On the Craig interpolation and the fixed point

On the Craig interpolation and the fixed point property for GLP Lev D. Beklemishev December 11, 2007 Abstract We prove the Craig interpolation and the fixed point property for GLP by finitary methods. Konstantin Ignatiev [4], among other things, established the Craig interpolation and the fixed point property for Japaridze s polymodal provability logic GLP. However, it remained open if these results could be established by finitary methods formalizable in Peano arithmetic PA. (The question concerning the Craig interpolation was stated e.g. in [3].) In this note we provide such proofs. These proofs are based on our previous paper [1] where a complete Kripke semantics for GLP is given. In that paper, using only finitary methods, the system GLP is reduced to a certain natural subsystem, denoted J. 1 J is sound and complete w.r.t. a natural class of finite Kripke frames [1]. (It is well-known that GLP is not complete w.r.t. any class of Kripke frames.) We establish the Craig interpolation and the fixed point properties for J, which also enables us to extend them to GLP. Apart from the reduction of GLP to J established in ref. [1] our methods are very standard. 1 Preliminaries The system J is given by the following axiom schemata and inference rules. Axioms: (i) Boolean tautologies; (ii) [n](ϕ ψ) ([n]ϕ [n]ψ); 1 Similarly, Ignatiev [4] used a reduction of GLP to a weaker subsystem I, however the reducibility has not been established by finitary methods. 1

(iii) [n]([n]ϕ ϕ) [n]ϕ; (iv) [m]ϕ [n][m]ϕ, for m n. (v) m ϕ [n] m ϕ, for m < n. (vi) [m]ϕ [m][n]ϕ, for m n. Rules: modus ponens, ϕ [n]ϕ. GLP is obtained from J by adding the monotonicity schema [m]ϕ [n]ϕ, for m n. Ignatiev s logic I is obtained from J by deleting axiom schema (vi). The system J is sound and complete w.r.t. the class of Kripke frames satisfying the following conditions: R k is a upwards well-founded, transitive ordering relation on W, for each k 0; x, y (xr n y z (xr m z yr m z)) if m < n; x, y (xr m y & yr n z xr m z) if m n. (I) (J) Such frames will be called J-frames. One of the main results of ref. [1] states that GLP is reducible to J as follows. Let M(ϕ) := i<s ([m i]ϕ i [m i + 1]ϕ i ), where [m i ]ϕ i for i < s are all subformulas of ϕ of the form [k]ψ. Let + ϕ := ϕ i n[i]ϕ, where n := max i<s m i, and let M + (ϕ) := + M(ϕ). Then, GLP ϕ J M + (ϕ) ϕ. (Red) This result is proved by finitary methods based on Kripke semantics. 2 Craig interpolation theorem for J In the proof of the Craig interpolation theorem we shall use notation similar to Tait-style sequent calculus, that is: 2

Formulas are built-up from constants,, propositional variables p i, i 0, and their negations p i using,, and modalities n, [n], for each n 0; Sequents are finite sets of formulas (denoted Γ,, etc.) understood as disjunctions of their elements. We write Γ if J Γ. Negation ϕ of a formula ϕ is defined by de Morgan s rules and the following identities: [n]ϕ := n ϕ, n ϕ := [n] ϕ. Implication ϕ ψ is defined by ϕ ψ. As usual we write Γ, for Γ and Γ, ϕ for Γ {ϕ}. We use the following abbreviations: n Γ := { n ϕ : ϕ Γ}, [n]γ := {[n]ϕ : ϕ Γ}. n Γ denotes the result of prefixing each formula from Γ by a modality of the form m for some m n (m can be different for each formula from Γ). n Γ is similarly defined. Lemma 2.1 Suppose is a set of formulas of the form m ψ and [m]ψ with m < n. If, n Γ, Γ, n ϕ, ϕ then, n Γ, [n]ϕ. Proof. Assume J (, Γ, n Γ, n ϕ, ϕ), then by propositional logic J ( Γ n Γ) ([n]ϕ ϕ). Denoting ϕ 1 :=, ϕ 2 := Γ, and ϕ 3 := n Γ we obtain: J [n](ϕ 1 ϕ 2 ϕ 3 ) [n]([n]ϕ ϕ) (1) [n]ϕ, by (iii). (2) However, if [m]ψ then J [m]ψ [n][m]ψ by (iv), and if m ψ, then J m ψ [n] m ψ by (v). Hence, J ϕ 1 [n]ϕ 1. Similarly, if [k]ψ n Γ then J [n]ψ [n](ψ [k]ψ), by (vi). Hence, We conclude J [n]ϕ 2 [n](ϕ 2 ϕ 3 ). J ϕ 1 [n]ϕ 2 [n](ϕ 1 ϕ 2 ϕ 3 ) [n]ϕ, by (2). 3

