Labelled tableaux for interpretability logics
aa r X i v : . [ m a t h . L O ] M a y Labelled tableaux for interpretability logics
Tuomas Hakoniemi Joost J. JoostenApril 2, 2018
Abstract
In is paper we present a labelled tableau proof system that serves awide class of interpretability logics. The system is proved sound and completefor any interpretability logic characterised by a frame condition given by a setof universal strict first order Horn sentences. As such, the current paper addsto a better proof-theoretical understanding of interpretability logics.
Provability logics like the G¨odel-L¨ob logic GL describe the structural behaviourof formalized provability in a simple propositional modal language. Inter-pretability logics are natural extensions of provability logics: they describe thestructural behaviour of relative interpretability.Essentially since Solovay’s landmark paper [17] we know that any Σ soundtheory that extends elementary arithmetic has the same provability logic GL .The situation is very different for interpretability logics. Basically, for twodifferent kind of theories we know the corresponding interpretability logics.On the one hand Shavrukov [15] and independently Berarducci [1] deter-mined the interpretability logic of any sound and essentially reflexive theorylike Peano Arithmetic to be IL M . On the other hand, Visser has proven in [18]that the interpretability logic of any sound and finitely axiomatised theory thatproves the totality of super-exponentiation –like IΣ – is IL P .In case the base theory is neither finitely axiomatizable nor essentially reflex-ive, the situation turns out to be much more difficult and actually, to determinethe interpretability logics in those situations remain open problems. Some par-tial results are known in the case of Primitive Recursive Arithmetic ([2]) or inthe case when we consider those modal principles that are provable in any rea-sonable arithmetical theory [11, 8, 9]. In this sense, interpretability logics arein need of more study compared to provability logics. A miracle happens writes Albert Visser as the first line of [19]: Whereasprovability is a Σ complete predicate, the logic GL that governs its structuralbehaviour is nice, well behaved and simple . The situation with interpretabilityseems even more extreme since Shavrukov has shown in [16] that interpretabilityis Σ complete and again, the modal logic describing the structural behaviouris nice. “Only” of complexity PSPACE. how nice they are. In particular, apart from some observationson the closed fragment ([10, 3]), close to nothing is known about the compu-tational complexity of interpretability logics. Also, very little is known aboutwell-behaved proof systems for interpretability logics with the sole exceptionof some work by Sasaki such as [14]. The current paper is intended to add tothe proof-theoretic understanding of interpretability logics by studying labelledtableaux proof systems for them.Tableaux proof systems are tightly related to sequent proof systems andare dual to them in many aspects. Rules in sequent proof systems typicallyhave possibly multiple antecedents and single conclusions/succedents. More-over, sequent-style proofs generally are trees that have the root at the bottomand are based on validity: the consequence of a rule is valid if (and often onlyif) all of the antecedents are valid.On the other hand, rules in tableaux systems typically have single antecedentand possibly multiple succedents. Moreover, Tableaux proofs generally are treesthat have the root at the top and are based on satisfiability: the antecedent of arule is satisfiable if and only if some of the ‘consequents’/succedents is satisfiable.Labelled tableaux introduce extra devices to the syntax that aim to repre-sent the accessibility relation in the corresponding Kripke-style semantics. Thisextra syntax allows us to give tableaux proof systems for many logics lacking atraditional one, where nodes of the tableaux carry only (sets of) formulas. Aprecursor for this idea of bringing a bit of semantics into the syntax appearsalready in [12], and labelled tableaux as they are now known were introducedprominently by Fitting in [5]. Standard references here are [6] and [7]. For moreon the history and development of tableaux systems for modal logics see e.g.[7]. Naturally, labels have also been incorporated into sequent calculi. We referthe reader to [13] for details on labelled sequent calculi for modal logics. Outline of the paper . After introducing the necessary preliminaries inSection 2, we use Section 3 to introduce the labelled tableaux system for allinterpretability logics IL X characterised by a set of first order Horn formulas.It is shown how a systematic tableau can be assigned to a finite set Γ of formulasso that the tableau contains all the necessary information as to the satisfiabilityof Γ.In Sections 4 and 5 we show that the tableau proofs are sound and completewith respect to IL X -validity. In the last section we remark that our resultsconcern most of the interpretability logics encountered in the literature, but notall. Interpretability logics are propositional modal logics with a unary modality (cid:3) whose dual modality ♦ is defined as ♦ := ¬ (cid:3) ¬ corresponding to provabilityand consistency respectively, and a binary modality ⊲ corresponding to relativeinterpretability. 2n this paper we shall work with the Boolean connectives ¬ and → . Thus,with Prop a countable set of propositional variables, the formulas F of inter-pretability logic are defined as F := Prop | ( ¬F ) | ( F → F ) | ( (cid:3) F ) | ( F ⊲ F ) . As always we will use the other connectives and Boolean constants freelysince they can be defined from ¬ and → . In order to use less parentheses weomit outer parentheses and shall say that ¬ , (cid:3) and ♦ bind strongest, followedby the equally strong binding ∨ and ∧ who bind stronger than ⊲ which in turnbinds stronger than → . Thus, for example, p ⊲ q → p ∧ (cid:3) r ⊲ q ∧ (cid:3) r is short for h ( p ⊲ q ) → (cid:16)(cid:0) p ∧ ( (cid:3) r ) (cid:1) ⊲ (cid:0) q ∧ ( (cid:3) r ) (cid:1)(cid:17) i . Definition 1.
