Modal Horn Logics Have Interpolation

Transcription

1 Modal Horn Logis Have Interpolation Marus Kraht Department of Linguistis, UCLA PO Box Hilgard Avenue Los Angeles, CA USA Abstrat We shall show that the polymodal Sahlqvist logis orresponding to a set of Horn formulae have interpolation. 1 Introdution This paper has been sparked off by a reent result by Rajeev Goré and Stéphane Demri that all grammar logis have interpolation. A grammar logi is a modal logi axiomatized by axioms of the form σ p i p, where σ is a string of modal operators. In this paper I show that this holds of a wider lass of logis, those whih are determined by a set of Sahlqvist formulae that are also Horn formulae. 2 Definitions Notation and terminology is as in [1]. Our language is that of polymodal logi i with any set B of basi modalities. We write x y in plae of x R( i ) y, and extend this to ompound modalities. Compound modalities are here simply sequenes σ = i 0 ; i 1 ; ; i p 1 where i j B for all i < p. Thus, x R( σ y is synonymous x σ y. Sahlqvist logis determine first-order onditions on frames. We 1

2 shall be interested in a very speifi lass. Reall that first-order Sahlqvist formulae have the following form. They are formed from atomi or onstant formulae using, and restrited existential and universal quantifiers suh that at least one variable in every matrix ondition is quantified by a universal quantifier that is not in the sope of an existential. In this paper we are interested in logis whose lass of frames is haraterised by sets of formulae that are both Sahlqvist and Horn formulae. It is not hard to see the following. We first do a first-order haraterisation and then proeed to a modal version. Call a formula a tree formula if it is formed from using and restrited existentials. Model theoretially suh a formula χ is satisfiable in a frame iff T(χ) is embeddable where T is defined from the formula by indution as follows. (a) T(χ) is a one-point tree (the relation is empty), (b) T( χ) is obtained by adding a node at the bottom of the tree to be the new root, () T(χ χ ) is obtained by fusing the two roots of T(χ) and T(χ ). Lemma 1 A Sahlqvist formula is a Horn formula iff it is formed from any number of tree formulae and a single matrix x R( i ) y using and restrited universal quantifiers. For a proof notie the following. A Horn formula has the form (1) ξ = ( x)( α i β) i<n where α i, i < n, and β are atomi. In order to onvert this Sahlqvist form, the α i must be restritors of the variables, sine they are negative. Hene they have the form w R( i ) w, where w does not our to the right of R( j ) in any of the α i (but it may our to the left). Hene, say that w immediately depends on w if there is an i suh that α i = w R( j ) w for some w, w and j B; let dependeny be the transitive losure of immediate dependeny. Then dependeny is a tree order with root x 0. Now, let β = x R( i ) y. Let x = x 0 x p 1. Let X be the set of variables on whih x depends, Y the set of variables on whih y depends, D := X Y. Let the variables be numbered suh that X Y = x 0 x q 1. Then (2) ξ ( x 0 x q 1 )(( x q x p 1 α i ) x R( i ) y) Now pik i < n. α = w R( j ) w. Two hoies arise: Case 1. w X Y. Then also w X Y. In this ase, α i an be moved outside the sope of the existentials. Moreover, w is bound by a universal quantifier, and α i is its restritor. Case 2. w X Y. Then w is bound by an existential, and α i is its restritor. This shows the laim. 2 i<n

3 We all a Sahlqvist Horn formula pure if it does not use any tree formula; equivalently, X Y = x 0 x p 1. Definition 2 Call L a Horn logi if it is Sahlqvist and determines a ondition on its frames that is haraterised by a set of Horn sentenes. This lass inludes the following axioms: reflexivity, transitivity, symmetry, tense axioms, and grammar logis. All these formulae are also pure. The theorem below therefore generalises a theorem given in [1]. Using ompound modalities, pure Sahlqvist Horn formulae an be defined as follows. They have the form (3) ( xyzw)((x ρ y y σ z y τ w) w i z) with ρ, σ and τ are sequenes. The orresponding Sahlqvist formula is as follows. (4) ρ ( σ p τ i p) We use an alternate of this formula, obtained by replaing p by and eliminating negations: (5) ρ ( τ i p σ p) We shall also all suh modal formulae Horn formulae. The general ase of a Sahlqvist Horn formula is as follows. Say that a onstant formula χ is a tree formula if it is onstant and uses only and i, i B. Now, in addition to ordinary boxes allow also the use of χ j, whih are defined as follows: χ j ϕ := j(χ ϕ) and all these tree restrited basi modalities. Extend the notion of ompound modality to inlude tree restrited basi modalities. Then (4) and (5) an be used verbatim. 3 The Main Theorem Theorem 3 Modal Sahlqvist Horn logis have interpolation. We use the method of onstrutive redution. Let L be Sahlqvist Horn. Reall that there is a funtion X L ( ) suh that (6) ; X L ( ; ϕ) K ϕ L ϕ 3