It follows that J derives (, n Γ, [n]ϕ), as required. Let Var(ϕ) denote the set of variables occurring in ϕ and Var(Γ) := ϕ Γ Var(ϕ). We say that θ interpolates a pair of sequents (Γ; ) if Var(θ) Var(Γ) Var( ) and Γ, θ and θ,. (Γ; ) is inseparable if it does not have an interpolant. The following theorem subsumes both the completeness theorem for J and the Craig interpolation theorem. Theorem 1 The following statements are equivalent: (i) (Γ; ) has an interpolant; (ii) Γ, ; (iii) For all (finite) J-models W, W (Γ ). Proof. The implications (i) (ii) and (ii) (iii) are easy. We prove (iii) (i). Call a finite set Φ of formulas adequate if it is closed under subformulas, negation, the following operation: [n]ϕ, [m]ψ Φ [m]ϕ Φ, and for each variable p Φ contains p p. Let Op(Φ) = {n ω : [n]ϕ Φ, for some ϕ}. Clearly, every finite set of formulas Ψ can be extended to a finite adequate set Φ Ψ such that Op(Φ) = Op(Ψ) and Var(Φ) = Var(Ψ). Let us fix some finite adequate Φ. Below we shall only consider sequents Γ over Φ, that is, Γ Φ. An inseparable pair (Γ 1 ; Γ 2 ) is maximal if for any other inseparable pair ( 1 ; 2 ) such that Γ 1 1 and Γ 2 2 one has Γ 1 = 1 and Γ 2 = 2. Lemma 2.2 Suppose (Γ 1 ; Γ 2 ) is maximal inseparable. Then, for all ϕ, ψ Φ, and i = 1, 2: (i) (ϕ ψ) Γ i ϕ Γ i or ψ Γ i ; (ii) (ϕ ψ) Γ i ϕ Γ i and ψ Γ i ; (iii) If Var(ϕ) Var(Γ i ) then either ϕ Γ i or ϕ Γ i ; 4

