θ and ( ϕ → ψ ) ∈ Θ then ψ ∈ Θ ; if Θ ⊢ S ϕ then ϕ ∈ Θ . III. V
ALIDITY IN ALMOST ALL FINITE IRREFLEXIVETRANSITIVE MODELS
The axiom system AX Φ , MGL has the same axioms and rulesas GL (see Section II) plus the following axioms: (cid:3)(cid:3)(cid:3) ⊥ (T3) ♦ ⊤ → ♦ A (C1) ♦♦ ⊤ → ♦ ( B ∧ ♦ C ) (C2)In the axiom schemes C1 and C2, the formulas A , B and C all stand for consistent conjunctions of literals over Φ .These axiom schemes have been inspired by Carnap’sconsistency axiom: ♦ ϕ for any ϕ that is a consistent propo-sitional formula [34], which has been used by Halpern andKapron [22] for axiomatizing almost sure model validities for K -models.Note that AX Φ , MGL is not a normal modal logic, because onecannot substitute just any formula for
A, B, C ; for example,substituting p ∧ ¬ p for A in C1 would make that formulaequivalent to ¬ ♦ ⊤ , which is clearly undesired. However, eventhough AX Φ , MGL is not closed under uniform substitution, it isstill a propositional theory, in the sense that it is closed undermodus ponens.
Example 1.
For
Φ = { p , p } , the axiom scheme C1 boilsdown to the following four axioms: ♦ ⊤ → ♦ ( p ∧ p ) (1) ♦ ⊤ → ♦ ( p ∧ ¬ p ) (2) ♦ ⊤ → ♦ ( ¬ p ∧ p ) (3) ♦ ⊤ → ♦ ( ¬ p ∧ ¬ p ) (4) The axiom scheme C2 covers 16 axioms, corresponding tothe possible choices of positive or negative literals, ascaptured by the following scheme, where “ [ ¬ ] ” is shorthandfor a negation being present or absent at the current location: ♦♦ ⊤ → ♦ ([ ¬ ] p ∧ [ ¬ ] p ∧ ♦ ([ ¬ ] p ∧ [ ¬ ] p )) The following definition of the canonical asymptotic Kripkemodel over a finite set of propositional atoms Φ is based onthe set of propositional valuations on Φ , namely, the functions v from the set of propositional atoms Φ to the set of truthvalues { , } . Definition 1.
Define M Φ GL = ( W, R, V ) , the canonical asymp-totic Kripke model over Φ , with W, R, V as follows: W = { b v | v a propositional valuation on Φ } ∪{ m v | v a propositional valuation on Φ } ∪{ u v | v a propositional valuation on Φ } R = {h b v , m v ′ i | v, v ′ propositional valuations on Φ } ∪{h m v , u v ′ i | v, v ′ propositional valuations on Φ } ∪{h b v , u v ′ i | v, v ′ propositional valuations on Φ } ; andfor all p i ∈ Φ and all propositional valuations v on Φ , themodal valuation V is defined by V b v ( p i ) = 1 iff v ( p i ) = 1 ; V m v ( p i ) = 1 iff v ( p i ) = 1 ; V u v ( p i ) = 1 iff v ( p i ) = 1 . If Φ were enumerably infinite, the definition could be adapted so thatprecisely those propositional valuations are used that make only finitely manypropositional atoms true, see also [22]. ote that the names of the worlds have been chosen formnemonic reasons to correspond to the Bottom ( b v ), Middle( m v ), and Upper ( u v ) layers.For the proof of the zero-one law for model validity, we willneed a completeness proof of AX Φ , MGL with respect to almostsure model validity, including use of Lindenbaum’s lemma andProposition 2, applied to AX Φ , MGL .The zero-one law for model validity will follow straightfor-wardly from the following theorem:
Theorem 1.
For every formula ϕ ∈ L (Φ) , the following areequivalent: M Φ GL | = ϕ ; AX Φ , MGL ⊢ ϕ ; lim n →∞ ν n, Φ ( ϕ ) = 1 ; lim n →∞ ν n, Φ ( ϕ ) = 0 .Proof. We show a circle of implications. Let ϕ ∈ L (Φ) . ⇒ By contraposition. Suppose that AX Φ , MGL ϕ , then ¬ ϕ is AX Φ , MGL -consistent. By Lindenbaum’s lemma, we can extend {¬ ϕ } to a maximal AX Φ , MGL -consistent set ∆ over Φ . We usea standard canonical model construction; here, we illustratehow that works for the finite set Φ = { p , p } , but the methodworks for any finite Φ = { p , . . . , p k } . We define the Kripkemodel
M C Φ GL = ( W ′ , R ′ , V ′ ) , which has: • W ′ = { w Γ | Γ is maximal AX Φ , MGL -consistent,based on Φ } . • R ′ = {h w Γ , w ∆ i | w Γ , w ∆ ∈ W ′ andfor all (cid:3) ψ ∈ Γ , it holds that ψ ∈ ∆ } • For each w Γ ∈ W ′ : V ′ w Γ ( p ) = 1 iff p ∈ Γ Because the worlds of this model correspond to the maximal AX Φ , MGL -consistent sets, it is easy to see that all worlds w Γ ∈ W ′ can be distinguished into three kinds, exhaustively andwithout overlap:U (cid:3) ⊥ ∈ Γ ; there are exactly four maximal consistentsets Γ of this form, determined by which of the fourconjunctions of relevant literals [ ¬ ] p ∧ [ ¬ ] p is anelement. These comprise the upper level U of themodel.M ¬ (cid:3) ⊥ ∈ Γ and (cid:3)(cid:3) ⊥ ∈ Γ ; there are exactly fourmaximal consistent sets Γ of this form, determinedby which of the four conjunctions of relevant literals [ ¬ ] p ∧ [ ¬ ] p is an element. By axiom C1 and Propo-sition 2, all these four maximal consistent sets con-tain the four formulas of the form ♦ ([ ¬ ] p ∧ [ ¬ ] p ) ;by definition of R ′ and using the fact that (cid:3)(cid:3) ⊥ ∈ Γ ,this means that all the four worlds in this middle levelM will have access to all the four worlds in the upperlevel U.B ¬ (cid:3) ⊥ ∈ Γ and ¬ (cid:3)(cid:3) ⊥ ∈ Γ and (cid:3)(cid:3)(cid:3) ⊥ ∈ Γ ; thereare exactly four maximal consistent sets Γ of thisform, determined by which of the four conjunctions For adapting to the enumerably infinite case, see [22, Theorem 4.15]. of relevant literals [ ¬ ] p ∧ [ ¬ ] p is an element.Because ♦♦ ⊤ ∈ Γ , by axiom C2 and Proposition 2,all these four maximal consistent sets contain the 16formulas ♦ ([ ¬ ] p ∧ [ ¬ ] p ∧ ♦ ([ ¬ ] p ∧ [ ¬ ] p )) . Bythe definition of R ′ , this means that all four worldsin this bottom level B will have direct access to allthe four worlds in middle level M as well as accessin two steps to all four worlds in upper level U.Note that R ′ is transitive because AX Φ , MGL extends GL , so forall maximal consistent sets Γ and all formulas ψ ∈ L (Φ) , wehave that (cid:3) ψ → (cid:3)(cid:3) ψ ∈ Γ . Also R ′ is irreflexive: Becauseeach world contains either (cid:3) ⊥ and ¬⊥ (for U), or (cid:3)(cid:3) ⊥ and ¬ (cid:3) ⊥ (for M), or (cid:3)(cid:3)(cid:3) ⊥ and ¬ (cid:3)(cid:3) ⊥ (for B), by definitionof R ′ , none of the worlds has a relation to itself.The next step is to prove by induction that a truth lemma holds: For all ψ in the language L (Φ) and for all maximal AX Φ , MGL -consistent sets Γ , the following holds: M C Φ GL , w Γ | = ψ iff ψ ∈ Γ .For atoms, this follows by the definition of V ′ . The stepsfor the propositional connectives are as usual, using theproperties of maximal consistent sets (see Proposition 2).For the (cid:3) -step, let Γ be a maximal AX Φ , MGL -consistent setand let us suppose as induction hypothesis that for somearbitrary formula χ , for all maximal AX Φ , MGL -consistent sets Π , M C Φ GL , w Π | = χ iff χ ∈ Π . We want to show that M C Φ GL , w Γ | = (cid:3) χ iff (cid:3) χ ∈ Γ .For the direction from right to left, suppose that (cid:3) χ ∈ Γ ,then by definition of R ′ , for all Π with w Γ R ′ w Π , we have χ ∈ Π , so by induction hypothesis, M C Φ GL , w Π | = χ . Therefore,by the truth definition, M C Φ GL , w Γ | = (cid:3) χ .For the direction from left to right, let us use contrapositionand suppose that (cid:3) χ Γ . Now we will show that the set { ξ | (cid:3) ξ ∈ Γ } ∪ {¬ χ } is AX Φ , MGL -consistent. For otherwise,there would be some ξ , . . . , ξ n for which (cid:3) ξ , . . . , (cid:3) ξ n ∈ Γ such that ξ , . . . , ξ n ⊢ AX Φ , MGL χ , so by necessitation, A2, andpropositional logic, (cid:3) ξ , . . . , (cid:3) ξ n ⊢ AX Φ , MGL (cid:3) χ , therefore bymaximal consistency of Γ and Proposition 2(iv), also (cid:3) χ ∈ Γ ,contradicting our assumption.Therefore, by Lindenbaum’s lemma there is a maximalconsistent set Π ⊇ { ξ | (cid:3) ξ ∈ Γ } ∪ {¬ χ } . It is clear bydefinition of R ′ that w Γ R ′ w Π , and by induction hypothesis, M C Φ GL , w Π | = ¬ χ , i.e., M C Φ GL , w Π = χ , so by the truthdefinition, M C Φ GL , w Γ = (cid:3) χ . This finishes the inductiveproof of the truth lemma.Finally, from the truth lemma and the fact stated at thebeginning of the proof of 2 ⇒ ¬ ϕ ∈ ∆ , we have that M C Φ GL , w ∆ = ϕ , so we have found our counter-model.It is clear that, with its three layers (Upper, Middle, andBottom) of four worlds each, corresponding to each consistent v p , p b v p , ¬ p b v ¬ p , p b v ¬ p , ¬ p m v p , p m v p , ¬ p m v ¬ p , p m v ¬ p , ¬ p u v p , p u v p , ¬ p u v ¬ p , p u v ¬ p , ¬ p Fig. 2. The canonical asymptotic Kripke model M Φ GL = ( W, R, V ) for Φ = { p , p } , defined in Definition 1. The accessibility relation is the transitiveclosure of the relation given by the arrows drawn in the picture. The four relevant valuations are v , v , v , v , given by v ( p ) = v ( p ) = 1 ; v ( p ) =1 , v ( p ) = 0 ; v ( p ) = 0 , v ( p ) = 1 ; v ( p ) = v ( p ) = 0 . conjunction of literals, the model M C Φ GL that we constructin the completeness proof above is isomorphic to thecanonical asymptotic Kripke model M Φ GL of Definition 1; for Φ = { p , p } , the latter model is pictured in Figure 2. ⇒ Suppose that AX Φ , MGL ⊢ ϕ . We will show that the axiomsof AX Φ , MGL hold in almost all irreflexive transitive Kleitman-Rothschild models of depth 3 (see Subsection I-A). First,it is immediate that GL is sound with respect to all finiteirreflexive transitive models, that axiom (cid:3)(cid:3)(cid:3) ⊥ is soundwith respect to those of depth 3, and that almost sure modelvalidity is closed under MP and Generalization. It remains toshow the almost sure model validity of axiom schemes C1 andC2 over finite irreflexive models of the Kleitman-Rothschildvariety.We will now show that the ‘Carnap-like’ axiom C1, namely ♦ ⊤ → ♦ A where A is a consistent conjunction of literalsover Φ , is valid in almost all irreflexive transitive models ofdepth 3 of the Kleitman-Rothschild variety. Let us supposethat Φ = { p , . . . , p k } , so there are k possible valuations.Let us consider a state s in such a model of n elementswhere ♦ ⊤ holds; then, being a Kleitman-Rothschild model, s has as direct successors approximately half of the states inthe directly higher layer, which contains asymptotically atleast of the model’s states. So s has asymptotically at least · n direct successors. The probability that a given state t is a direct successor of s with the right valuation to make A true is therefore at least · k = k +3 . Thus, the probabilitythat s does not have any direct successors in which A holdsis at most (1 − k +3 ) n . Therefore, the probability that thereis at least one s in a Kleitman-Rothschild model not havingany direct successors satisfying A is at most n · (1 − k +3 ) n .It is known that lim n →∞ n · (1 − k +3 ) n = 0 (cf [22]), soC1 is valid in almost all Kleitman-Rothschild models, i.e., lim n →∞ ν n, Φ ( ♦ ⊤ → ♦ A ) = 1 . Similarly, we can show that