Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1--18, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... on chains of functional roles) and ALC ;ffi;t (extending ALC with transitive closure, composition, and union of roles) For both DLs, the subsumption of concept descriptions is known to be decidable [18, 29, 6] However, the subsumption problem for their union ALCF ;ffi;t is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs [5, 4, 33] But even fusions behave badly if the class of models is not closed under disjoint unions, which is the case when nominals or negation of roles are required [5, 4] or if we combine ....

....that A satisfiability is not preserved under E connections. More precisely, the following negative result holds, where ALCF is the extension of ALC with functional roles and the same as constructor (it is known that A satisfiability for ALCF is decidable [18, 24] while satisfiability is not [3]) Theorem 8. Let S be an arbitrary ADS. Then the A satisfiability problem for any E connection C(ALCF ; S) with a non empty E is undecidable. Before presenting the proofs, we illustrate the notion of E connection with illuminative examples. 4 Examples of E connections In this section we give ....

[Article contains additional citation context not shown here]

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic and Computation, (2):1--18, 1993.

....8 is identical to the one presented for Lemma 4. Again, summing up, we have the following theorem. Theorem 2 The modified tableaux algorithm is a decision procedure for the satisfiability and subsumption of ALCHI R concepts. 5. 1 General Concept Inclusion Axioms In [Baader,1991; Schild,1991; Baader et al. 1993],theinternalisation of terminological axioms is introduced. This technique is used to reduce reasoning with respect to a (possibly cyclic) terminology to satisfiability of concepts. In [Horrocks Gough,1997] we saw how role hierarchies can be used to reduce satisfiability and subsumption with ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information,2:1-- 18, 1993.

....that Inv(R) R if R is a role name, and Inv(R) S if R = S . 2. Obviously, a role R is transitive i Inv(R) is transitive. We therefore de ne a function Trans which returns true i R is a transitive role. More precisely, Trans(R) true i R 2 R or Inv(R) 2 R . In [ Baader1990; Schild1991; Baader et al..1993 ] the internalisation of terminological axioms is introduced, a technique that reduces reasoning with respect to a (possibly cyclic) terminology to satis ability of concepts. In [ Horrocks1998b ] we saw how role hierarchies can be used for this reduction. In the presence of inverse roles, ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1-18, 1993. 24 Practical Reasoning for Very Expressive Description Logics

....single clause s topic x s comp obj x: Kaplan Maxwell III (1988) have shown that the satisfiability problem of the pure existential fragment (i.e. the satisfiability of formulae built with A(x) xLy and equations x : y) is decidable, provided that a certain acyclicity condition is met. (Baader et al. 1993) have shown that satisfiability is undecidable if we add unrestricted negation. It has, however, remained an open problem whether satisfiability of the purely existential fragment is decidable in the absence of additional conditions (such as acyclicity) In this paper we will show that it is ....

....on page 425, the combination of both types of cycle makes our problem hard. Thus, while Baader can split the problem into independent subproblems, we cannot use a similar technique because of the structure of our problem (to be more concrete, the technique of internalization has been used in (Baader et al. 1993) to prove undecidability of functional uncertainty with negation) Acknowledgements I would like to thank Jochen Dorre, Joachim Niehren, Stephen Spackman and Ralf Treinen for helpful discussions and reading draft versions of the paper. In particular, I am grateful to Joachim Niehren for his ....

Baader, F., Burckert, H.-J., Nebel, B., Nutt, W., Smolka, G. (1993). On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18.

....We have stated above that a quasi normal extension of a subframe logic is decidable if finitely axiomatizable. We will see that this is not so for normal extensions. The tool to prove this is to code Thue processes. See [ Kracht, 1995a ] for the connection with modal logic as described here and [ Baader et al. 1993 ] for similar undecidability result for feature logics. A Thue process T is a finite set of equations v i w i , i k, over a finite alphabet L = f 1 ; m g. One can think of the Thue process as a specification of a semigroup, with generators i and certain equations. In mathematical ....

.... modal apparatus have been considered ( Blackburn, 1993 ] Blackburn and Spaan, 1993a ] Blackburn and Spaan, 1993b ] Gazdar et al. 1988 ] In addition there is the classical language of Kasper and Rounds (see [ Kasper and Rounds, 1990 ] with it s various extensions, for example [ Baader et al. 1993 ] We will discuss them in turn concentrating on the aspects of definitional strength and their decidability. Kasper and Rounds in their work advance a language L KR that has conjunction, disjunction, arc modalities h j i and a device that is equivalent to having path equations v w. Indeed, ....

[Article contains additional citation context not shown here]

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty and sort equations. Journal of Language, Logic and Information, 2:1 -- 18, 1993.

....of Lemma 8 is identical to the one presented for Lemma 4. Again, summing up, we have the following theorem. Theorem 2 The tableaux algorithm is a decision procedure for the satisfiability and subsumption of ALCHI R concepts. 3. 1 General Concept Inclusion Axioms In [ Baader,1991; Schild,1991; Baader et al. 1993 ] the internalisation of terminological axioms is introduced. This technique is used to reduce reasoning with respect to a (possibly cyclic) terminology to satisfiability of concepts. In [ Horrocks Gough,1997 ] we saw how role hierarchies can be used to reduce satisfiability and subsumption ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

No context found.

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1--18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1--18, 1993.

No context found.

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1-18, 1993.

No context found.

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. DFKI Report RR-9101, German Research Center for Artificial Intelligence (DFKI), Saarbrucken, January 1991.

