Stephan Tobies. A PSPACE algorithm for graded modal logic. In H. Ganzinger, editor, Proc. of the 16th Int. Conf. on Automated Deduction (CADE'99), volume 1632 of Lecture Notes in Artificial Intelligence, pages 52-- 66. Springer-Verlag, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....procedures for ALC as been pioneered in [66] where only the Pspace satis ability case without global axioms is dealt with. More recent works have extended the calculus to deal with additional constructs such as inverse or transitive roles (see e.g. the works of Horrocks, Sattler and Tobies 40 [40, 43, 44, 70]) but have not dealt with the algorithmic complexity of dealing with global axioms. For instance in [44] a calculus for an EXPtime complete logic including ALC is given but complexity results are only shown for a Pspace fragment of the logic. For EXPtime completeness results they refer to De ....

Tobies, S. A PSpace algorithm for graded modal logic. In Proceedings of the 16th International Conference on Automated Deduction (CADE-99) (1999), H. Ganzinger, Ed., vol. 1632 of LNAI, Springer-Verlag, pp. 52-66. 47

....for ALCN with decimal notation does no longer work: it is not sufficient to introduce just one successor as representative for the role successors required by at least restrictions. Nevertheless, it is possible to design a PSPACE algorithm for ALCQ also w.r.t. decimal notation of numbers [52]. Like the PSPACE algorithm for ALC, this algorithm treats the successors separately. It uses appropriate counters (and a new type of clashes) to check whether qualified number restrictions are satisfied. By combining the pre completion approach of [25] with this algorithm, we also obtain a ....

Stephan Tobies. A PSPACE algorithm for graded modal logic. In Proc. of CADE99, volume 1632 of LNCS. Springer-Verlag, 1999.

....would imply that NP =EXPTIME , so this is likely to be hard. We would conjecture instead that L (8;2) is exponentially more succinct than L (C1 ;C2 ; We have focussed here on S5n structures. There exist results on the complexity of graded modal logic for other classes of Kripke structures [15, 14], but it is not clear whether similar expressibility and succinctness questions are appropriate in these cases. It is certainly not obvious what the corresponding notion of local proposition would be in the case of structures other than S5n structures. For classes of models in which the relations ....

Stephan Tobies. A PSPACE algorithm for graded modal logic. In CADE-16 The 16th International Conference on Automated Deduction, LNAI. Springer-Verlag, July 1999.

....that it is not rewarding to search for better alternatives. Last, if one is only interested in decidability of concepts w.r.t. acyclic TBoxes, unfolding is a technique which is easy to use and always applicable. For many DLs, reasoning without TBoxes is PSpace complete (see, e.g. 9 ] 12 ] [ 18 ] ) Although the complexity of reasoning with acyclic TBoxes is rarely addressed formally, it is common knowledge in the DL community that, if reasoning without TBoxes is in PSpace, then taking into account TBoxes does usually not increase complexity. This knowledge has been exploited for ....

S. Tobies. A PSpace algorithm for graded modal logic. In Proceedings of CADE-16, LNCS, 1999. Springer-Verlag, Berlin { Heidelberg { New York, 1999.

....would imply that NP =EXPTIME , so this is likely to be hard. We would conjecture instead that L (8;2) is exponentially more succinct than L (C1 ;C2 ; We have focussed here on S5n structures. There exist results on the complexity of graded modal logic for other classes of Kripke structures [15, 14], but it is not clear whether similar expressibility and succinctness questions are appropriate in these cases. It is certainly not obvious what the corresponding notion of local proposition would be in the case of structures other than S5n structures. For classes of models in which the relations ....

Stephan Tobies. A PSPACE algorithm for graded modal logic. In CADE-16 The 16th International Conference on Automated Deduction, LNAI. Springer-Verlag, July 1999.