axiom C2, namely ♦♦ ⊤ → ♦ ( B ∧ ♦ C ) where B, C are consistent conjunctions of literalsover Φ , is valid in almost all irreflexive transitive Kleitman-Rothschild models of depth 3. Let Φ = { p , . . . , p k } . Again,let us consider a state s in such a model of n elementswhere ♦♦ ⊤ holds, then s is in the bottom of the three layers;therefore, the model being of Kleitman-Rothschild type, s hasas direct successors approximately half of the states in themiddle layer, which contains asymptotically at least of themodel’s states. So s has asymptotically at least · n directsuccessors.The probability that a given state t is a direct successor of s with the right valuation to make B true is therefore at least · k = k +2 . Similarly, given such a t , the probability thata given state t ′ in the top layer is a direct successor of t inwhich C holds is asymptotically at least k +2 · k +3 = k +5 Therefore, the probability that for the given s there are no t, t ′ with sRtRt ′ with B true at t and C true at t ′ is at most (1 − k +5 ) n . Summing up, the probability that there is atleast one s in a Kleitman-Rothschild model not having anypair of successors sRtRt ′ with B true at t and C true at t ′ isat most n · (1 − k +5 ) n . Again, lim n →∞ n · (1 − k +5 ) n = 0 ,so C2 holds in almost all Kleitman-Rothschild models, i.e. lim n →∞ ν n, Φ ( ♦♦ ⊤ → ♦ ( B ∧ ♦ C )) = 1 . ⇒ Straightforward, because = 1 . ⇒ By contraposition. Suppose as before that
Φ = { p , . . . , p k } .Now suppose that the canonical asymptotic Kripke model M Φ GL = ϕ for some ϕ ∈ L (Φ) , for example, M Φ GL , s = ϕ ,for some s ∈ W . We claim that this counter-model to ϕ can be copied into almost every Kleitman-Rothschild modelas they grow large enough, which we will now proceedto show. Consider a large finite Kleitman-Rothschild typeirreflexive transitive model M ′ = ( W ′ , R ′ , V ′ ) of three layers.Asymptotically, we will be able to find in M ′ a world s ′ thats situated at the same layer (top, middle or bottom) as thelayer where s is in M Φ GL and that has the same valuation forall atoms p , . . . , p k . Let us look at the three cases, layer bylayer. If s is in the top layer, this already ensures that M Φ GL , s and M ′ , s ′ satisfy the same formulas (including (cid:3) ⊥ ). If s is in the middle layer, we only need to show that for largeenough M ′ , there will be an s ′ in the middle layer such that s ′ has access to at least k different states in the top layer of M ′ that each correspond to one of the k possible valuationson Φ . Also in this case, M Φ GL , s and M ′ , s ′ satisfy the sameformulas. Finally, if s is in the bottom layer, then for almostall large enough M ′ of Kleitman-Rothschild form, we canfind an s in the bottom layer that has direct access to atleast k states in the middle layer corresponding one-by-oneto each valuation; and each of these has direct access to atleast k states in the top layer that correspond state by stateto each valuation. Again, it is clear that for such a state s ′ ,the two pointed models M Φ GL , s and M ′ , s ′ satisfy the sameformulas. Summing up, this means that in all three cases, M ′ , s ′ = ϕ , so M ′ = ϕ for almost all Kleitman-Rothschildmodels, as n grows large. Conclusion: lim n →∞ ν n, Φ ( ϕ ) = 0 .We can now conclude that all of 1, 2, 3, 4 are equivalent.Therefore, each modal formula in L (Φ) is either almost surelyvalid or almost surely invalid over finite models in GL .This concludes our investigation of validity in almost allmodels. For almost sure frame validity, it turns out that there istransfer between validity in the countable irreflexive KleitmanRothschild frame and almost sure frame validity.IV. T HE COUNTABLE RANDOM IRREFLEXIVE K LEITMAN -R OTHSCHILD FRAME
Differently than for the system K [5], it turns out that inlogics for transitive partial (strict) orders such as GL , wecan prove transfer between validity of a sentence in almostall relevant finite frames and validity of the sentence inone specific frame, namely the countable random irreflexiveKleitman Rothschild frame. Let us start by introducing thisframe step by step. Definition 2 (Finite and countable random irreflexive Kleit-man-Rothschild frames) . Following [5], for each n ∈ N ,a random labelled frame of size n is a frame obtained byrandom and independent assignments of truth/falsity to thebinary direct successor relation R on every pair ( x, y ) fromthe set { , . . . , n } with probability .This definition can be restricted to three-layer strictly or-dered frames, in which the set of worlds { , . . . , n } has beenpartitioned into three levels L (bottom), L (middle) and L (upper). A finite random irreflexive three-layer frame can beobtained by independent assignments of truth/falsity to the(irreflexive, asymmetric) immediate successor relation R onevery pair ( x, y ) with x ∈ L and y ∈ L or with x ∈ L and y ∈ L with probability . Then, the relation < is thetransitive closure of R . This definition can be extended to the infinite, countablerandom irreflexive three-layer Kleitman-Rothschild frame onthe set N . Let us call this frame F KR . The following definition specifies a first-order theory in thelanguage of strict (irreflexive asymmetric) partial orders. Wehave adapted it from Compton’s [7] set of extension axioms T as (where the subscript “ as ” stands for “ almost sure”) forreflexive partial orders of the Kleitman-Rothschild form, whichwere in turn inspired by Fagin’s extension axioms for almostall first-order models with a binary relation [3]. Definition 3 (Extension axioms) . The theory T as - irr in-cludes the axioms for strict partial orders, namely, ∀ x ¬ ( x Straightforward adaptation from Compton’s re-flexive to our irreflexive orders of his proof that his T as is ℵ -categorical and therefore also complete [4, Theorem 3.1]. Proposition 4. Each of the sentences in T as - irr has labeledasymptotic probability 1 in the class of finite strict (irreflexive)partial orders. Proof sketch Straightforward adaptation to our irreflexiveorders of Compton’s proof that his T as has labeled asymptoticprobability 1 in reflexive partial orders [4, Theorem 3.2].Now that we have shown that the extension axioms hold inthe countable random irreflexive Kleitman Rothschild frameas well as in almost all finite strict partial orders (i.e., F KR | = T as - irr ), we have enough background to be able toprove the modal zero-one law with respect to the class of finiteirreflexive transitive frames corresponding to provability logic.V. V ALIDITY IN ALMOST ALL FINITE IRREFLEXIVETRANSITIVE FRAMES Take Φ = { p , . . . , p k } or Φ = { p i | i ∈ N } . The axiomsystem AX Φ , FGL corresponding to validity in almost all finiteframes of provability logic has the same axioms and rules as GL , plus the following axiom schemas, for all k ∈ N , whereall ϕ i ∈ L (Φ) : (cid:3)(cid:3)(cid:3) ⊥ (T3) ♦♦ ⊤ ∧ ^ i ≤ k ♦ ( ♦ ⊤ ∧ (cid:3) ϕ i ) → (cid:3) ( ♦ ⊤ → ♦ ( ^ i ≤ k ϕ i )) (DIAMOND-k) ♦♦ ⊤ ∧ ^ i ≤ k ♦ ( (cid:3) ⊥ ∧ ϕ i ) → ♦ ( ^ i ≤ k ♦ ϕ i ) (UMBRELLA-k)Here, UMBRELLA-0 is the formula ♦♦ ⊤ ∧ ♦ ( (cid:3) ⊥ ∧ ϕ ) → ♦♦ ϕ , which represents the property that direct successors ofbottom layer worlds are never endpoints but have at least onesuccessor in the top layer.The formula DIAMOND-0 has been inspired by the well-known axiom ♦(cid:3) ϕ → (cid:3)♦ ϕ that characterizes confluence,also known as the diamond property: for all x, y, z , if xRy and xRz , then there is a w such that yRw and zRw .Note that in contrast to the theory AX Φ , MGL introduced inSection III, the axiom system AX Φ , FGL gives a normal modallogic, closed under uniform substitution.Also notice that AX Φ , FGL is given by an infinite set ofaxioms. It turns out that if we base our logic on an infiniteset of atoms Φ = { p i | i ∈ N } , then for each k ∈ N ,DIAMOND-k+1 and UMBRELLA-k+1 are strictly strongerthan DIAMOND-k andUMBRELLA-k, respectively. So wehave two infinite sets of axioms that both strictly increase instrength, thus by a classical result of Tarski, the modal theorygenerated by AX Φ , FGL is not finitely axiomatizable.For the proof of the zero-one law for frame validity, wewill again need a completeness proof, this time of AX Φ , FGL with respect to almost sure frame validity, including use of Lindenbaum’s lemma and finitely many maximal AX Φ , FGL -consistent sets of formulas, each intersected with a finite setof relevant formulas Λ .Below, we will define the closure of a sentence ϕ ∈ L (Φ) .You can view this closure as the set of formulas that arerelevant for making a (finite) countermodel against ϕ . Definition 4 (Closure of a formula) . The closure of ϕ withrespect to AX Φ , FGL is the minimal set Λ of AX Φ , FGL -formulassuch that: ϕ ∈ Λ . (cid:3)(cid:3)(cid:3) ⊥ ∈ Λ . If ψ ∈ Λ and χ is a subformula of ψ , then χ ∈ Λ . If ψ ∈ Λ and ψ itself is not a negation, then ¬ ψ ∈ Λ . If ♦ ψ ∈ Λ and ψ itself is not of the form ♦ ξ or ¬ (cid:3) χ ,then ♦♦ ψ ∈ Λ , and also (cid:3) ¬ ψ , (cid:3)(cid:3) ¬ ψ ∈ Λ . If (cid:3) ψ ∈ Λ and ψ itself is not of the form ♦ ξ or ¬ (cid:3) χ ,then (cid:3)(cid:3) ψ ∈ Λ , and also ♦ ¬ ψ , ♦♦ ¬ ψ ∈ Λ . Note that Λ is a finite set of formulas, of size polynomial inthe length of the formula ϕ from which it is built. Definition 5. Let Λ be a closure as defined above and let ∆ , ∆ , ∆ be a maximal AX Φ , FGL -consistent sets. Then wedefine: • ∆ Λ := ∆ ∩ Λ ; • ∆ ≺ ∆ iff for all (cid:3) χ ∈ ∆ , we have χ ∈ ∆ ; • ∆ Λ1 ≺ ∆ Λ2 iff ∆ ≺ ∆ . Theorem 2. For every formula ϕ ∈ L (Φ) , the following areequivalent: AX Φ , FGL ⊢ ϕ ; F KR | = ϕ ; lim n →∞ µ n, Φ ( ϕ ) = 1 ; lim n →∞ µ n, Φ ( ϕ ) = 0 .Proof. We show a circle of implications. Let ϕ ∈ L (Φ) . ⇒ Suppose AX Φ , FGL ⊢ ϕ . Because finite irreflexive Kleitman-Rothschild frames are finite strict partial orders that have nochains of length > , the axioms and theorems of GL + (cid:3)(cid:3)(cid:3) ⊥ hold in all Kleitman-Rothschild frames, therefore theyare valid in F KR | = ϕ .So we only need to check the validity of the DIAMOND-kand UMBRELLA-k axioms in F KR for all k ≥ .DIAMOND-k-1: Fix k ≥ , take sentences ϕ i ∈ L (Φ) for i = 1 , . . . , k − and let ϕ = ♦♦ ⊤ ∧ V i ≤ k − ♦ ( ♦ ⊤ ∧ (cid:3) ϕ i ) → (cid:3) ( ♦ ⊤ → ♦ ( V i ≤ k − ϕ i )) . By Propositions 3 and 4, we knowthat each of the extension axioms of the form (b) holds in F KR . We want to show that ϕ is valid in F KR .To this end, let V be any valuation on the set of labelledstates W = N of F KR and let M = ( F KR , V ) . Nowtake an arbitrary b ∈ W and suppose that M, b | = ♦♦ ⊤ ∧ V i ≤ k − ♦ ( ♦ ⊤ ∧ (cid:3) ϕ i ) . Then b is in the bottom layer L andhere are worlds