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‘How can we recognize, given axioms and inference rules of a calculus, whether the calculus has such-and-such property?’ A question of this kind arises whenever we deal with a new logic system. For large families of logics, this question... more
‘How can we recognize, given axioms and inference rules of a calculus, whether the calculus has such-and-such property?’ A question of this kind arises whenever we deal with a new logic system. For large families of logics, this question may be considered as an algorithmic problem, and a property is called decidable in a given family if there exists an algorithm which is capable of deciding, for a finite axiomatics of a calculus in the family, whether or not it has the property. In the class of intermediate propositional logics, for instance, nontrivial properties such as the tabularity, pretabularity, and interpolation property (Maksimova [1972, 1977]) are decidable. However, for many other important properties—decidability, finite model property, disjunction property, Halldén-completeness, etc.—effective criteria were not found in spite of considerable efforts. In this paper we show that the difficulties in investigating these properties in the classes of intermediate logics and normal modal logics containing S4 are of principal nature, since all of them turn out to be algorithmically undecidable. In other words, there are no algorithms which, given a finite set of axioms of an intermediate or modal calculus, can recognize whether or not it is decidable, Halldén-complete, has the finite model or disjunction property. The first results concerning the undecidability of properties of calculi seem to have been obtained by Linial and Post [1949], who proved the undecidability of the problem of equivalence to classical calculus in the class of all propositional calculi with the same language as the classical one and the two inference rules: modus ponens and substitution. Kuznetsov [1963] generalized this result having proved the undecidability of the problem of equivalence to any fixed intermediate calculus (for instance, to intuitionistic calculus or even the inconsistent one). However, these results will not hold if we confine ourselves only to the class of intermediate logics, though the problem of equivalence to the undecidable intermediate calculus of Shehtman [1978] is clearly undecidable in this class as well.
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Here we solve a number of major open problems concerning computational properties of products and commutators of two 'transitive' (but not 'symmetric') standard modal logics (such as, e.g., K4, S4, S4.1, Grz, or GL) by... more
Here we solve a number of major open problems concerning computational properties of products and commutators of two 'transitive' (but not 'symmetric') standard modal logics (such as, e.g., K4, S4, S4.1, Grz, or GL) by showing that all of them are undecidable and lack the (abstract) finite model property. Some of these products turn out to be even not recursively
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In this paper, we construct and investigate a hierarchy of spatio-temporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic PTL, the spatial logics... more
In this paper, we construct and investigate a hierarchy of spatio-temporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic PTL, the spatial logics RCC-8, BRCC-8, S4u and their fragments. The obtained results give a clear picture of the trade-off between expressiveness and computational realisability within the hierarchy. We demonstrate how
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this paper.each frame of the class.) For example, Kis the logic of all n-ary product frames. It is nothard to see that S5is the logic of all n-ary products of universal frames having the sameworlds, that is, frames hU; R i i with R i = U... more
this paper.each frame of the class.) For example, Kis the logic of all n-ary product frames. It is nothard to see that S5is the logic of all n-ary products of universal frames having the sameworlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kindas cubic universal product
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The paper considers the set $\mathscr{M}\mathscr{L}_1$ of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability... more
The paper considers the set $\mathscr{M}\mathscr{L}_1$ of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in $\mathscr{M}\mathscr{L}_1$, which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of
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In this paper, we construct a new concept description language intendedfor representing dynamic and intensional knowledge. The mostimportant feature distinguishing this language from its predecessors inthe literature is that it allows... more
In this paper, we construct a new concept description language intendedfor representing dynamic and intensional knowledge. The mostimportant feature distinguishing this language from its predecessors inthe literature is that it allows applications of modal operators to allkinds of syntactic terms: concepts, roles and formulas. Moreover, thelanguage may contain both local (i.e., state-dependent) and global (i.e.,state-independent) concepts, roles and objects. All
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this paper argues for the richworld of representation that lies between these twoextremes."Levesque and Brachman (1985)1 IntroductionTime and space belong to those few fundamental concepts that always puzzledscholars from almost all... more
this paper argues for the richworld of representation that lies between these twoextremes."Levesque and Brachman (1985)1 IntroductionTime and space belong to those few fundamental concepts that always puzzledscholars from almost all scientific disciplines, gave endless themes to sciencefiction writers, and were of vital concern to our everyday life and commonsensereasoning. So whatever approach to AI one takes [ Russell and