The axioms of the basic interpretability logic IL are, apart fromall substitution instances (in the language of interpretability) of all propositionaltautologies, given by the following axiom schemata L (cid:3) ( A → B ) → ( (cid:3) A → (cid:3) B ); L (cid:3) ( (cid:3) A → A ) → (cid:3) A ; J (cid:3) ( A → B ) → A ⊲ B ; J A ⊲ B ) ∧ ( B ⊲ C ) → A ⊲ C ; J A ⊲ C ) ∧ ( B ⊲ C ) → A ∨ B ⊲ C ; J A ⊲ B → ( ♦ A → ♦ B ); J ♦ A ⊲ A .The rules are Modus Ponens and Necessitation: A/ (cid:3) A .The following lemma collects two easily obtainable and well-known proper-ties of IL that will play prominent role in our tableaux systems. Lemma 1. IL ⊢ ♦ A → ♦ ( A ∧ (cid:3) ¬ A ) ;2. IL ⊢ B ⊲ B ∧ (cid:3) ¬ B . The logic GL is the fragment of IL where the modal language is restricted to (cid:3) . We shall consider various extensions of IL . By IL X we denote the logic thatarises by adding the axiom scheme(s) X to IL . The extensions of IL obtainedby the following axiom schemes play a prominent role in the literature. P : A ⊲ B → (cid:3) ( A ⊲ B ); M : A ⊲ B → A ∧ (cid:3) C ⊲ B ∧ (cid:3) C .Interpretability logics allow for a relational semantics very much in the senseas GL does. 3 efinition 2. An IL -frame is a triple h W, R, S i where W is a non-empty do-main set, whose members are often called worlds , and R is a binary relation on W that is transitive and Noetherian (no infinite chains x Rx Rx . . . ). S is aternary relation on W that is often considered as a collection { S x } x ∈ W of binaryrelations by fixing the first argument x of the ternary S . It is required that each S x is a transitive and reflexive binary relation on { y ∈ W : xRy } satisfying thefollowing property: if xRyRz , then yS x z. An IL -model is a quadruple h W, R, S, V i , where h W, R, S i is an IL -frame and V is a function assigning a collection V ( p ) of worlds to a propositional variable p . Given an IL -model h W, R, S, V i we define a forcing relation (cid:13) between worldsand formulas as usual: • M, x (cid:13) p ⇔ x ∈ V ( p ); • M, x (cid:13) ¬ A ⇔ M, x A ; • M, x (cid:13) A → B ⇔ M, x A or M, x (cid:13) B ; • M, x (cid:13) (cid:3) A ⇔ ∀ y ( xRy ⇒ M, y (cid:13) A ); • M, x (cid:13) A ⊲ B ⇔ ∀ y (cid:0) xRy ∧ M, y (cid:13) A ⇒ ∃ z ( yS x z ∧ M, z (cid:13) B ) (cid:1) .We shall write x ∈ M whenever M = h W, R, S, V i with x ∈ W and likewisefor frames. We write M (cid:15) A to denote that M, x (cid:13) A for all x ∈ M . The abovedefined semantics is good in that one can prove completeness for IL as was firstdone in [4]: IL ⊢ A ⇔ ∀ M M (cid:15) A. An extension L of IL can be specified either axiomatically or semanticallyby restricting the class of models for example by specifying so called frameconditions . We say that a frame F := h W, R, S i validates A and we write F (cid:15) A whenever for all valuations V on F we have h W, R, S, V i (cid:15) A .A set of first or higher order sentences C in the language { R, S } with R abinary and S a ternary first-order relation symbol is called a frame condition for a logic L extending IL whenever we have h W, R, S i (cid:15) L ⇐⇒ h W, R, S i (cid:15) fo / ho C , where in the right-hand side the interpretations of R and S are R and S ,respectively. Then we also say that C characterises the logic L . As always, h W, R, S i (cid:15) L denotes that h W, R, S i (cid:15) A for any theorem A of L and we use asimilar convention for models. From now on we will use the same symbol R for R and its interpretation and likewise for S .In case an axiomatic extension IL X of IL is characterised by a set of strictuniversal Horn sentences in the language { R, S } we say that IL X is a Horn ∀ . . . ∀ ( ϕ ∧ . . . ∧ ϕ n → ψ ) , where n ≥ ϕ , . . . , ϕ n and ψ are atomic formulas and ∀ . . . ∀ denotes the uni-versal closure. In case that IL X is a Horn logic, we shall denote the correspond-ing frame condition by C X and call an IL -frame satisfying C X an IL X -frame.For example, the logic IL P is characterised by the (universal closure of the)first order formula xRy ∧ yRz ∧ zS x u → zS y u, and IL is characterised by the empty frame condition (or ∀ x ( xRx → xRx ) forthat matter).In this paper, we shall – given a frame – reduce the binary modality ⊲ toa series of unary ones. We will do so, so that the corresponding tableaux rulesbecome more amenable. Thus, given an IL -frame F = h W, R, S i , we introducenew unary modal operators (cid:3) x for each x ∈ W and give the following truthdefinition for the operators in a model M on the frame F M, y (cid:13) (cid:3) x A ⇔ ∀ z ( yS x z ⇒ M, z (cid:13) A ) . Now it is easy to verify that for any IL -model M, x (cid:13) A ⊲ B ⇔ M, x (cid:13) (cid:3) ( A ⇒ ¬ (cid:3) x ¬ B ) . (1) In this section we define a tableau proof method for interpretability logics whichare Horn. Moreover, we will give a systematic tableau procedure for such IL X that yields a canonical tableau given a finite set of formulas.As always, our tableaux will be downward growing trees. Each node of thetree carries a labelled formula. A labelled formula is a pair with a label and aformula. The label corresponds to a possible world where the formula is to besatisfied. We will show the unsatisfiability of a finite set of formulas in case allbranches in the systematic tableau close (precise definition follow). In case thesystematic tableau contains an open branch, that branch will carry informationabout a satisfying model. Definition 3.
Labels are strings composed of non-negative integers and lettersR and S. The set of all labels is defined recursively as follows: • • If σ is a label, then σRn is a label for all n ∈ N ; • If σ and ρ are labels and ρ is a strict non-empty prefix of σ , then σS ρ n isa label for all n ∈ N .Now that we have a sufficiently large set of labels we will describe how wegenerically build (almost) frames from them. Definition 4 ( IL X -label structure) . Given a Horn logic IL X and a set of labelsΛ, we define relations R Λ IL X and S Λ IL X on the set Λ as the least relations on Λsuch that: 5. If σ, σRn ∈ Λ, then h σ, σRn i ∈ R Λ IL X for all labels σ and n ∈ N ;2. If h σ, τ i ∈ R Λ IL X and h τ, ρ i ∈ R Λ IL X , then h σ, ρ i ∈ R Λ IL X ;3. If σ, ρ, σS ρ n ∈ Λ, then h ρ, σ, σS ρ n i ∈ S Λ IL X for all labels σ and ρ and all n ∈ N ;4. If h σ, τ i ∈ R Λ IL X , then h σ, τ, τ i ∈ S Λ IL X ;5. If h ρ, σ i ∈ R Λ IL X and h σ, τ i ∈ R Λ IL X , then h ρ, σ, τ i ∈ S Λ IL X ;6. If h ρ, σ, τ i ∈ S Λ IL X and h ρ, τ, υ i ∈ S Λ IL X , then h ρ, σ, υ i ∈ S Λ IL X ;7. If h ρ, σ, τ i ∈ S Λ IL X , then h ρ, σ i ∈ R Λ IL X and h ρ, τ i ∈ R Λ IL X ;8. h Λ , R Λ IL X , S Λ IL X i (cid:15) fo C X .Note that the least relations exist since C X is a set of strict first order Hornsentences. If the context allows us so, we will drop both the sub- and thesuperscripts in R Λ IL X and S Λ IL X . Moreover, when h ρ, σ, τ i ∈ S we will denote thisby σ S ρ τ and likewise for R .Note that R is irreflexive in case IL X is consistent and Λ sufficiently nice.Moreover, apart from R being Noetherian, all the other properties of IL X -framesare satisfied: R is transitive; S ρ is a relation on { σ ∈ Λ | ρ R σ } that is transitiveand reflexive so that ρ R σ R τ ⇒ σ S ρ τ .We will now define the generating rules for tableaux for Horn interpretabilitylogics. As mentioned, the nodes carry labelled formulas which consist of a pair σ :: A , where σ is a label and A is a formula. Recall that the idea of the labels is,that they will correspond to worlds in a model where the corresponding formulawill be satisfied if satisfiable.The rules that we present are not entirely local since, for example, we haveto guarantee that new labels have not yet been used in relevant parts of thetableau so far. Thus, we define the rules relative to a set of labels. Definition 5 (Tableau rules) . Let IL X be a Horn interpretability logic and letΛ be a set of labels. The IL X -tableau rules with respect to Λ are as follows:Propositional rules: σ :: ¬¬ A ( ¬ ) ; σ :: Aσ :: A → B ( → ) ; σ :: ¬ A | σ :: B :: ¬ ( A → B )( ¬ → ) . σ :: Aσ :: ¬ B ( ν )-rules: σ :: (cid:3) A ( ν (cid:3) , Λ) , when σ R τ ; τ :: Aσ :: (cid:3) ρ A ( ν S , Λ) , when σ S ρ τ ; τ :: Aσ :: A ⊲ B ( ν ⊲ , Λ) , when σ R τ . τ :: ¬ A | τ :: ¬ (cid:3) σ ¬ B ( π )-rules: σ :: ¬ (cid:3) A ( π (cid:3) , Λ) , where n ∈ N is such that σRn / ∈ Λ; σRn :: ¬ AσRn :: (cid:3) Aσ :: ¬ (cid:3) ρ A ( π S , Λ) , where n ∈ N is such that σS ρ n / ∈ Λ; σS ρ n :: ¬ AσS ρ n :: (cid:3) Aσ :: ¬ ( A ⊲ B )( π ⊲ , Λ) , where n ∈ N is such that σRn / ∈ Λ. σRn :: AσRn :: (cid:3) σ ¬ BσRn :: (cid:3) ¬ A We call the labelled formula above the line in the rules above the antecedent and the labelled formula(s) under the line succedent(s) .Some clarifying remarks on the tableau rules seem in order. First we notethat we use the symbol “ | ” in the rules ( → ), and ( ν ⊲ , Λ) to denote branchingin proof-trees. Next, we observe that various non-branching rules have multiplesuccedents such as the rules ( ¬ → ), ( π (cid:3) , Λ) and ( π ⊲ , Λ). These succedents areto be understood as different nodes one placed under the other. Lemma 1.1is reflected in the rules ( π (cid:3) , Λ) and ( π ⊲ , Λ) and Lemma 1.2 is reflected in the( π S , Λ) rule.Another non-local feature of the tableaux proof system will be that we willallow to apply rules to any node σ :: A in a branch, not necessarily only tobottom-nodes. Upon application of the rule, the succedent(s) with possiblebranching can be appended to the bottom of any branch passing through σ :: A .If B is a branch in a tree whose nodes are labelled formulas, by lab( B ) we denotethe collection of labels that occur in B .7 efinition 6 ( IL X -Tableaux, open and closed) . Given a Horn logic IL X anda finite set Γ of formulas, an IL X -tableau for Γ is a binary irreflexive directeddownward growing tree with nodes carrying labelled formulas defined induc-tively as follows: • A single node tree T with 0 :: A as the sole node for some formula A ∈ Γis an IL X -tableau for Γ. • If T is an IL X -tableau for Γ, then a tree T ′ obtained by extending (ap-pending below) any branches of T with 0 :: A for some formula A ∈ Γ isan IL X -tableau for Γ. • Let T be an IL X -tableau for Γ, B be a branch of T , and let ( ρ ) be arule w.r.t. lab( B ). If some labelled formula σ :: A that occurs in B is theantecedent of an instance of ( ρ ), then the tree T ′ obtained by extending B with the appropriate succedents of ( ρ ) in any particular ordering (withpossible branching) is an IL X -tableau for Γ.A branch B of an IL X -tableau T for Γ is called closed if there is σ and A such that σ :: A ∈ B and σ :: ¬ A ∈ B . Otherwise the branch is open . An IL X -tableau T for Γ is closed if all of its branches are closed. Otherwise T isopen.Given a Horn logic IL X , we are now ready to assign to a finite set of for-mulas Γ what we call a systematic IL X -tableau for Γ which will contain all theinformation as to the satisfiability of Γ. The systematic tableau method givenbelow follows closely the procedure given in [7].