x , . . . , x k − (not necessarily distinct) in themiddle layer L such that for all i ≤ k − , we have b < x i and M, x i | = (cid:3) ϕ i . Now take any x k in L with b < x k . Then,by the extension axiom (b), there is an element z in the upperlayer L such that V i ≤ k x i < z . Now for that z , we havethat M, z | = V i ≤ k − ϕ i . But then M, x k | = ♦ ( V i ≤ k − ϕ i ) ,so because x k is an arbitrary direct successor of b , we have M, b | = (cid:3) ( ♦ ⊤ → ♦ ( V i ≤ k − ϕ i )) . To conclude, M, b | = ♦♦ ⊤∧ ^ i ≤ k − ♦ ( ♦ ⊤∧ (cid:3) ϕ i ) → (cid:3) ( ♦ ⊤ → ♦ ( ^ i ≤ k − ϕ i )) , so because b and V were arbitrary, we have F KR | = ♦♦ ⊤∧ ^ i ≤ k − ♦ ( ♦ ⊤∧ (cid:3) ϕ i ) → (cid:3) ( ♦ ⊤ → ♦ ( ^ i ≤ k − ϕ i )) , as desired.UMBRELLA-k-1: Fix k ≥ , take sentences ϕ i ∈ L (Φ) for i = 1 , . . . , k − and let ϕ = ♦♦ ⊤ ∧ V i ≤ k − ♦ ( (cid:3) ⊥ ∧ ϕ i ) → ♦ ( V i ≤ k − ♦ ϕ i ) . By Propositions 3 and 4, we know that eachof the extension axioms of the form (c) holds in F KR . Wewant to show that ϕ is valid in F KR .To this end, let V be any valuation on the set of labelledstates W = N of F KR and let M = ( F KR , V ) . Nowtake an arbitrary b ∈ W and suppose that M, b | = ♦♦ ⊤ ∧ V i ≤ k − ♦ ( (cid:3) ⊥∧ ϕ i ) . Then b is in the bottom layer L and thereare accessible worlds x , . . . , x k − (not necessarily distinct)in the upper layer L such that for all i ≤ k − , we have b < x i and M, x i | = ϕ i . By the extension axiom (c) fromDefinition 3, there is an element z in the middle layer L such that b < z and for all i ≤ k − , z < x i . But that meansthat M, z | = V i ≤ k − ♦ ϕ i , therefore M, b | = ♦ ( V i ≤ k − ♦ ϕ i ) .In conclusion, M, b | = ♦♦ ⊤ ∧ ^ i ≤ k − ♦ ( (cid:3) ⊥ ∧ ϕ i ) → ♦ ( ^ i ≤ k − ♦ ϕ i ) , so because b and V were arbitrary, we have F KR | = ♦♦ ⊤ ∧ ^ i ≤ k − ♦ ( (cid:3) ⊥ ∧ ϕ i ) → ♦ ( ^ i ≤ k − ♦ ϕ i ) , as desired. ⇒ Suppose F KR | = ϕ . Using Van Benthem’s translation (seeSubsection I-B), we can translate this as a Π sentence beingtrue in F KR (viewed as model of the relevant second-orderlanguage): Universally quantify over predicates correspondingto all propositional atoms occurring in ϕ , to get a sentenceof the form χ := ∀ P , . . . , P n ∀ xϕ ∗ , where ∀ xϕ ∗ is afirst-order sentence. Now the claim is that χ follows froma finite set of the extension axioms. For if not, then everyfinite set of the extension axioms is satisfiable togetherwith ¬ χ , hence by compactness, the full set of extensionaxioms is satisfiable together with ¬ χ . But then ¬ χ is true insome P , . . . , P n -extension of F KR , contradicting our earlierassumption. This proof is an adaptation of the result for the general random framein [5, Proposition 5], which was in turn based on [10]. ⇒ Straightforward, because = 1 . ⇒ By contraposition. Let ϕ ∈ L (Φ) and suppose that AX Φ , FGL ϕ . Then ¬ ϕ is AX Φ , FGL -consistent. We will do a completenessproof by the finite step-by-step method (see, for example, [35],[36]), but based on infinite maximal consistent sets, eachof which is intersected with the same finite set of relevantformulas Λ , so that the constructed counter-model remainsfinite (see [37], [38, footnote 3]).In the following, we are first going to construct a model M ϕ = h W, R, V i that will contain a world where ¬ ϕ holds(Step 4 ⇒ lim n →∞ µ n, Φ ( ϕ ) = 0 (Step 4 ⇒ Step 4 ⇒ By the Lindenbaum Lemma, we can extend {¬ ϕ } to amaximal AX Φ , FGL -consistent set Ψ . Now define Ψ Λ := Ψ ∩ Λ ,where Λ is as in Definition 4.We distinguish three cases for the step-by-step construction: U (upper layer), M (middle layer), and B (bottom layer). Case U, with (cid:3) ⊥ ∈ Ψ Λ :In this case we are done: a one-point counter-model suffices. Case M, with (cid:3) ⊥ 6∈ Ψ Λ , (cid:3)(cid:3) ⊥ ∈ Ψ Λ :Let ♦ ψ , . . . , ♦ ψ n be an enumeration of all the formulas of theform ♦ ψ in Ψ Λ . Note that for all these formulas, ♦♦ ψ i Ψ Λ ,because (cid:3)(cid:3) ⊥ ∈ Ψ Λ . Take an arbitrary one of the ψ i for which ♦ ψ i ∈ Ψ Λ . Claim: the set ∆ i := { (cid:3) χ, χ | (cid:3) χ ∈ Ψ } ∪ { ψ i , (cid:3) ¬ ψ i } is AX Φ , FGL -consistent. For if not, then { (cid:3) χ, χ | (cid:3) χ ∈ Ψ } ⊢ AX Φ , FGL (cid:3) ¬ ψ i → ¬ ψ i . Because proofs are finite, there is a finite set χ , . . . , χ k with (cid:3) χ , . . . (cid:3) χ k ∈ Ψ and { (cid:3) χ j , χ j | j ∈ { , . . . , k }} ⊢ AX Φ , FGL (cid:3) ¬ ψ i → ¬ ψ i . Using necessitation, we get { (cid:3)(cid:3) χ j , (cid:3) χ j | j ∈ { , . . . , k }} ⊢ AX Φ , FGL (cid:3) ( (cid:3) ¬ ψ i → ¬ ψ i ) . Because we have ⊢ AX Φ , FGL (cid:3) χ j → (cid:3)(cid:3) χ j for all j = 1 , . . . , k and ⊢ AX Φ , FGL (cid:3) ( (cid:3) ¬ ψ i → ¬ ψ i ) → (cid:3) ¬ ψ i , we can conclude: { (cid:3) χ | (cid:3) χ ∈ Ψ } ⊢ AX Φ , FGL (cid:3) ¬ ψ i . Using Proposition 2(4) and the fact that (cid:3) ¬ ψ i ∈ Λ , this leadsto (cid:3) ¬ ψ i ∈ Ψ Λ , contradicting our assumption that ♦ ψ i ∈ Ψ Λ .Also note that because (cid:3)(cid:3) ⊥ ∈ Ψ , by definition, (cid:3) ⊥ ∈ ∆ i .We can now extend ∆ i to a maximal AX Φ , FGL -consistent set Ψ i by the Lindenbaum Lemma, and we define for each i ∈{ , . . . , n } the set Ψ Λ i := Ψ i ∩ Λ (see Definition 5).herefore, we have for all i ∈ { , . . . , n } that Ψ Λ ≺ Ψ Λ i as well as ψ i , (cid:3) ¬ ψ i ∈ Ψ Λ i . Case B, with (cid:3)(cid:3) ⊥ 6∈ Ψ Λ :In this case, we also look at all formulas of the form ♦ ψ ∈ Ψ Λ .We first divide this into two sets, as follows:1) The set of ♦ -formulas in Ψ Λ for which we have that ♦ ξ k +1 , . . . , ♦ ξ l ∈ Ψ Λ but ♦♦ ξ k +1 , . . . , ♦♦ ξ l Ψ Λ forsome l ∈ N , so (cid:3)(cid:3) ¬ ξ k +1 , . . . , (cid:3)(cid:3) ¬ ξ l ∈ Ψ Λ . 