(iv) For no ϕ both ϕ, ϕ Γ i. Proof. By the obvious symmetry it is sufficient to prove both claims for i = 1. (i) Assume ϕ, ψ Γ 1. We claim that at least one of the following two pairs is inseparable: (Γ 1, ϕ; Γ 2 ) and (Γ 1, ψ; Γ 2 ). Indeed, if θ 1 interpolates the first pair and θ 2 interpolates the second pair, then whence Γ 1, ϕ, θ 1 θ 1, Γ 2 Γ 1, ψ, θ 2 θ 2, Γ 2, Γ 1, ϕ ψ, θ 1 θ 2 θ 1 θ 2, Γ 2. Hence, θ 1 θ 2 interpolates (Γ 1 ; Γ 2 ), a contradiction. It follows that (Γ 1 ; Γ 2 ) is not maximal. (ii) Assume ϕ / Γ 1 then (Γ 1, ϕ; Γ 2 ) is inseparable. Otherwise, if θ interpolates this pair, then hence Γ 1, ϕ, θ and θ, Γ 2 Γ 1, ϕ ψ, θ and θ, Γ 2, that is, θ interpolates (Γ 1 ; Γ 2 ), a contradiction. It follows that (Γ 1 ; Γ 2 ) is not maximal. The case ψ / Γ 1 is similar. (iii) Assume Var(ϕ) Var(Γ 1 ) and ϕ, ϕ / Γ 1. Then one of the pairs (Γ 1, ϕ; Γ 2 ) and (Γ 1, ϕ; Γ 2 ) is inseparable. Otherwise, if then Γ 1, ϕ, θ 1 θ 1, Γ 2, Γ 1, ϕ, θ 2 θ 2, Γ 2, Γ 1, θ 1 θ 2 θ 1 θ 2, Γ 2. Hence, θ 1 θ 2 interpolates (Γ 1 ; Γ 2 ), a contradiction. It follows that (Γ 1 ; Γ 2 ) is not maximal. (iv) If ϕ, ϕ Γ 1 then Γ 1, and, Γ 2, which is impossible. Consider the following Kripke frame. Let W := {(Γ 1 ; Γ 2 ) : (Γ 1 ; Γ 2 ) is maximal inseparable over Φ}. For any x = (Γ 1 ; Γ 2 ) and y = ( 1 ; 2 ) in W let xr n y if the following conditions hold, for i = 1, 2: 5

1. Var(Γ i ) = Var( i ); 2. For any n ϕ Γ i, ϕ i and 3. For any m < n, k n (k Op(Φ) k ϕ i ); m ϕ Γ i m ϕ i ; 4. For some j {1, 2}, there is a n ϕ j such that n ϕ / Γ j. Lemma 2.3 W is a J-frame. Proof. Condition 4 guarantees the irreflexivity of the relations R n. Assume (Γ 1 ; Γ 2 )R n ( 1 ; 2 ), (Γ 1 ; Γ 2 )R m (Σ 1 ; Σ 2 ) and m < n; we prove ( 1 ; 2 )R m (Σ 1 ; Σ 2 ). Indeed, Var( i ) = Var(Γ i ) = Var(Σ i ), for i = 1, 2. If m ϕ i then m ϕ Γ i, since m < n. Hence ϕ, k ϕ Σ i, for k m, k Op(Φ). If k < m then k ϕ i k ϕ Γ i k ϕ Σ i. Finally, we have m ψ Σ j, m ψ / Γ j, for some ψ, j. Hence, m ψ / j because m < n. Assume (Γ 1 ; Γ 2 )R n ( 1 ; 2 ), ( 1 ; 2 )R m (Σ 1 ; Σ 2 ) and m n; we prove (Γ 1 ; Γ 2 )R m (Σ 1 ; Σ 2 ). Indeed, if m ϕ Γ i then m ϕ i, since m n. Hence ϕ, k ϕ Σ i, for k m, k Op(Φ). If k < m then k ϕ Γ i k ϕ i k ϕ Σ i. Finally, we have m ψ Σ j, m ψ / j, for some ψ, j. Hence, m ψ / Γ j because m n. Assume (Γ 1 ; Γ 2 )R m ( 1 ; 2 ), ( 1 ; 2 )R n (Σ 1 ; Σ 2 ) and m n; we prove (Γ 1 ; Γ 2 )R m (Σ 1 ; Σ 2 ). Let k Op(Φ). If m k n, then m ϕ Γ i implies k ϕ i and k ϕ Σ i. If k n, then m ϕ Γ i implies n ϕ i and ϕ, k ϕ Σ i. Finally, there is a ψ such that m ψ j, m ψ / Γ j. Since m n we also have m ψ Σ j, and we are done. We define the evaluation of propositional variables on W by letting (Γ 1 ; Γ 2 ) p p / Γ 1 Γ 2. ( ) Lemma 2.4 For any ϕ Γ 1 Γ 2 one has (Γ 1 ; Γ 2 ) ϕ. Proof. Induction on the length of ϕ. We consider the following cases. Case 1: ϕ =. If Γ 1 then Γ 1, and, Γ 2, hence (Γ 1 ; Γ 2 ) is not inseparable. Thus, / Γ 1 and similarly / Γ 2. Case 2: ϕ =. We always have (Γ 1 ; Γ 2 ). 6