Definition 7 (Systematic IL X -tableau) . For a Horn logic IL X , a systematic IL X -tableau for a finite set Γ of formulas is constructed in stages. Throughoutthe stages, the nodes in the tree T i will be marked with exactly one of awake,asleep or finished . The marked version of T i will be denoted by µ ( T i ). Stage 0:
Form the initial tableau T with 0 :: A for all A ∈ Γ in some order ontop of each other and mark them all awake.
Stage n+1:
Look for an awake σ :: A in µ ( T n ) closest to the root of the tableau;if there are several with the same distance, choose the leftmost one. If A = p or A = ¬ p for some propositional variable p , then T n +1 and µ ( T n +1 ) are as T n and µ ( T n ) respectively except that we mark the node σ :: A as finished and weend Stage n+1.Otherwise we obtain T n +1 and µ ( T n +1 ) as follows: • If A = ¬¬ B for some B , for every open branch B that passes through σ :: A , extend B with σ :: B marking it awake and marking σ :: A asfinished. Here and below ‘extending B ’ means ‘appending new nodes tothe bottom of B ’. • If A = ( B → C ) for some B and C , for every open branch B that passesthrough σ :: A , split the end of B and extend the left fork with σ :: ¬ B and the right fork with σ :: C . Both new nodes will be marked awake and σ :: A will be marked as finished. • If A = ¬ ( B → C ) for some B and C , for every open branch B that passesthrough σ :: A , extend B with σ :: B and σ :: ¬ C in whatever order. Bothnew nodes will be marked awake and σ :: A will be marked as finished.8 If A = (cid:3) B for some B , for every open branch B that passes through σ :: A and for all τ ∈ lab( B ), if σ R τ , then extend B with τ :: B . Thesenew nodes will be marked awake and σ :: A will be marked as asleep. • If A = ¬ (cid:3) B for some B , for every open branch B that passes through σ :: A , extend B with σRn :: ¬ B and σRn :: (cid:3) B , where n ∈ N is the leastnumber such that σRn / ∈ lab( B ). Mark both σRn :: ¬ B and σRn :: (cid:3) B awake and σ :: A finished. Moreover, mark as awake every τ :: (cid:3) B ∈ B and τ :: B ⊲ C ∈ B whenever τ R σRn and mark awake every τ :: (cid:3) ρ B ∈ B whenever τ S ρ σRn . • If A = (cid:3) ρ B for some ρ and B , for every branch B that passes through σ :: A and for all τ ∈ lab( B ), if σ S ρ τ , then extend B with τ :: B . Mark τ :: B awake and σ :: A asleep. • If A = ¬ (cid:3) ρ B for some ρ and B , for every open branch B that passesthrough σ :: A , extend B with σS ρ n :: ¬ B and σS ρ n :: (cid:3) B , where n ∈ N is the least number such that σS ρ n / ∈ lab( B ). Mark both σS ρ n :: ¬ B and σS ρ n :: (cid:3) B awake and mark σ :: A finished. Moreover, mark awakeevery τ :: (cid:3) B ∈ B and τ :: B ⊲ C ∈ B such that τ R σS ρ n and every τ :: (cid:3) ρ B ∈ B such that τ S ρ σS ρ n . • If A = B ⊲ C for some B and C , for every open branch B that passesthrough σ :: A and every τ ∈ lab( B ), if σ R τ , split the end of B and extendthe left fork with τ :: ¬ B and the right fork with τ :: ¬ (cid:3) σ ¬ C . Both newnodes are marked awake and σ :: A will be marked asleep. • If A = ¬ ( B ⊲ C ) for some B and C , for every open branch B that passesthrough x , pick the smallest n ∈ N such that σRn / ∈ lab( B ) and extend B with σRn :: B , with σRn :: (cid:3) σ ¬ C and with σRn :: (cid:3) ¬ B in whateverorder you like. All new nodes are marked awake and σ :: A finished.Moreover, mark awake every τ :: (cid:3) B ∈ B and τ :: B ⊲ C ∈ B such that τ R σRn and every τ :: (cid:3) ρ B ∈ B such that τ S ρ σRn .By this procedure we construct a chain h T i : i ∈ ω i of IL X -tableaux for Γ.We call S i ∈ ω T i a systematic IL X -tableau for X . Remark 1.