2) The set of ♦♦ -formulas with ♦♦ ξ , . . . , ♦♦ ξ k ∈ Ψ Λ .Note that for these formulas, we also have ♦ ξ , . . . , ♦ ξ k ∈ Ψ Λ , because GL ⊢ ♦♦ ξ i → ♦ ξ i .We will treat these pairs ♦♦ ξ i , ♦ ξ i for i = 1 , . . . , k atthe same go.Note that (1) and (2) lead to disjoint sets which togetherexhaust the ♦ -formulas in Ψ Λ . Altogether, that set nowcontains { ♦ ξ , . . . , ♦ ξ k , ♦♦ ξ , . . . , ♦♦ ξ k , ♦ ξ k +1 , . . . , ♦ ξ l } .Let us first check the formulas of type (1): ♦ ξ k +1 , . . . , ♦ ξ l ∈ Ψ Λ , but (cid:3)(cid:3) ¬ ξ k +1 , . . . , (cid:3)(cid:3) ¬ ξ l ∈ Ψ Λ . We can now show bysimilar reasoning as in Case M that for each i ∈ { k +1 , . . . , l } , ∆ i = { (cid:3) χ, χ | (cid:3) χ ∈ Ψ } ∪ { ξ i , (cid:3) ¬ ξ i } is AX Φ , FGL -consistent,so we can extend them to maximal AX Φ , FGL -consistent sets Ψ i and define Ψ Λ i := Ψ i ∩ Λ with Ψ ≺ Ψ i , and therefore Ψ Λ ≺ Ψ Λ i , for all i ∈ { k + 1 , . . . , l } .We now claim that for all i ∈ { k + 1 , . . . , l } , the world Ψ Λ i is not in the top layer of the model with root Ψ Λ . To derive acontradiction, suppose that it is in the top layer, so (cid:3) ⊥ ∈ Ψ Λ i .Then also (cid:3) ⊥ ∧ ξ i ∈ Ψ i for i ∈ { k + 1 , . . . , l } , so because Ψ and all the Ψ i for i ∈ { k + 1 , . . . , l } are maximal AX Φ , FGL -consistent and each Ψ i contains χ for all formulas χ with (cid:3) χ ∈ Ψ , we have ♦ ( (cid:3) ⊥ ∧ ξ i ) ∈ Ψ for all i ∈ { k + 1 , . . . , l } .By UMBRELLA-0, we know for all i ∈ { k + 1 , . . . , l } that ⊢ AX Φ , FGL ♦♦ ⊤ ∧ ♦ ( (cid:3) ⊥ ∧ ξ i ) → ♦♦ ξ i . Also having ♦♦ ⊤ ∈ Ψ , we can now use Proposition 2(4) toconclude that ♦♦ ξ i ∈ Ψ . Therefore, because ♦♦ ξ i ∈ Λ , wealso have ♦♦ ξ i ∈ Ψ Λ , contradicting our starting assumptionthat ♦ ξ i is a type (1) formula. We conclude that (cid:3) ⊥ 6∈ Ψ Λ i ,therefore, Ψ Λ i is in the middle layer for all i in k + 1 , . . . , l .Let us now look for each of these Ψ Λ i with i in k + 1 , . . . , l ,which direct successors in the top layer they require. Anyformulas of the form ♦ χ ∈ Ψ Λ i have to be among the formulas ♦ ξ , . . . , ♦ ξ k of type (2), for which ♦♦ ξ , ♦♦ ξ k ∈ Ψ . Suppose ♦ ξ j ∈ Ψ i for some j in , . . . , k and i in k + 1 , . . . , l . Thenwe can show (just like in Case M) that there is a maximalconsistent set X i,j with Ψ i ≺ X i,j and ξ j , (cid:3) ⊥ ∈ X i,j .The corresponding world in the top layer will be called X Λ i,j = X i,j ∩ Λ . Because X Λ i,j is finite, we can describe it by (cid:3) ⊥ and a finite conjunction of literals, which we representas χ i,j . For ease of reference in the next step, let us define: The required formulas of the form (cid:3)(cid:3) ¬ ξ j are in Λ because of the finaltwo clauses of Definition 4. A := {h i, j i | there are i in k +1 , . . . , l and j in , . . . , k s.t. ♦ ξ j ∈ Ψ i } .For the formulas of type (2), we have ♦♦ ξ i ∈ Ψ Λ . Moreover,we have for each i ∈ { , . . . , k } : GL + (cid:3)(cid:3)(cid:3) ⊥ ⊢ ♦♦ ξ i → ♦ ( (cid:3) ⊥ ∧ ξ i ) . Therefore, by maximal AX Φ , FGL -consistency of Ψ , we have byProposition 2 that ♦ ( (cid:3) ⊥ ∧ ξ i ) ∈ Ψ for each i ∈ { , . . . , k } .Similarly, for the formulas χ i,j constructed in the last part ofthe step for formulas of type (1), we have for all h i, j i ∈ A that ♦ ( (cid:3) ⊥ ∧ χ i,j ) ∈ Ψ . We also have ♦♦ ⊤ ∈ Ψ . UMBRELLA-know gives us Ψ ⊢ AX Φ , FGL ♦♦ ⊤∧ ^ i =1 ,...,k ♦ ( (cid:3) ⊥∧ ξ i ) ∧ ^ h i,j i∈ A ♦ ( (cid:3) ⊥∧ χ i,j ) → ♦ ( ^ i =1 ,...,k ♦ ξ i ∧ ^ h i,j i∈ A ♦ χ i,j ) We may conclude from maximal AX Φ , FGL -consistency of Ψ and Proposition 2(4) that ♦ ( V i =1 ,...,k ♦ ξ i ∧ V h i,j i∈ A ♦ χ i,j ) ∈ Ψ .This means that we can construct one direct successor of Ψ Λ containing all the ♦ ξ i for i ∈ { , . . . , k } and all the ♦ χ i,j for h i, j i ∈ A . To this end, let ∆ := { (cid:3) χ, χ | (cid:3) χ ∈ Ψ }∪{ ♦ ξ , . . . , ♦ ξ k }∪{ ♦ χ i,j | h i, j i ∈ A } Claim: ∆ is AX Φ , FGL -consistent. For if not, we would have: { (cid:3) χ, χ | (cid:3) χ ∈ Ψ } ⊢ AX Φ , FGL ¬ ( ^ i =1 ,...,k ♦ ξ i ∧ ^ h i,j i∈ A ♦ χ i,j ) But then by the same reasoning as we used before (“boxingboth sides” and using GL ⊢ (cid:3) χ → (cid:3)(cid:3) χ ) we conclude that { (cid:3) χ | (cid:3) χ ∈ Ψ } ⊢ AX Φ , FGL (cid:3) ¬ ( ^ i =1 ,...,k ♦ ξ i ∧ ^ h i,j i∈ A ♦ χ i,j ) . This directly contradicts ♦ ( V i =1 ,...,k ♦ ξ i ∧ V h i,j i∈ A ♦ χ i,j ) ∈ Ψ , which we showed above. Now that we know ∆ tobe AX Φ , FGL -consistent, we can extend it by the LindenbaumLemma to a maximal AX Φ , FGL -consistent set, which we call Ψ ⊇ ∆ . Define Ψ Λ1 := Ψ ∩ Λ . Note that by Defini-tion 5, Ψ ≺ Ψ so Ψ Λ ≺ Ψ Λ1 . Moreover, (cid:3)(cid:3) ⊥ ∈ Ψ Λ1 because (cid:3)(cid:3)(cid:3) ⊥ ∈ Ψ Λ . Therefore, by Proposition 2(4), (cid:3)(cid:3) ¬ ξ , . . . , (cid:3)(cid:3) ¬ ξ k ∈ Ψ Λ1 but also ♦ ξ , . . . , ♦ ξ k ∈ Ψ Λ1 .Now we can use the same method as in Case M to find therequired k direct successors of Ψ Λ1 . Namely, we find maximal AX Φ , FGL -consistent sets Ξ i and define Ξ Λ i := Ξ i ∩ Λ such that Ψ Λ1 ≺ Ξ Λ i and ξ i ∈ Ξ Λ i for all i in , . . . , k .We have now handled making