D. Gabbay and V. B. Shehtman. Products of modal logics, part i. Logic Journal of the IGPL, 6(1):73--146, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from well behaved calculi for the component logics. In this paper, we do not apply the method as there are interaction axioms between K and B , and the calculi considered in [3] are labeled tableau systems. The combination of modal logics has gained a lot of attention in the past years (see e.g. [12, 19, 5, 8, 13, 9, 4]) The logics considered in this paper are fusions of the component logics (with some interaction axioms) In the way of [4] the logic K B C is denoted by S5 KD45, and K B 5C by S4 KD4. The formulation of our systems is based on the work by Gor e [10] We use a similar technique to prove ....

D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic Journal of the IGPL, 6(1):73-146, 1998.

....including soundness, completeness, the nite model property and decidability. Another form of combination of two logics is their product. With products the situation is more varied and complicated than with fusions. First of all, products can be de ned in two ways: axiomatically and semantically [3]. Whereas in fusions there is no interdependence between the operators of the di erent modal dimensions, in products the modal operators are commuting. This complicates matters, so that for products there is no preservation theorem of the generality as for fusions. In fact, the particular type of ....

....This research is supported by EPSRC Research Grant GR M88761. 1 modal opertors in products makes it much more dicult to obtain positive results regarding completeness, the nite model property and decidability for products. However advanced technique to deal with products have been developed [3], see also [8] These are techniques for investigating the properties of products of modal logics. The product of PDL and S5 is a combination of a dynamic logic and a modal logic. Therefore, the techniques developed for standard modal logics need to be elaborated and can be applied to products ....

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D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic J. IGPL, 6(1):73-146, 1998.

....properties on models. 8i R i is an equivalence relation (eq) 8i8s9t (s; t) 2 R i (br) 8i8 (R i Q( i ) Q( i ) R i ) com 8i8 (Q( i ) R i R i Q( i ) com Main results. The main results in this paper are the following. The proofs use well known methods of modal and dynamic logic [7, 3, 5, 4], but require nontrivial modi cations due to the presence of the new informational test operator. Theorem 2 (Completeness) Let L be an extension of BDL by any combination of the axioms T, D, NL, and PR. Then L is complete with respect to the class of all models restricted by the corresponding ....

.... [1] does not completely work for BDL but nevertheless we could prove the expected model properties for the axioms of permutability of the informational modalities and action modalities (perfect recall and no learning) These extensions of BDL are therefore closely related to the products of logics [5]. It was shown in [5] that, usually, results on decidability is hard to obtain for the products and some advanced techniques must be developed for each particular instance. Nevertheless, the decidability for such extensions of BDL was shown in this paper. For further work, it would be interesting ....

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D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic J. IGPL, 6(1):73-146, 1998.

....Strand, London WC2R 2LS, U.K. fkuag,mzg dcs.kcl.ac.uk 1 Introduction One may think of many ways of combining modal logics representing various aspects of an application domain. Two canonical constructions, supported by a well developed mathematical theory, are fusions [17, 6, 7] and products [8, 7]. The fusion L L n of n 2 normal propositional unimodal logics L i with the boxes 2 i is the smallest multimodal logic in the language with n boxes 2 1 ; 2 n (and their duals 3 1 ; 3 n ) that contains all the L i . This means that if the L i are axiomatised by sets Ax i ....

.... G n such that every G i has the same set of worlds and each G i is a frame for L i . Indeed, take a cardinal max 1 i n jF i j and let G i be the disjoint union of many copies of F i . Then jG i j = and F i is a p morphic image of G i , whenever 1 i n. Thus it is easy to see (cf. e.g. [8]) that the product of these p morphisms gives a p morphism from G onto F. Since all the G i have the same cardinalities, we may assume that they are frames over the same set of worlds. 2 4 Expanding and decreasing relativisations First order modal and intuitionistic logics as well as modal ....

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Journal of the IGPL,

....# is defined as usual. If # = O(# 1 , # n ) with O OP(T i ) we define 2 # by replacing every occurrence of M,x in the definition of s i T i # (i 2 ) by x, s 2 (if i = 1) or s 1 , x (if i = 2) In our presentation of the join of logics, we have followed [6] in [8] a slightly different but equivalent construction is studied: the product of modal logics. 4 Model Checking for Combined Logics In this section we consider model checking procedures for each of the modes of combining logics considered in Section 3. 4.1 Temporalization. We first define the ....