A systematic IL X -tableau T for a finite set Γ of formulas is not ingeneral an IL X -tableau in the sense of Definition 6. However, if T is finite it isan IL X -tableau. In particular, if T closes, then T is an IL X -tableau. Moreover,if there is a closed IL -tableau T ′ for Γ, then there is a closed systematic tableaufor Γ. Lemma 2 (Fairness) . If σ :: A is awake at stage n+1, the systematic IL X -tableau procedure visits σ :: A at some later stage.Proof. Straightforward and similar to Lemma 6.4.4 in [7]. An example is when Γ = { ♦ p, p ⊲ q, q ⊲ p } : The ♦ p yields a world where p holds, and sincewe do not reuse worlds the circular p ⊲ q, q ⊲ p keep on creating new worlds. Soundness
As usual, for a Horn logic IL X , we call a formula A IL X -tableau provablewhenever the systematic IL X -tableaux for {¬ A } closes. In this section we shallshow that this notion of provability is sound with respect to IL X frames. Definition 8.
A set X of labelled formulas is IL X -satisfiable if there exists an IL X -model M = h W, R, S, V i and an interpretation I : lab( X ) → W such that(i) If σ, τ ∈ lab( X ) and σ R τ , then I ( σ ) RI ( τ );(ii) If ρ, σ, τ ∈ lab( X ) and σ S ρ τ , then I ( σ ) S I ( ρ ) I ( τ );(iii) M, I ( σ ) (cid:13) A for all σ :: A ∈ X ;An IL X -tableau T is IL X -satisfiable, if there is a branch B of T such that B is IL X -satisfiableThe next lemma tells us that satisfiable IL X -tableaux are closed under ap-plying the rules to them for Horn logics IL X . Lemma 3.
Let IL X be a Horn logic and let T be a satisfiable IL X -tableau.Then for any rule, the tableau T ′ obtained by the application of the rule is also IL X -satisfiable.Proof. Suppose B is an IL X -satisfiable branch of T . We show that if we applysome rule to a some labelled formula in B , we obtain a branch that is IL X -satisfiable. The cases for the propositional rules are trivial.Suppose σ :: (cid:3) A ∈ B and consider the branch C obtained by an applicationof the ( ν (cid:3) ) -rule with τ :: A added to the branch for some τ ∈ lab( B ) suchthat σ R τ . By assumption there is an IL X -model M = h W, R, S, V i and aninterpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) (cid:3) A . Now τ ∈ lab( B ) and σ R τ . Hence I ( σ ) RI ( τ ) and so M, I ( τ ) (cid:13) A .Suppose σ :: (cid:3) ρ A ∈ B and consider the branch C obtained by an applicationof the ( ν S ) rule with τ :: A added to the branch for some τ ∈ lab( B ) suchthat σ S ρ τ . By assumption there is an IL X -model M = h W, R, S, V i and aninterpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) (cid:3) I ( ρ ) A . Now since τ, ρ ∈ lab( B ) and σ S ρ τ , we also have that I ( σ ) S I ( ρ ) I ( τ ). Hence M, I ( τ ) (cid:13) A .Suppose σ :: A ⊲ B ∈ B and consider the two branches obtained by an application of the ( ν ⊲ ) rule with τ :: ¬ A in the left branch and τ :: ¬ (cid:3) σ ¬ B in the right branch for some τ ∈ lab( B ) such that σ R τ . Now by assumptionthere is an IL X -model M = h W, R, S, V i and an interpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) A ⊲ B . Now since τ ∈ lab( B ) and σ R τ , we have that I ( σ ) RI ( τ ). If M, I ( τ ) (cid:13) ¬ A , then the left branch is satisfiable with I . If on theother hand M, I ( τ ) (cid:13) A , then there exists some x ∈ W such that I ( τ ) S I ( σ ) x and M, x (cid:13) B . Hence M, I ( τ ) (cid:13) ¬ (cid:3) I ( σ ) ¬ B .Suppose σ :: ¬ (cid:3) A ∈ B and consider the branch C obtained by an ap-plication of the ( π (cid:3) ) rule with σRn :: ¬ A and σRn :: (cid:3) A added to thebranch for some n ∈ N such that σRn / ∈ lab( B ). By assumption there is an IL X -model M = h W, R, S, V i and an interpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) ¬ (cid:3) A . Hence there is some x ∈ W such that I ( σ ) Rx and M, x (cid:13) ¬ A M, x (cid:13) (cid:3) A . Now, since σRn / ∈ lab( B ), we can extend I to I ′ by putting I ′ ( σRn ) = x . Now define R I ′ and S I ′ on lab( B ) ∪ { σRn } by h τ, ρ i ∈ R I ′ ⇔ I ′ ( τ ) RI ′ ( ρ ) , h υ, τ, ρ i ∈ S I ′ ⇔ I ′ ( τ ) S I ′ ( υ ) I ′ ( ρ ) . Now R I ′ and S I ′ satisfy conditions (1)-(8) in Definition 4. Hence R ⊆ R I ′ and S ⊆ S I ′ , and so I ′ is an interpretation from C to M .Suppose σ :: ¬ (cid:3) ρ A ∈ B and consider the branch C obtained by an applica-tion of the ( π S ) rule with σS ρ n :: ¬ A and σS ρ n :: (cid:3) A added to the branch B for some n ∈ N . By assumption there is an IL X -model M = h W, R, S, V i and aninterpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) ¬ (cid:3) I ( ρ ) A . Now there is some x ∈ W such that I ( σ ) S I ( ρ ) x and M, x (cid:13) ¬ A . Now if M, x (cid:13) (cid:3) A , we extend I to I ′ by putting I ′ ( σS ρ n ) = x . On the other hand if M, x (cid:13) ¬ (cid:3) A , then there is y ∈ W such that xRy and M, y (cid:13) ¬ A and M, y (cid:13) (cid:3) A . Now I ( ρ ) RxRy and so xS I ( ρ ) y . Hence I ( σ ) S I ( ρ ) y . Now extend I to I ′ by putting I ′ ( σS ρ n ) = y . Now I ′ is again an interpretation from C to M .Suppose finally σ :: ¬ ( A ⊲ B ) ∈ B and consider the branch obtained by anapplication of the ( π ⊲ ) rule with σRn :: A , σRn :: (cid:3) σ ¬ B and σRn :: (cid:3) ¬ A added to the branch for some n ∈ N . By assumption there is an IL X -model M = h W, R, S, V i and an interpretation I : lab( B ) → W such that M, I ( σ ) (cid:13) ¬ ( A ⊲ B ). Hence there is x ∈ W such that I ( σ ) Rx , M, x (cid:13) A and M, x (cid:13) (cid:3) I ( σ ) ¬ B . If M, x (cid:13) (cid:3) ¬ A , we may extend I to I ′ by putting I ′ ( σRn ) = x . On the other handif M, x (cid:13) ¬ (cid:3) ¬ A , then there is y ∈ W such that M, y (cid:13) A and M, y (cid:13) (cid:3) ¬ A . Butnow since I ( σ ) RxRy , we have that xS I ( σ ) y and so M, y (cid:13) (cid:3) I ( σ ) ¬ B . So now wemay extend I to I ′ by putting I ′ ( σRn ) = y . Again, I ′ is an interpretation from C to M . Theorem 1.
Let IL X be a Horn logic. If a systematic IL X -tableau for a set offormulas Γ closes, then Γ is IL X -unsatisfiable.Proof. Suppose a systematic tableau T for Γ is closed, but that there is an IL X -model M = h W, R, S, V i and x ∈ W such that M, x (cid:13)
Γ. Now consider theinitial tableau T for Γ with 0 :: A for all A ∈ Γ.Now, by assumption, I = {h , x i} is an interpretation from lab( T ) to W .By the above lemma, every tableau obtained from the initial tableau is IL X -satisfiable. In particular, the closed tableau T obtained by the systematic pro-cedure is IL X -satisfiable. A contradiction since, for any branch B of T there is σ and A such that σ :: A ∈ B and σ :: ¬ A ∈ B .For the sake of being explicit let us formulate the soundness of our tableauxas an immediate corollary. Corollary 1.