direct successors of Ψ Λ for allthe formulas of type (1) and type (2). We can then finish off thestep-by-step construction for Case B by populating the upperlayer U with one appropriate restriction to Λ of a maximalconsistent set Ξ , as follows. We note that (cid:3) ¬ ξ i ∈ Ψ Λ i for i n k + 1 , . . . , l , and that (cid:3)(cid:3) ⊥ ∈ Ψ Λ1 . Let us take the followinginstance of the DIAMOND-(l-k) axiom scheme: ♦♦ ⊤ ∧ ^ i ∈{ k +1 ,...,l } ♦ ( ♦ ⊤ ∧ (cid:3) ¬ ξ i ) → (cid:3) ( ♦ ⊤ → ♦ ( ^ i ∈{ k +1 ,...,l } ¬ ξ i )) Now we have ♦♦ ⊤ ∈ Ψ Λ . Because Ψ ≺ Ψ i and ♦ ⊤∧ (cid:3) ¬ ξ i ∈ Ψ i for all i in k + 1 , . . . , l , we derive that ^ i ∈{ k +1 ,...,l } ♦ ( ♦ ⊤ ∧ (cid:3) ¬ ξ i ) ∈ Ψ . Now by one more application of Proposition 2(4), we have (cid:3) ( ♦ ⊤ → ♦ ( ^ i ∈{ k +1 ,...,l } ¬ ξ i )) ∈ Ψ . Because Ψ ≺ Ψ j and ♦ ⊤ ∈ Ψ j for all j in , k + 1 , . . . , l , weconclude that ♦ ⊤ → ♦ ( ^ i ∈{ k +1 ,...,l } ¬ ξ i ) ∈ Ψ j for all j ∈ { , k + 1 , . . . , l } . Now we can find one world Ξ such that for all j in , k + 1 , . . . , l , we have Ψ j ≺ Ξ , therefore Ψ Λ j ≺ Ξ Λ0 . Andmoreover, ¬ ξ i ∈ Ξ Λ0 for all i in k + 1 , . . . , l .We have now finished creating our finite counter-model M Φ ,FGL = ( W, R, V ) , which has: • W = { Ψ Λ , Ψ Λ1 , Ψ Λ k +1 , . . . , Ψ Λ l }∪{ Ξ Λ i | i ∈ { , . . . , k }} ∪ { X Λ i,j | h i, j i ∈ A } . • R = ≺ (see Definition 5). • For each p ∈ Φ and Γ Λ ∈ W : V ΛΓ ( p ) = 1 iff p ∈ Γ Λ Now we can relatively easily prove a truth lemma, restrictedto formulas from Λ , as follows. Truth Lemma For all ψ in Λ and all sets Γ Λ in W : M Φ ,FGL , Γ Λ | = ψ iff ψ ∈ Γ Λ . Proof By induction on the construction of the formula. Foratoms p ∈ Λ , the fact that M Φ ,FGL , Γ Λ | = p iff p ∈ Γ Λ followsby the definition of V . Induction Hypothesis : Suppose for some arbitrary χ, ξ ∈ Λ ,we have that for all sets ∆ Λ in W : M Φ ,FGL , ∆ Λ | = χ iff χ ∈ ∆ Λ and M Φ ,FGL , ∆ Λ | = ξ iff ξ ∈ ∆ Λ . Inductive step : • Negation : Suppose ¬ χ ∈ Λ . Now by the truth definition, M Φ ,FGL , ∆ Λ | = ¬ χ iff M Φ ,FGL , ∆ Λ = χ . By the inductionhypothesis, the latter is equivalent to χ ∆ Λ . But thisin turn is equivalent by Proposition 2(1) to ¬ χ ∈ ∆ Λ . • Conjunction : Suppose χ ∧ ξ ∈ Λ . Now by the truthdefinition, M Φ ,FGL , ∆ Λ | = χ ∧ ξ iff M Φ ,FGL , ∆ Λ | = χ and M Φ ,FGL , ∆ Λ | = χ . By the induction hypothesis, the latter is equivalent to χ ∈ ∆ Λ and ξ ∈ ∆ Λ , which byProposition 2(2) is equivalent to χ ∧ ξ ∈ ∆ Λ . • Box : Suppose (cid:3) χ ∈ Λ . We know by the loaded inductionhypothesis that for all sets ∆ Λ in W , M Φ ,FGL , ∆ Λ | = χ iff χ ∈ ∆ Λ . We want to show that M Φ ,FGL , Γ Λ | = (cid:3) χ iff (cid:3) χ ∈ Γ Λ .For one direction, suppose that (cid:3) χ ∈ Γ Λ , then bydefinition of R , for all ∆ Λ with Γ Λ R ∆ Λ , we have χ ∈ ∆ Λ , so by induction hypothesis, for all these ∆ Λ , M Φ ,FGL , ∆ Λ | = χ . Therefore by the truth definition, M Φ ,FGL , Γ Λ | = (cid:3) χ .For the other direction, suppose that (cid:3) χ ∈ Λ but (cid:3) χ Γ Λ . Then (by Definition 4 and Proposition 2(4)),we have ♦ ¬ χ ∈ Γ Λ . Then in the step-by-stepconstruction, in Case M or Case B , we have constructeda maximal AX Φ , FGL -consistent set Ξ with Γ ≺ Ξ andthus Γ Λ R Ξ Λ and ¬ χ ∈ Ξ thus ¬ χ ∈ Ξ Λ , respectively ξ ∈ Ξ , thus ξ ∈ Ξ Λ . Now by the induction hypothesis,we have in both cases M Φ ,FGL , Ξ Λ = χ , so by the truthdefinition, M Φ ,FGL , Ξ Λ = (cid:3) χ .Finally, from the truth lemma and the fact above that ¬ ϕ ∈ Ψ Λ , we have M Φ ,FGL , Ψ Λ = ϕ , so we have found ourcounter-model. Step 4 ⇒ Now we need to show that lim n →∞ µ n, Φ ( ϕ ) = 0 . Thereare three cases, corresponding to Case U, Case M, andCase B of the step-by-step construction of the counter-modelin Step 4 ⇒ ϕ is not valid. Case U The one-point counter-model against ϕ , let us call it M with W = { Ψ Λ } , can be turned into a counter-model on everythree-layer Kleitman-Rothschild frame F as follows. Take aworld u in the top layer and take a valuation on L (Φ) thatcorresponds on that world with the valuation of world Ψ Λ in M and is arbitrary everywhere else. Then this world providesa counterexample showing F = ϕ . Case M The two-layer model M can be embedded into almost allKleitman-Rothschild frames. Take a world m in the middlelayer of the Kleitman-Rothschild frame with sufficiently manysuccessors in the top layer, and take care that all valuationson L (Φ) corresponding to Ψ Λ1 , . . . , Ψ Λ n appear as valuationsof the successors of m , while no other valuations appear. Case B The three-layer model M Φ ,FGL = ( W, R, V ) with W = or an appropriate logically equivalent ♦ ξ Ψ Λ , Ψ Λ1 , Ψ Λ k +1 , . . . , Ψ Λ l , Ξ Λ0 , Ξ Λ1 , . . . , Ξ Λ k } ∪ { X Λ i,j | h ij i ∈ A } can be embedded into almost all sufficiently large Kleitman-Rothschild frames. Take different nodes m , m k +1 , . . . , m l in the middle layer L . Then by extension axiom (a) thereis a b in the bottom layer L such that V i ∈{ ,k +1 ,...,l } b < m i . Now by extension axiom (b) one can take different nodes u , u , . . . , u k and u i,j for all h i, j i ∈ A in the upper layer L such that V i ∈{ ,k +1 ,...,l } m i < u and V i ∈{ ,...,k } m < u i aswell as V h i,j i∈ A m < u i,j and V h i,j i∈ A m i < u i,j , but ^ i ∈{ k +1 ,...,l, } ,j ∈{ ,...,k } ¬ ( m i < u j ) and ^ i ∈{ k +1 ,...,l, } , h