D. Gabbay and V. Shehtman. Products of modal logics, Part 1. Logic Journal of the IGPL, 6:73--146, 1998.

....in the absence of interaction among the component logics. Such positive result is usually based on a divide and conquer strategy: split problems into sub problems and delegate these to the components [41, 76] However, the transfer can easily fail when the components are allowed to interact [45, 57]. In this chapter we describe the combining approach for temporal logics and we propose a similar approach for finite state automata. In Section 3.1 we introduce three well known modes for combining temporal logics: temporalization, independent combination and join. In Section 3.2 we propose an ....

....counterparts. 3.1 Combining methods Various forms of logic combination have been proposed in the literature. Temporalization, independent combination (or fusion) and join (or product) are probably the most popular ones as well as the ones that have been studied most extensively [41, 44, 45, 54, 57, 76, 112]. They have been successfully applied in several areas, including databases [42, 43, 46] artificial intelligence [37, 39, 64, 91, 3, 131] and system specification and verification [51] We are mainly interested in this last application of combined logics. In the following, we introduce syntax ....

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D. M. Gabbay and V. Shehtman. Products of modal logics, part I. Logic Journal of the IGPL, 6:73--146, 1998.

....propositional variables. Truth is de ned in the usual way the nonpropositional cases are: for every x 2 U 0 U 1 , M;x 3 ( M; y for some (x; y) 2 H M;x 3 ( M; y for some (x; y) 2 V: 1 L. J. of the IGPL, Vol. 0 No. 0, pp. 1 12 0000 c Oxford University Press We recall from [3] that the logic K is nitely axiomatizable by the standard K axioms in both dimensions together with commutativity (Comm) and con uence (Conf) PT: enough) propositional tautologies, DB: 2( 2 2 ) where 2 2 f2; 2g, Comm: 33 33 , Conf: 32 23 , and the usual rules of Modus ....

....modal predicate logics (with constant domains) can be viewed as products in which one dimension is S5. The S5 diamond will be the existential quanti er. A good reference is the forthcoming book on products [4] As an example of the kind of theorems in this area we mention a powerful result from [3] (Theorem 7.12) on axiomatizability of binary products. This will be used later in our decision procedures. We need one de nition. A pseudo transitive formula is one of the form r2 k p 4p where 4 is a sequence of (possibly di erent) boxes and r a sequence of (possibly di erent) diamonds. A PTC ....

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D. Gabbay and V. Shehtman. Products of modal logics, Part 1. Logic Journal of the IGPL, 6(1):73-146, 1998.

....modal logic S5 is widely studied, under a variety of names. Its algebraic counterpart is the variety Df 2 = RDf 2 of (representable) diagonal free cylindric algebras of dimension two [4] Segerberg [9] discusses an expansion under the name of two dimensional modal logic. In Gabbay and Shehtman [2], S5 is studied as a special case of taking products of modal logics. One of the sources of interest for S5 is that it corresponds to a clean equality free fragment of rst order logic with two variables. Our work is best motivated from this angle. These results were obtained during a visit ....

....be represented as products of two S5 frames. Recall that for given two S5 frames F = W; E) and F = W ) the product F F is de ned as the triple (W W ; E 1 ; E 2 ) where ) i w = v and w E ; i wEv and w = v is complete with respect to both classes (cf. 4] or [2]) that is for every formula of the language of S5 we have i is valid in every S5 frame i is valid in every product frame: Note that in [4] by RDf 2 is denoted the class of algebras generated by product frames. One of the main interests towards S5 is that S5 axiomatizes ....

D. Gabbay and V. Shehtman. Products of modal logics, Part 1. Logic Journal of the IGPL, 6(1):73-146, 1998.

....modal logic S5 is widely studied, under a variety of names. Its algebraic counterpart is the variety Df 2 = RDf 2 of (representable) diagonal free cylindric algebras of dimension two [4] Segerberg [9] discusses an expansion under the name of two dimensional modal logic. In Gabbay and Shehtman [2], S5 is studied as a special case of taking products of modal logics. One of the sources of interest for S5 is that it corresponds to a clean equality free fragment of rst order logic with two variables. Our work is best motivated from this angle. These results were obtained during a ....