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The paper considers the standard concept descriptionlanguage ALC augmented with variouskinds of modal operators which can beapplied to concepts and axioms. The mainaim is to develop methods of proving decidabilityof the satisfiability... more
The paper considers the standard concept descriptionlanguage ALC augmented with variouskinds of modal operators which can beapplied to concepts and axioms. The mainaim is to develop methods of proving decidabilityof the satisfiability problem for thislanguage and apply them to description logicswith most important temporal and epistemicoperators, thereby obtaining satisfiabilitychecking algorithms for these logics.We deal with the possible world semanticsunder the
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We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates, as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of... more
We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates, as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable
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Topological logics are a family of languages for repre- senting and reasoning about topological data. In this paper, we consider propositional topological logics able to express the property of connectedness. The satisfia- bility problem... more
Topological logics are a family of languages for repre- senting and reasoning about topological data. In this paper, we consider propositional topological logics able to express the property of connectedness. The satisfia- bility problem for such logics is shown to depend not only on the spaces they are interpreted in, but also on the subsets of those spaces over which
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In this paper we prove some results on the computational complexity of standard quantier- free spatial logics with the connectedness predicate interpreted over the Euclidean spaces R and R2. Topological logics with connectedness. A... more
In this paper we prove some results on the computational complexity of standard quantier- free spatial logics with the connectedness predicate interpreted over the Euclidean spaces R and R2. Topological logics with connectedness. A topological logic is a formal language whose vari- ables range over subsets of topological spaces, and whose non-logical primitives denote xed topo- logical properties and operations
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We analyse DL-Lite logics with role inclusions and present a complete classification of the trade-off between their expressiveness and computational complexity. In particular, we show that in logics with role inclusions the data... more
We analyse DL-Lite logics with role inclusions and present a complete classification of the trade-off between their expressiveness and computational complexity. In particular, we show that in logics with role inclusions the data complexity of instance checking becomes P-hard in the presence of functionality constraints, and coNP-hard if arbitrary number restrictions are allowed, even with a very primitive form of
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We present a formal framework for (minimal) mod- ule extraction based on an abstract notion of in- separability w.r.t. a signature between ontologies. Two instances of this framework are discussed in detail for DL-Lite ontologies: concept... more
We present a formal framework for (minimal) mod- ule extraction based on an abstract notion of in- separability w.r.t. a signature between ontologies. Two instances of this framework are discussed in detail for DL-Lite ontologies: concept inseparabil- ity, when ontologies imply the same complex con- cept inclusions over the signature, and query in- separability, when they give the same answers
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Databases and related information systems can bene- fit from the use of ontologies to enrich the data with general background knowledge. The DL-Lite family of ontology languages was specifically tailored towards such ontology-based data... more
Databases and related information systems can bene- fit from the use of ontologies to enrich the data with general background knowledge. The DL-Lite family of ontology languages was specifically tailored towards such ontology-based data access, enabling an imple- mentation in a relational database management system (RDBMS) based on a query rewriting approach. In this paper, we propose an alternative approach
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Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major... more
Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the !-reachability problem for lossy channel systems) that the dynamic topological logic over arbitrary
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... item Jacques Jayez and Lucia M. Tovena DRAFT for the PICS project - June 10, 2003 ... and K ato, w asuhik o (eds), N e g ation and Polarity¨¤ ¢ xford :|¢ xford U niversity P ress, 147-192 . J aye z , J acques & Tovena, L... more
... item Jacques Jayez and Lucia M. Tovena DRAFT for the PICS project - June 10, 2003 ... and K ato, w asuhik o (eds), N e g ation and Polarity¨¤ ¢ xford :|¢ xford U niversity P ress, 147-192 . J aye z , J acques & Tovena, L ucia ( 2003 ). Free Choiceness and N on Indi-viduation. ...
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Abstract: The aim of this paper is to summarize and analyze some results obtained in 2000-2001 about decidable and undecidable fragments of various first-order temporal logics, give some applications in the field of knowledge... more
Abstract: The aim of this paper is to summarize and analyze some results obtained in 2000-2001 about decidable and undecidable fragments of various first-order temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community' to a number of interesting open problems.