Case 3: ϕ = p. By ( ). Case 4: ϕ = p. Suppose p Γ 1. If p Γ 1, then Γ 1, and, Γ 2, a contradiction. If p Γ 2, then Γ 1, p and p, Γ 2, also contradicting the inseparability of (Γ 1 ; Γ 2 ). Hence, p / Γ 1 Γ 2 which entails (Γ 1 ; Γ 2 ) p and (Γ 1 ; Γ 2 ) p. Case 5: ϕ = ϕ 1 ϕ 2. If ϕ Γ i then by Lemma 2.2 either ϕ 1 Γ i or ϕ 2 Γ i. Hence, (Γ 1 ; Γ 2 ) ϕ 1 or (Γ 1 ; Γ 2 ) ϕ 2. Therefore, (Γ 1 ; Γ 2 ) ϕ 1 ϕ 2. Case 6: ϕ = ϕ 1 ϕ 2. This is established dually by the same lemma. Case 7: ϕ = n ϕ 0. Assume ϕ Γ 1. If (Γ 1 ; Γ 2 )R n ( 1 ; 2 ) then ϕ 0 1 and by the induction hypothesis ( 1 ; 2 ) ϕ 0. Since this holds for all such ( 1 ; 2 ), we have (Γ 1 ; Γ 2 ) n ϕ 0. Case 8: ϕ = [n]ϕ 0. This is the central case. Assume [n]ϕ 0 Γ 1. Let i for i = 1, 2 denote the union of the following sets of formulas: 1. Φ i 1 := { m ψ : m ψ Γ i, m < n}; 2. Φ i 2 := {[m]ψ : [m]ψ Γ i, m < n}; 3. Φ i 3 := { k ψ, ψ : n ψ Γ i, k n, k Op(Φ)}; 4. Φ i 4 := {p p : p Var(Γ i)}. We show that the pair ( 1, n ϕ 0, ϕ 0 ; 2 ) is inseparable. Assume otherwise, then for some θ, where and 1, n ϕ 0, ϕ 0, θ and θ, 2, Var(θ) Var( 1, n ϕ 0, ϕ 0 ) = Var(Γ 1 ) Var(θ) Var( 2 ) = Var(Γ 2 ). The equalities hold because of the components Φ i 4. Since Φ i 4 is equivalent to and can be dropped from a disjunction, we obviously have Φ 1 1, Φ 1 2, Φ 1 3, n θ, θ, n ϕ 0, ϕ 0 and hence Φ 1 1, Φ 1 2, { n ψ : ψ Γ 1 }, n θ, [n]ϕ 0, 7