....descriptions in logic programming has been advocated and studied [3, 4, 5, 6, 21] Essentially, feature descriptions provide a logical version of records, a data structure found in many programming languages. Feature descriptions have been proposed in various forms with various formalizations [1, 2, 14, 18, 11, 20, 7, 12]. We will follow the logical approach pioneered by [20] which accommodates feature descriptions as standard first order formulae interpreted in first order structures. In this approach, a semantics for feature descriptions can be given by means of a feature theory (i.e. a set of closed feature ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Research Report RR-91-01, German Research Center for Artificial Intelligence (DFKI), Stuhlsatzenhausweg 3, D-6600 Saarbr ucken, Germany, January 1991.

....concept C 0 to be tested for satisfiability. Then we extend C 0 to the concept C 0 : C 0 C#D#T D) # #u. C # D) We can then show that C 0 is satisfiable w.r.t. the extended role hierarchy i# the original concept C 0 is satisfiable w.r.t. the TBox and the original role hierarchy [1, 73, 3, 53]. Consequently, it is su#cient to design an algorithm that can decide satisfiability w.r.t. role hierarchies and transitive roles. This problem is known to be ExpTime complete [77] In fact, ExpTime hardness can be shown by an easy adaptation of the ExpTime hardness proof for satisfiability in ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

....on chains of functional roles; and (ii) ALC ;ffi;t , which extends ALC by transitive closure, composition, and union of roles. For both DLs, subsumption of concept descriptions is known to be decidable [9, 11, 1] However, their union ALCF ;ffi;t has an undecidable subsumption problem [2]. This undecidability result depends on the fact that, in ALCF ;ffi;t , the role constructors transitive closure, composition, and union can be applied to functional roles that also appear within the same as constructor. This is not allowed in the fusion ALCF Omega ALC ;ffi;t . Of course, ....

....ADSs must satisfy the stronger requirement that the relativized satisfiability problem is decidable, which may not always be the case. For example, the theorem cannot be applied for the fusion of ALCF and ALC ;ffi;t since the relativized satisfiability problem for ALCF is already undecidable [2]. However, the satisfiability problem is decidable for both DLs. Before we can formulate a transfer result for the satisfiability problem, we need to introduce an additional notion, which generalizes the notion of a normal modal logics. Definition 9. Let (L; M) be an ADS and f be a function ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic and Computation, 2:1--18, 1993.

....general terminological axioms is known to be EXPTIME complete [48] Blocking does not work for all extensions of ALC that have a tableau based satisfiability algorithm. An example is again the DL ALCF , for which satisfiability is decidable, but satisfiability w.r.t. general TBoxes undecidable [40, 3]. 6 Expressive roles The DLs considered until now allowed for atomic roles only. There are two ways of extending the expressivity of DLs w.r.t. roles: adding role constructors and allowing to constrain the interpretation of roles. Role constructors can be used to build complex roles from atomic ....

....v s for roles r; s. An interpretation I satisfies the role hierarchy H iff r I s I holds for each r v s 2 H. DLs with transitive roles and role hierarchies have the nice property that reasoning w.r.t. TBoxes can be reduced to reasoning without TBoxes using a technique called internalisation [3, 30, 32]. Like in Section 5.2, we may assume that TBoxes are of the form T = f b C : g. In SH, the extension of ALC with transitive roles and role hierarchies, we introduce a new transitive role name u and assert in the role hierarchy that u is a super role of all roles occurring in b C and the ....

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1--18, 1993.

....(agreement) on chains of functional roles; and (ii) ALC ; t , which extends ALC by transitive closure, composition, and union of roles. For both DLs, subsumption of concept descriptions is known to be decidable [9, 11, 1] However, their union ALCF ; t has an undecidable subsumption problem [2]. This undecidability result depends on the fact that, in ALCF ; t , the role constructors transitive closure, composition, and union can be applied to functional roles that also appear within the same as constructor. This is not allowed in the fusion ALCF ALC ; t . Of course, failure of a ....

....ADSs must satisfy the stronger requirement that the relativized satis ability problem is decidable, which may not always be the case. For example, the theorem cannot be applied for the fusion of ALCF and ALC ; t since the relativized satis ability problem for ALCF is already undecidable [2]. However, the satis ability problem is decidable for both DLs. Before we can formulate a transfer result for the satis ability problem, we need to introduce an additional notion, which generalizes the notion of a normal modal logics. De nition 9. Let (L; M) be an ADS and f be a function symbol ....

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic and Computation, 2:1-18, 1993.

....for least and greatest fixpoint semantics, this characterization also led directly to sound and complete subsumption algorithms for these cases. In addition, this result gave rise to the idea of extending the expressive power of terminological logics by adding regular expressions over roles [Baader, 1991]. 4 Thue Systems For feature logics, the computational complexity was analyzed quite early. The feature logic described in Sect. 2 without union t and complement : which give essentially the terms mentioned above, was shown to have a quasi linear satisfiability problem [A it Kaci, 1984] ....

....closely related to terminological logics that permit regular expressions over roles, one would expect that undecidability would show up again in this case. In fact, if the feature logic specified in Sect. 2 is extended by functional uncertainty, then satisfiability of feature terms is undecidable [Baader et al. 1991]. 2 An even stronger result can be shown. Satisfiability of a feature term relative to a set of arbitrary axioms can be reduced to satisfiability of a feature term without axioms [Baader et al. 1991] 3 However, these results strongly depend on the presence of the complement operator. Thus, ....

[Article contains additional citation context not shown here]

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. DFKI Report RR-9101, German Research Center for Artificial Intelligence (DFKI), Saarbrucken, January 1991.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1-18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1-18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1-18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. J. of Logic, Language and Information, 2:1--18, 1993.

No context found.

Franz Baader, Hans-Jurgen Burckert, Bernhard Nebel, Werner Nutt, and Gert Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1--18, 1993.

No context found.

F. Baader, H.-J. Burckert, B. Nebel, W. Nutt, and G. Smolka. On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic, Language and Information, 2:1-18, 1993.

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