....Reasoning Carsten Lutz LTCS Report 99 04 RWTH Aachen LuFg Theoretische Informatik http: www lti.informatik.rwth aachen.de Ahornstr. 55 52074 Aachen Germany On the Complexity of Terminological Reasoning Carsten Lutz RWTH Aachen, LuFG Theoretical Computer Science Ahornstr. 55, 52074 Aachen April 28, 1999 Abstract TBoxes are an important component of knowledge representation systems based on description logics (DLs) since they allow for a natural representation of terminological knowledge. Largely due to a classical result given by Nebel [ 20 ] complexity analyses for DLs have, until now, ....

....practical applications, the worst case is almost never encountered. Largely due to these results, complexity analyses of reasoning with TBoxes have long been neglected. On the other hand, for a lot of description logics, reasoning without TBoxes is PSpace complete (see, e.g. 10 ] 22 ] 27 ] and it is common knowledge in the DL community that, if reasoning without TBoxes is in PSpace, then taking into account TBoxes does usually not increase complexity. This knowledge has been exploited for e cient practical reasoning with TBoxes [ 5 ] but has, to the best of our ....

S. Tobies. A PSpace algorithm for graded modal logic. In Proceedings of CADE-16, LNCS, 1999. to appear.

....) in PSPACE. Using an extension of these techniques we obtain a PSPACE algorithm for the logic Gr(K R 1 ) which extends Gr(KR ) by inverse relations and intersection of relations. This solves an open problem from [DLNN97] This paper is an significantly extended and improved version of [Tob99]. 2 Preliminaries In this section we introduce the graded modal logic Gr(KR ) the extension of the multimodal logic KR with graded modalities, first introduced in [Fin72] DEFINITION 2.1 (SYNTAX AND SEMANTICS OF Gr(KR ) Let P = fp 0 ; p 1 ; g be a set of propositional atoms and R a set ....

S. Tobies. A PSpace algorithm for graded modal logic. In H. Ganzinger, editor, Automated Deduction -- CADE-16, 16th International Conference on Automated Deduction, LNAI 1632, pages 52--66, Trento, Italy, July 7--10, 1999. Springer-Verlag. 23

....) in PSpace. Using an extension of these techniques we obtain a PSpace algorithm for the logic Gr(K R 1 ) which extends Gr(KR ) by inverse modalities and intersection of relations. This solves an open problem from [DLNN97] This paper is an signi cantly extended and improved version of [Tob99]. 2 Preliminaries In this section we introduce the graded modal logic Gr(KR ) the extension of the multi modal logic KR with graded modalities, rst introduced in [Fin72] De nition 1 (Syntax and Semantics of Gr(KR ) Let P = fp 0 ; p 1 ; g be a set of propositional atoms and R a set of ....

S. Tobies. A PSpace algorithm for graded modal logic. In H. Ganzinger, editor, Automated Deduction { CADE-16, 16th International Conference on Automated Deduction, LNAI 1632, pages 52-66, Trento, Italy, July 7{ 10, 1999. Springer-Verlag. 24

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Tobies, S. (1999c). PSpace algorithms for graded modal logics. Submitted to the Journal for Logic and Computation. Available online from http://www-lti.informatik.rwthaachen. de/~tobies/publications.html.

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Tobies, S. (1999b). A PSpace algorithm for graded modal logic. In Ganzinger, H. (Ed.), Automated Deduction { CADE-16, 16th International Conference on Automated Deduction, LNAI 1632, pp. 52-66 Trento, Italy. Springer-Verlag.

....In the following we present, in detail, a tableaux algorithm for SHIQ, which extends SHIF with qualifying number restrictions. The algorithm combines the methods already used in the previous section with techniques that are used to deal with qualifying number restrictions [ HB91; BBH96; Tob99 ] In order to avoid references to definitions and proofs in all previous sections, we make this section self contained. Of course, this leads to various repetitions, yet, it increases the readability. 5.1 Syntax and Semantics Definition 11 Let R be a set of role names with both transitive and ....