Let IL X be a Horn logic. If a systematic IL X -tableau for {¬ A } closes, then A is IL X -valid. In this section we shall show that our proof system is also complete w.r.t. IL X -frames. 11 efinition 9. A set X of labelled formulas is a IL X -Hintikka set if the followinghold:(i) There is no σ and A such that σ :: A ∈ X and σ :: ¬ A ∈ X ;(ii) If σ :: ¬¬ A ∈ X , then σ :: A ∈ X ;(iii) If σ :: A → B ∈ X , then σ :: ¬ A ∈ X or σ :: B ∈ X ;(iv) If σ :: ¬ ( A → B ) ∈ X , then σ :: A ∈ X and σ :: ¬ B ∈ X ;(v) If σ :: A ⊲ B ∈ X , then τ :: ¬ A ∈ X or τ :: ¬ (cid:3) σ ¬ B ∈ X for all τ ∈ lab( X )such that σ R τ ;(vi) If σ :: ¬ ( A ⊲ B ) ∈ X , then there is τ ∈ lab( X ) such that τ :: A ∈ X , τ :: (cid:3) σ ¬ B ∈ X and σRτ ;(vii) If σ :: (cid:3) A ∈ X , then τ :: A ∈ X for each τ ∈ lab( X ) such that σ R τ ;(viii) If σ :: ¬ (cid:3) A ∈ X , then there is τ ∈ lab( X ) such that τ :: ¬ A ∈ X and σ R τ ;(ix) If σ :: (cid:3) ρ A ∈ X , then τ :: A ∈ X for each τ ∈ lab( X ) such that σ S ρ τ ;(x) If σ :: ¬ (cid:3) ρ A ∈ X , then there is τ ∈ lab( X ) such that τ :: ¬ A ∈ X and σ S ρ τ .Hintikka sets contain all the needed information to extract a model fromthem. This is clearly manifested in the proof of the following lemma. Lemma 4 (Truth Lemma) . Let IL X be a Horn logic. If X is an IL X -Hintikkaset and R lab( X ) IL X is Noetherian, then X is IL X -satisfiable.Proof. As mentioned before, we will omit various sub and superscripts. Thus,if R is Noetherian, then F = h lab( X ) , R , S i is clearly an IL X -frame. Define avaluation V on lab( X ) by putting V ( p ) = { σ ∈ lab( X ) : σ :: p ∈ X } for all propositional variables p. Now we can prove by an easy induction on the complexity of formulas that forall σ and A : σ :: A ∈ X ⇒ hF , V i , σ (cid:13) A and σ :: ¬ A ∈ X ⇒ hF , V i , σ A. Hence hF , V i satisfies X with the identity interpretation.12 emma 5. If B is an open branch in a systematic IL X -tableau for a finite set Γ , then B is a Hintikka set and R is Noetherian.Proof. That B is a Hintikka set follows easily from the fairness of the systematic IL X -tableau procedure.Notice that if σ :: (cid:3) A ∈ B for some σ and A , then A is either a subformulaof a formula from Γ or the negation of a subformula of a formula from Γ. Let˜Γ = sub[Γ] ∪ {¬ A : A ∈ sub[Γ] } .Now suppose towards a contradiction that there is an ascending R -chain h σ i : i ∈ ω i in lab( B ). Without loss of generality we may assume that σ = 0and σ i = 0 for all i > i ∈ ω there is A i such that σ i +1 :: (cid:3) A i ∈ B , but σ j :: (cid:3) A i / ∈ B for all j ≤ i .If σ i +1 = τ Rn for some τ ∈ lab( B ) and n ∈ N , then σ i +1 is introduced eitherwith a ( π (cid:3) )-rule applied to some τ :: ¬ (cid:3) A ∈ B or by a ( π ⊲ )-rule applied tosome τ :: ¬ ( A ⊲ B ) ∈ B . In the first case σ i +1 :: (cid:3) A ∈ B , but σ j :: (cid:3) A / ∈ B forall j ≤ i . In the second case σ i +1 :: (cid:3) ¬ A ∈ B , but σ j :: (cid:3) ¬ A / ∈ B for all j ≤ i .If σ i +1 = τ S ρ n for some τ, ρ ∈ lab( B ) and n ∈ N , then σ i +1 is introducedwith a ( π S )-rule applied to some τ :: ¬ (cid:3) ρ A . Now σ i +1 :: (cid:3) A ∈ B , but σ j :: (cid:3) A / ∈ B for all j ≤ i .Now for large enough m , |{ A : A ∈ ˜Γ and σ m :: A ∈ B}| > | ˜Γ | . We have the following corollaries from this lemma.
Corollary 2.
Let IL X be a Horn logic. If a systematic IL X -tableau for a finite Γ has an open branch, then Γ is IL X -satisfiable. In particular, we can formulate completeness of our proof systems.
Corollary 3.
Let IL X be a Horn logic. If A is IL X -valid, then any systematic IL X -tableau for {¬ A } closes. Our results apply to all interpretability logics extending IL that are Horn. Inparticular the results apply to the most important systems IL M and IL P . Atfirst sight, the restriction of the logic being Horn might seem quite severe. How-ever, most logics that occur in the literature turn out to be Horn. In particularalso the logics based on R : A ⊲ B → ¬ ( A ⊲ ¬ C ) ⊲ B ∧ (cid:3) C ;and the two series of generalizations of this principle as presented in [9] are Hornlogics. An important logic that falls out of the scope of this paper is IL W sincethe corresponding frame condition is second order.13 cknowledgement We would like to thank Rajeev Gor´e for his comments on a draft version ofthis paper. Further thanks go to Volodya Shavrukov and an anonymous refereefor helping to improve the paper. The second author was supported by theGeneralitat de Catalunya under grant number 2014 SGR 437 and from theSpanish Ministry of Science and Education under grant number MTM2014-59178-P.
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