4 and for all : var(x L ( )) var( ). Also, the sets an be hosen finite if, but we do not assume that X L ( ) is finite. Moreover, if X L has the property (7) X L (ϕ; δ) = X L (ϕ) X L (δ) then L has interpolation. We shall show that a funtion X L satisfying (7) an be found. The proof is simple. Suppose ϕ L ψ. Then ϕ; X L (ϕ; ψ) L ψ, and so ϕ; X L (ϕ); X L (ψ) K ψ. Now, either X L (ϕ) and X L (ψ) are already finite or else we may at this point hoose finite subsets of them and ontinue with those in plae of the original sets. From this we get ϕ; X L (ϕ) K XL (ψ) ψ. Now, var(ϕ; X L (ϕ)) = var(ϕ) and var( X L (ψ) ψ). K has interpolation, and so there exists a χ suh that var(χ) var(ϕ) var(ψ) and ϕ; X L (ϕ) K χ and χ K XL (ψ) ψ. From this we get ϕ; X L (ϕ) L χ and χ L XL (ψ) ψ. Sine X L ( ) is a set of theorems of L, ϕ L χ as well as χ L ψ. Let be given. We know that has a model based on a frame whih is a tree of depth bounded by the modal depth of. Let this tree be T = T, R with root w 0. Lemma 4 Let L be a Sahlqvist Horn logi. For every Kripke frame W, R there is a least R R suh that W, R is an L frame. Proof. Clearly, the elementary axioms do not ask for the addition of new points; rather, they ask for the introdution of new relations between points. It is easy to define a funtion on the set of all funtions B W 2 suh that if R does not yet satisfy the axioms, then f (R) onsists in the addition of a single transition that is fored by the axioms. Then R is defined either as the limit of f n (R), or as the intersetion of all f losed sets. So, the proess of adding relations is iterative. Notie that the Horn formulae say something like this: there is a tree T suh that if this tree is embeddable into the frame then one additional relation obtains. Suppose that we start with a tree T 0. Then the first set of onsequenes are derived from embedding T into T 0 and then adding the new relations. This gives T 1, whih may embed T in more ways, yielding additional patterns. This an be flattened into a single operation by losing the Horn formulae under derivability. This will mean that there is no easy way to hoose a finite redution set, but this is atually of no relevane for interpolation. At this point it is perhaps instrutive to see why pure Sahlqvist Horn formulae are not suitable for the proof (and why we have to move to the more omplex Sahlqvist Horn formulae). Let δ 1 = ( xyz)((x a;b y x 4 b z) (y z)),

5 δ 2 = ( xy)(x ; y x y) and δ 3 = ( xy)(x y y x). Let the a b b following tree be given: w 0 w 1 w 2, w 0 w 3. Then, using δ 1 we get w 0 w 3, using δ 3 we get w 3 w 0, and finally through δ 2 we get w 2 w 2. Now, if the node w 2 had not been there, we would not have to add the transition w 2 w 2. Whether or not w 2 w 2 depends not only on the path between them b but also on the existene of some additional path, here w 0 w 3. Now we enter the definition of X L. First, let H be the set of all Sahlqvist Horn formulae valid in L. We assume χ H to be of the form (5). Now set (8) X L ( ) := { ρ ( τ i δ σ δ) : i δ Sf( ), ρ ( τ i p σ p) H} This set has the property (7). (For i δ Sf(ϕ; δ) iff i δ Sf(ϕ) or i δ Sf(δ).) We have to show that it also satisfies (6). The diretion from left to right is lear. Therefore, assume (9) ; X L ( ; ϕ) K ϕ Then there is a frame W, R suh that (10) W, R, w 0, β ; X L ( ; ϕ); ϕ Let R be defined as follows. R is the set of all transitions w following is satisfied: (11) w 0 ρ y; y σ z; y τ w i z suh that the Clearly, these transitions must be added in order to satisfy the first order ondition. Also, it is enough to add those, sine these are all the transitions that an be dedued to exist. We show by indution on χ Sf( ; ϕ) that (12) W, R, β, z χ W, R, β, z χ There is only one problemati ase, namely χ = i δ. ( ) is lear. ( ). So assume that we have (13) W, R, β, z i δ Pik w suh that z R ( i ) w. Two ases arise: z R( i ) w and so already W, R, β, w. i The seond ase is when the transition z w has been added. By assumption this is the ase beause W, R satisfies (14) w 0 ρ y; y σ z; y τ w 5

6 where ρ, σ and τ are sequenes of tree restrited basi modalities. This means that (15) W, R, β, w 0 ρ ( τ i δ σ δ) From this we get (16) W, R, β, y τ i δ σ δ By assumption, sine y τ z and W, R, β, z i δ we now have (17) W, R, β, y σ δ Now, sine y σ w, we must have (18) W, R, β, w δ as required. It follows that for all w suh that z R ( i ) w we have W, R, β, w δ. By indution hypothesis W, R, β, w δ, for all w suh that x R( i ) w, from whih W, R, β, z i δ. This onludes the proof. Referenes [1] Marus Kraht, Tools and Tehniques in Modal Logi, Studies in Logi, no. 142, Elsevier, Amsterdam,