S. Tobies. A PSpace algorithm for graded modal logic. In H. Ganzinger, editor, Automated Deduction -- CADE-16, 16th International Conference on Automated Deduction, LNAI 1632, pages 52--66, Trento, Italy, July 7--10, 1999. Springer-Verlag.

....1999 Revised Version Abstract The description logic ALCQI extends the standard description logic ALC by qualifying number restrictions and converse roles. We show that concept satis ability for this DL is still decidable in polynomial space. The presented algorithm combines techniques from [Tob99a] to deal with qualifying number restrictions and from [HST99] to deal with converse roles. Additionally, we extend the result to ALCQIR, which extends ALCQI by role intersections. This solves an open problem from [DLNN97] The result for ALCQI has already been presented in the seperate technical ....

....of both role quanti cation and standard number restrictions that are present in almost all implementations of DL systems. They provide an expressive means to describe objects by the number of other objects they are related to and are necessary for reasoning with semantic data models [CLN94] In [Tob99a] we have shown that at least for ALC number restrictions can be replaced by qualifying number restrictions without increasing the (worst case) complexity of the satis ability problem. In this section we extend this result to converse roles. De nition 1 (The DL ALCQI) Let NC be a set of atomic ....

[Article contains additional citation context not shown here]

S. Tobies. A PSpace algorithm for graded modal logic. In H. Ganzinger, editor, Automated Deduction { CADE-16, 16th International Conference on Automated Deduction, LNAI 1632, pages 52-66, Trento, Italy, July 7-10, 1999. Springer-Verlag.

....a further degradation in empirical tractability. As far as complexity is concerned, we have already been successful in extending the PSpace result for SI to SIN [HST98] Currently we are working on an extension of this result to SIQ combining the techniques from this paper with those presented in [Tob99] ....

S. Tobies. A PSpace algorithm for graded modal logic. In Proc. of CADE-16, LNCS. Springer, 1999.

....a common generalisation of both role quantification and standard number restrictions that are present in almost all DL systems. They provide an expressive means to describe objects by the number of other objects they are related to and are necessary for reasoning with semantic data models [5] In [16] we have shown that at least for ALC number restrictions can be replaced by qualifying number restrictions without increasing the (worst case) complexity of the satisfiability problem. In this paper we extend this result to inverse roles. Definition 1 (The DL ALCQI) Let NC be a set of atomic ....

....amount of memory needed for the n successors is polynomial in the size of the input. This changes if we assume binary coding of numbers: then n consumes only log 2 n bits in the input, making the amount of memory required to store n successors potentially exponential in the size of the input. In [16] we give an algorithm derived from the one presented in [10] that is capable of deciding SAT(ALCQ) in PSPACE, even if binary coding of numbers in the input is allowed. While still generating n successors for a concept ( n R C) non deterministic guessing of an assignment of relevant constraints ....

[Article contains additional citation context not shown here]

S. Tobies. A PSPACE algorithm for graded modal logic. In Proceedings of CADE-99, LNCS. Springer-Verlag, 1999. To appear.

....degradation in empirical tractability. As far as complexity is concerned, we have already been successful in extending the PSpace result for SI to SIN [ HST98 ] Currently we are working on an extension of this result to SIQ combining the techniques from this paper with those presented in [ Tob99 ] ....

S. Tobies. A PSpace algorithm for graded modal logic. In Proc. of CADE-16, LNCS. Springer, 1999.

No context found.

Stephan Tobies. A PSPACE algorithm for graded modal logic. In H. Ganzinger, editor, Proc. of the 16th Int. Conf. on Automated Deduction (CADE'99), volume 1632 of Lecture Notes in Artificial Intelligence, pages 52-- 66. Springer-Verlag, 1999.

No context found.

Stephan Tobies. A PSPACE algorithm for graded modal logic. In H. Ganzinger, editor, Proc. of the 16th Int. Conf. on Automated Deduction (CADE'99), volume 1632 of Lecture Notes in Arti cial Intelligence, pages 52-66. Springer, 1999.

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