....of two S5 frames. 1 INTRODUCTION 3 Recall that for given two S5 frames F = W; E) and F = W ) the product F F is de ned as the triple (W W ; E 1 ; E 2 ) where ) i w = v and w E ; i wEv and w = v is complete with respect to both classes (cf. 4] or [2]) that is for every formula of the language of S5 we have i is valid in every S5 frame i is valid in every product frame: Note that in [4] by RDf 2 is denoted the class of algebras generated by product frames. One of the main interests towards S5 is that S5 ....

D. Gabbay and V. Shehtman. Products of modal logics, Part 1. Logic Journal of the IGPL, 6(1):73-146, 1998.

.... have that L = L 0 Ln 1 always includes L i (i n) and for every product frame F = F 0 Fn 1 , F is a frame for L i F i is a frame for L i ; for all i n: Products of modal logics have been studied in both pure modal logic (see Segerberg [16] Shehtman [17] Gabbay Shehtman [5]) and in computer science applications (see Wolter Zakharyaschev [19] 20] Gabbay et al. . 3] and the references therein) Product logics are also relevant to nite variable fragments of modal and intermediate predicate logics, see Gabbay Shehtman [4] Axiomatization, decision and complexity ....

....[19] 20] Gabbay et al. . 3] and the references therein) Product logics are also relevant to nite variable fragments of modal and intermediate predicate logics, see Gabbay Shehtman [4] Axiomatization, decision and complexity problems of two dimensional products were thoroughly investigated in [5], Marx [14] Spaan [18] In higher dimensions n 3 from now on the rst results related to product logics were obtained in algebraic logic. This is due to the fact that the modal algebras corresponding to S5 n are well known in this area: the representable diagonal free cylindric algebras ....

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D.M. Gabbay and V.B. Shehtman. Products of modal logics, part I. Logic Journal of the IGPL, 6:73-146, 1998.

....showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using non structural Gabbay type inference rules. Keywords: Modal logic, Kripke frame, Cartesian product, decidability. ....

....in order to reflect interactions between modal operators representing time, space, knowledge, actions, etc. Products of modal logics (i.e. sets of multi modal formulas that are valid in the Cartesian products of Kripke frames for those logics) have been studied in both pure modal logic (see e.g. [23, 24, 18, 7, 15, 19, 31]) and applications in computer science and artificial intelligence (see e.g. 20, 3, 1, 21, 4, 27, 28, 29] since the 1970s. It would not be an exaggeration to say that now multi dimensional logics in general and Cartesian products in particular are becoming the subject of one of the most ....

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Journal of the IGPL, 6:73--146, 1998.

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D. Gabbay and V. B. Shehtman. Products of modal logics, part i. Logic Journal of the IGPL, 6(1):73--146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic Journal of the IGPL, 6(1):73--146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics. I. Logic Journal of the IGPL, 6(1):73--146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic J. IGPL, 6(1):73-146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Logic Journal of the IGPL, 6:73--146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Logic Journal of the IGPL, 6:73--146, 1998. 9

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D. Gabbay and V. Shehtman. Products of modal logics. I. Logic Journal of the IGPL, 6(1):73--146, 1998.

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D. M. Gabbay and V. B. Shehtman, Products of modal logics. I, Log. J. IGPL 6 (1998), no. 1, 73--146.

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D. Gabbay and V. Shehtman. Products of modal logics, Part 1. Logic Journal of the IGPL, 6(1):73--146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Journal of the IGPL, 6:73-146, 1998.

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D. M. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic Journal of the IGPL, 6(1):73-146, 1998.

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Dov M. Gabbay and Valentin B. Shehtman. Products of Modal Logics, Part 1. Logic Journal of the IGPL, 6:73-146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics. Part I. Journal of the IGPL, 6:73-146, 1998.

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D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic J. IGPL, 6(1):73--146, 1998.

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