F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. J. Symbolic Logic, 66:1415--1438, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....constant domains in which the class of first order temporal structures, where FOTL formulae are interpreted, is restricted to structures M = hD n ; I n i, n 2 N, such that D i = D j for all i; j 2 N. The notions of truth and validity are defined similarly to the expanding domain case. It is known [19] that satisfiability over expanding domains can be reduced to satisfiability over constant domains. Example 1 The formula 8xP(x) 8xP(x) 8xQ(x) Q(c) is unsatisfiable over both expanding and constant domains; the formula 8xP(x) 8x(P(x) Q(x) Q(c) is unsatisfiable over ....

....and removing all occurrences of the U and W operators [9, 5] 4. Completeness Calculus A resolution like procedure for the monodic fragment over constant domains has been introduced in [5] Although satisfiability over expanding domains can be reduced to satisfiability over constant domains [19], it has been proved in [6] that a simple modification of the procedure can be directly applied to the expanding domain case. We sketch the monodic temporal resolution system here to make the paper self contained. We use this completeness calculus to show relative completeness of the calculus ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 66:1415--1438, 2001.

.... A natural way to reduce the interaction between the dimensions is to restrict first order quantification to state formulas that is to work with the language Q PCT L and to limit the scope of the path quantifiers (hence also of the temporal operators) to formulas with 1 free variable; cf. [14, 24]. DEFINITION 6 (MONODIC FORMULAS) Let Q PCT L 1 be the set of all Q PCT L formulas j such that any subformula of j of the form Ay, y 1 Uy 2 , or y 1 Sy 2 has at most one free variable. Such formulas j will be called monodic and Q PCT L 1 the monodic fragment of Q PCT L . It should ....

....of first order branching time logics. We have obtained decidability results of two kinds. The most striking bad news is that the one variable fragments of logics containing BQCTL F are undecidable, which contrasts with the situation in many other modal and temporal first order logics (cf. [14, 24]) The good news is that there are still ways of obtaining decidable fragments with nontrivial interaction between first order quantifiers, path quantifiers and temporal operators. In this paper, we considered the case when the first order quantifiers are applied only to state formulas, while the ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. J. Symbolic Logic, 66:1415--1438, 2001.

....and Cartesian products in particular are becoming the subject of one of the most important and interesting research fields in pure and applied modal logic. See e.g. applications of results and techniques developed in multi dimensional logic to first order classical, modal and temporal logics in [8, 30, 13]. Unfortunately, as was observed by Gabbay and Shehtman [7] there is no general transfer theorem that could guarantee the preservation of such properties of logics as decidability or axiomatizability under the formation of products. If we consider only 2D products of standard modal systems, ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 2001. (In print.)

....fact, finitely axiomatizable or even decidable and yet expressive fragments with interesting interactions between the common knowledge operator and quantifiers. A promising approach to singling out such kind of fragments of firstorder modal and temporal logics has been proposed in [12,25]. The idea is to restrict attention to the class of monodic 1 formulas which allow applications of modal or temporal operators only to formulas with at most one free variable. In the epistemic context, monodicity means, in particular, that: we have the full expressive power of common ....

....in G(x; y) as well, then the formula 8y(G(x; y) x; y) is in G. Note that although the guarded fragment of classical first order logic is decidable (see [1] the guarded fragment of L(F a ) turns out to be undecidable. For more details and an idea of the proof the reader is referred to [12,25]. Actually, no non trivial decidable fragments of epistemic predicate logics have been constructed before. It maybe also of interest to note that these decidability results make it possible to construct various decidable description logics with common knowledge and other epistemic operators ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 2000.