j,k i∈ A,i = j ¬ ( m i < u j,k ) . Give b the valuation corresponding to Ψ Λ on L (Φ) . Now takecare that: • the valuations of all successors m of b in the middle layerthat are direct predecessors of all of u , . . . , u k and u i,j for all h i, j i ∈ A (so such m include m ) correspond tothe valuation of Ψ Λ1 ; • the valuations of m k +1 , . . . , m l correspond one by oneto the valuations of Ψ Λ k +1 , . . . , Ψ Λ l ; and • all other successors of b in the middle layer that are not smaller than all of u , . . . , u k and all u i,j for h i, j i ∈ A also correspond to Ψ Λ k +1 , . . . , Ψ Λ m , in such a way thatall these valuations are covered and no other valuationsappear.Likewise, for the u i , take care that: • the valuation on u corresponds to that of Ξ Λ0 , whichshould also be the valuation of all other nodes that aresuccessors of all of m , m k +1 , . . . , m l ; • the valuations of u , . . . , u k correspond one by one tothe valuations of Ξ Λ1 , . . . , Ξ Λ k , which should also be thevaluation of any other nodes that are direct successors of m but not of all of m k +1 , . . . , m l and the valuations ofthe u i,j for all h i, j i ∈ A correspond one by one to thevaluations of the X i,j ; • For all other successors of the middle layer worlds, takecare that their valuations correspond to Ξ Λ0 , Ξ Λ1 , . . . , Ξ Λ k and the X i,j for h i, j i ∈ A , in such a way that all thesevaluations are covered and no other valuations appear.Now it is the case that in this large enough KleitmanRothschild frame and under such a valuation leading to amodel M as described above, we get ( M, b ) = ϕ .To conclude, all of 1, 2, 3, and 4 are equivalent.VI. C OMPLEXITY OF ALMOST SURE MODEL AND FRAMESATISFIABILITY It is well known that the satisfiability problem and thevalidity problem for GL are PSPACE-complete (for a proofsketch, see [31]), just like for other well-known modal logics such as K and S4 . In contrast, for enumerably infinite vo-cabulary Φ , the problem whether lim n →∞ ν n, Φ ( ϕ ) = 0 is in ∆ p (for the dag-representation of formulas), by adapting [22,Theorem 4.17]. If Φ is finite, the decision problem whether lim n →∞ ν n, Φ ( ϕ ) = 0 is even in P , because you only need tocheck validity of ϕ in the fixed finite canonical model M Φ GL .For example, for Φ = { p , p } , this model contains only 16worlds, see Figure 2.The problem whether lim n →∞ µ n, Φ ( ϕ ) = 0 is in NP, moreprecisely, NP-complete for enumerably infinite vocabulary Φ . To show that it is in NP, suppose you need to decidewhether lim n →∞ µ n, Φ ( ϕ ) = 0 . By the proof of part 4 ⇒ < | ϕ | , a model on it and a world in that model, and check(in polynomial time) whether ϕ is not true in that world. NP-hardness is immediate for Φ infinite: for propositional ψ , wehave ψ ∈ SAT iff lim n →∞ µ n, Φ ( ϕ ) = 0 .In conclusion, if the polynomial hierarchy does not collapseand in particular (as most complexity theorists believe) ∆ p = PSPACE and NP = PSPACE, then the problems of decidingwhether a formula is almost always valid in finite models orframes of provability logic are easier than deciding whetherit is always valid. For comparison, remember that for first-order logic the difference between validity and almost surevalidity is a lot starker still: Grandjean [39] proved that thedecidability problem of almost sure validity in the finite is onlyPSPACE-complete, while the validity problem on all structuresis undecidable [40], [41].VII. C ONCLUSION AND FUTURE WORK We have proved zero-one laws for provability logic withrespect to both model and frame validity. On the way, we haveaxiomatized validity in almost all relevant finite models andin almost all relevant finite frames, leading to two differentaxiom systems. If the polynomial hierarchy does not collapse,the two problems of ‘almost sure model/frame validity’ areless complex than ‘validity in all models/frames’.Among finite frames in general, partial orders are prettyrare – using Fagin’s extension axioms, it is easy to show thatalmost all finite frames are not partial orders. Therefore, resultsabout almost sure frame validities in the finite do not transferbetween frames in general and strict partial orders. Indeed,the logic of frame validities on finite irreflexive partial ordersstudied here is quite different from the modal logic of thevalidities in almost all finite frames [5], [27]. One of the mostinteresting results in [5] is that frame validity does not transferfrom almost all finite K -frames to the countable random frame,although it does transfer in the other direction. In contrast, wehave shown that for irreflexive transitive frames, validity doestransfer in both directions between almost all finite frames andthe countable random irreflexive Kleitman-Rothschild frame. A. Future work Currently, we are proving similar 0-1 laws for logics ofreflexive transitive frames, such as S4 and Grzegorczyk logic,xiomatizing both almost sure model validity and almostsure frame validity. It turns out that Halpern and Kapron’sclaim that there is a 0-1 law for S frame validity can stillbe salvaged, albeit with a different, stronger axiom system,containing two infinite series of umbrella and diamond axiomssimilar to the ones in the current paper. 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