by Lemma 2.1. All the formulas in this sequent except for n θ belong to Γ 1, hence Γ 1, n θ. On the other hand, from θ, 2 we similarly obtain Φ 2 1, Φ 2 2, { n ψ : ψ Γ 2 }, [n] θ and hence Γ 2, n θ. It follows that n θ interpolates (Γ 1 ; Γ 2 ), which is impossible. Thus, ( 1, n ϕ 0, ϕ 0 ; 2 ) is inseparable and can be extended to a maximal inseparable pair ( 1 ; 2 ) such that Var( i ) = Var( i) = Var(Γ i ) for i = 1, 2. We observe that (Γ 1 ; Γ 2 )R n ( 1 ; 2 ). Indeed, Conditions 1, 2 and 4 are obviously satisfied. Also, if m ψ Γ i and m < n, then m ψ i i. On the other hand, if m < n and m ψ i then Var( m ψ) Var(Γ i) and hence either m ψ Γ i or m ψ Γ i, by Lemma 2.2 (iii). Yet, [m] ψ Γ i implies [m] ψ i, whence i contains both m ψ and its negation contradicting Lemma 2.2 (iv). Thus, we conclude m ψ Γ i, as required. Since ϕ 0 1, by the induction hypothesis we obtain ( 1 ; 2 ) ϕ 0. Hence, (Γ 1 ; Γ 2 ) [m]ϕ 0. From the previous lemma we obtain a proof of Theorem 1 in a standard way. Assume (Γ; ) is inseparable. Extend Γ to a finite adequate set Φ and build the corresponding model W. Let x be any maximal inseparable pair of sequents over Φ containing (Γ; ). By Lemma 2.4, W, x (Γ ). Corollary 2.5 (Craig interpolation for J) If J ϕ ψ, then there is a formula θ such that Var(θ) Var(ϕ) Var(ψ) and J ϕ θ and J θ ψ. Corollary 2.6 Craig interpolation property holds for GLP. Proof. If GLP ϕ ψ then J M + (ϕ ψ) (ϕ ψ) by (Red). Since every subformula [i]ξ of ϕ ψ belongs either to ϕ or to ψ, we have J M + (ϕ) ϕ (M + (ψ) ψ). Let θ be the corresponding interpolant. Then obviously GLP ϕ θ, GLP θ ψ, and Var(θ) Var(ϕ) Var(ψ). 8

Open questions: 1. Can we also obtain an interpolant θ satisfying an additional condition Op(θ) Op(ϕ) Op(ψ), that is, if modalities occurring in θ occur both in ϕ and in ψ? The given proof of Theorem 1 only implies that Op(θ) is contained in Op(ϕ) Op(ψ). The stronger interpolation theorem obviously fails for the GLP, as the example [0]p [1]p shows. 2. Does the sequential inference rule formulated in Lemma 2.1 provide a complete cut-free sequent calculus for J, taken together with a standard Tait-style axiomatization of propositional logic? 3. Does J satisfy uniform interpolation? Lindon interpolation? 3 Fixed points As a standard corollary of interpolation we obtain Beth definability property for J and GLP. Corollary 3.1 (Beth definability for J) If q does not occur in ϕ(p) and J ϕ(p) ϕ(q) (p q), then there is a ψ such that Var(ψ) = Var(ϕ(p)) \ {p} and J ϕ(p) (p ψ). Proof. Let ψ be the interpolant of the implication J ϕ(p) p (ϕ(q) q), A similar property obviously holds for GLP. We obtain the fixed point property for J and GLP using the method of Smoryński and Bernardi (cf. [2]). First, we prove the so-called Bernardi lemma for J. Lemma 3.2 Suppose q does not occur in ϕ(p) and p only occurs in ϕ(p) within the scope of a modality. Then J proves the following formula B ϕ : + (p ϕ(p)) + (q ϕ(q)) (p q). 9