....and the employed techniques as clear as possible. The developed methods can be extended to more 3 sophisticated logics, say, S4 or temporal logics based on ALC. Moreover, the approach developed in this paper can be generalized to the monodic fragments of first order modal and temporal logics [16, 39] by combining tableau procedures for their first order and modal components in a modular way; for details visit http: www.dcs.kcl.ac.uk staff mz. It is worth also noting that KALC is closely related to the Cartesian product K Theta S5 (cf. 13, 33] Thus we obtain for free a tableau based ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 2001. 30

....1 ; ym ) 22 Note that although the guarded fragment of classical first order logic is decidable (see [ Andr eka et al. 1998 ] the guarded fragment of L(F a ) turns out to be undecidable. For more details and an idea of the proof the reader is referred to [ Hodkinson et al. 2000; Wolter and Zakharyaschev, 2001 ] Actually, no non trivial decidable fragments of epistemic predicate logics have been constructed before. It maybe also of interest to note that these decidability results make it possible to construct various decidable description logics with common knowledge and other epistemic operators ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 2001. 25

....of using common knowledge predicate logic Still there exist manageable fragments with non trivial interaction between the common knowledge operator and quantifiers. A promising approach to singling out non trivial decidable fragments of first order modal and temporal logics has been proposed in [9, 20]. The idea is to restrict attention to the class of monodic 2 formulas which allow applications of modal or temporal operators only to formulas with at most one free variable. In the epistemic context, monodicity means, in particular, that ffl we have the full expressive power of first order ....

....following fragments are decidable: ffl the monadic fragment of L 1 (F) ffl the two variable fragment of L 1 (F) ffl the guarded fragment of L 1 (F) 12 (Note, however, that the guarded fragment of L(F a ) is undecidable. For more details and an idea of the proof the reader is referred to [9, 20]. Actually, no non trivial decidable fragments of epistemic predicate logics have been constructed before. It maybe also of interest to note that these decidability results make it possible to construct various decidable description logics with common knowledge and other epistemic operators ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. 2000. 15

....immediately. So reasoning in first order common knowledge logics is highly undecidable even for rather weak fragments. The question arises wether there are interesting decidable fragments of (standard) first order common knowledge logics at all. This question was answered in the positive in [15] for the following fragment CL: let 2 CEL = be a formula containing local predicate symbols and global constants only. Then 2 CL iff contains no subformula of the form 2 , 2 2 fC; K 1 ; K 2 g, such that contains more than one free variable. CL is undecidable since its first order ....

....predicate symbols and global constants only. Then 2 CL iff contains no subformula of the form 2 , 2 2 fC; K 1 ; K 2 g, such that contains more than one free variable. CL is undecidable since its first order non epistemic fragment is undecidable. However, using the technique developed in [15] one can prove: THEOREM 5.8 Let L be a standard first order common knowledge logic. ffl Let Sigma CL be the fragment of CL containing formulas with monadic predicates and constants only. Then Sigma L is decidable. ffl Let Sigma CL be the fragment of CL containing formulas with two ....

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F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Submitted, 1999. 22

....TL(N) and TL fin (N) respectively; similar notation is used for hZ; i, hQ ; i, and hR; i. Remark 1. In this paper we consider only models with constant domains. Satisfiability in models with expanding domains is known to be reducible to satisfiability in models with constant domains (see [47]) 5 2.1 Undecidable fragments of T L The following two theorems indicate some limits outside which one cannot hope to find decidable fragments of first order temporal logics. For , let T L be the variable fragment of T L (i.e. every formula in T L contains at most distinct ....

....not known whether the finite model reasoning in it is decidable. So we cannot say whether the satisfiability problem for CIQ US formulas is decidable in models with finite domains. For more information on the connection between multi dimensional description logics and first order modal logic, see [47]. 9 Open questions We end the paper with some problems arising from the work above. 1. Do our results extend to the flow of time hR; i with arbitrary domains Or with countable domains (The logic here is different see Theorem 25. 2. Can our results be extended to first order temporal logic ....

F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Submitted. See http://www.informatik.uni-leipzig.de/wolter, 1999. 45

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F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. J. Symbolic Logic, 66:1415--1438, 2001.

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F. Wolter, M. Zakharyaschev, Decidable fragments of first-order modal logics, Journal of Symbolic Logic 66 (2001) 1415--1438.

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F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. Journal of Symbolic Logic, 66:1415--1438, 2001.

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