Proof. We show that B ϕ is valid in all finite J-models W. With every x W we associate a sequence of numbers D(x) := d 0 (x), d 1 (x),..., d n (x), where d i (x) denotes the depth of x in W w.r.t. relation R i inductively defined by d i (x) := sup{d i (y) + 1 : xr i y}, and n is the maximal number such that R n is non-empty on W. We consider a lexicographic ordering of such sequences. Lemma 3.3 For all x, y W and any k, if xr k y then D(x) < D(y). Proof. Suppose xr k y. For each i < k we have d i (x) = d i (y), since by (I) the same points z are R i -accessible from x and from y. Also, obviously d k (x) > d k (y), hence the result. Suppose W is given and W B ϕ. By considering a suitable generated submodel we may assume that W p ϕ(p), q ϕ(q) ( ) and W p q. Select x W such that p and q have have different evaluations at x and D(x) is the minimal possible. By ( ) we have that ϕ(p) and ϕ(q) have different evaluations at x. Since p only occurs within the scope of modality in ϕ(p), ϕ(p) is a boolean combination of formulas of the form [k]ψ(p) and variables different from p, q. Hence, there must exist a subformula [k]ψ(p) of ϕ(p) such that [k]ψ(p) and [k]ψ(q) have different evaluations at x. It follows that for some y such that xr k y the formulas ψ(p) and ψ(q) have different evaluations at y. Let W y denote the submodel of W generated by y. For each z W y one has xr i z, for some i. (If yr m z and m < k then xr m z by (I), and if m k then xr k z by (J).) Hence, for all z W y, D(z) < D(x). Therefore, by the choice of x, W y p q. It follows that for all subformulas θ(p) of ϕ(p), W y θ(p) θ(q). In particular, a contradiction. W, y ψ(p) ψ(q), Corollary 3.4 (Fixed points in J) Suppose q does not occur in ϕ(p) and p only occurs in ϕ(p) within the scope of a modality. Then there is a ψ (a fixed point of ϕ(p)) such that Var(ψ) = Var(ϕ(p)) \ {p} and J ψ ϕ(ψ). Moreover, any two fixed points of ϕ(p) are provably equivalent in J. 10

Proof. Apply Beth definability property for the formula + (p ϕ(p)). Then we obtain a formula ψ such that J + (p ϕ(p)) (p ψ). We show that ψ is the required fixed point. Lemma 3.5 J + (p ψ) (p ϕ(p)). Proof. Consider a finite J-model W and a node x W with the minimal D(x) such that W, x + (p ψ) and W, x p ϕ(p). As before, we obviously have W x p ψ. Let p be a fresh variable evaluated as follows: W, y p iff W, y p, for all y x, and W, x p iff W, x p. If y W x and y x then W, y p ϕ(p ), since p and p have the same evaluation above x and D(x) was chosen minimally. Since p occurs within the scope of a modality in ϕ(p) we have W, x ϕ(p) iff W, x ϕ(p ). Therefore, W, x p ϕ(p ), since p and p have opposite evaluations at x. We conclude that W, x + (p ϕ(p )) and by the choice of ψ we must have W, x p ψ. This implies W, x p ψ p, quod non. As an immediate corollary of this lemma (substituting ψ for p) we obtain J ψ ϕ(ψ). If ψ 1 and ψ 2 are two fixed points of ϕ(p), then obviously J + (ψ i ϕ(ψ i )), for i = 1, 2. Hence, by Bernardi s lemma J ψ 1 ψ 2. Corollary 3.6 The fixed-point property holds for GLP. Proof. Given a formula ϕ(p) in which p only occurs within the scope of a modality, we obtain a ψ such that J ψ ϕ(ψ). Obviously, the same equivalence also holds in a stronger system GLP. To show the uniqueness, assume GLP ψ 1 ϕ(ψ 1 ), for another formula ψ 1. Denoting θ := ψ 1 ϕ(ψ 1 ) we obtain by (Red): It follows that J M + (θ) (ψ 1 ϕ(ψ 1 )). J + M + (θ) + (ψ 1 ϕ(ψ 1 )). Since we also have J + (ψ ϕ(ψ)), this implies J + M + (θ) (ψ 1 ψ), by Bernardi s lemma. Taking into account that GLP + M + (θ), for any formula θ, this implies GLP ψ ψ 1. 11

References [1] L.D. Beklemishev. Kripke semantics for Japaridze s provability logic. Logic Group Preprint Series 260, University of Utrecht, November 2007. http://preprints.phil.uu.nl/lgps/. [2] G. Boolos. The Logic of Provability. Cambridge University Press, Cambridge, 1993. [3] K.N. Ignatiev. The closed fragment of Dzhaparidze s polymodal logic and the logic of Σ 1 -conservativity. ITLI Prepublication Series X 92 02, University of Amsterdam, 1992. [4] K.N. Ignatiev. On strong provability predicates and the associated modal logics. The Journal of Symbolic Logic, 58:249 290, 1993. 12