Harald Ganzinger, Christoph Meyer, and Margus Veanes. The twovariable guarded fragment with transitive relations. In Proc. 14th IEEE Symposium on Logic in Computer Science, pages 24--34. IEEE Computer Society Press, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the principle and the insights underlying the axiomatic translation is applicable to a wide range of non standard axiom schemata, in particular, those found in agent based systems, description logics, and related fragments of first order logic such as the monadic guarded fragment introduced in [7]. ....

H. Ganzinger, C. Meyer, and H. de Nivelle. The two-variable guarded fragment with transitive relations. In Proc. LICS, pp. 24--34. IEEE Computer Society, 1999.

....throughout the paper) Transitivity axioms cannot be expressed in GF: Adding transitivity axioms to a GF formula causes undecidability. See [Gr a99a] Because of the apparent insuciency of GF to capture some of the basic modal logics, various extensions of GF have been proposed and studied. In [GMV99], it has been shown that GF with transitivity axioms is decidable, on the condition that binary predicates occur only in guards. The complexity bound given there is non elementary. In [ST01] the complexity bound for GF with transitive guards is improved to 2EXPTIME and also shown NEXPTIME ....

.... of [Niv99] and [Dem01] In this paper, we use a translation into decidable rst order fragments instead of a translation into propositional dynamic logic as done in [Dem01] The frame conditions we consider in this work can be de ned by the closure operators that are MSO de nable considered in [GMV99]. By contrast to what is done in [GMV99] we get decidability and complexity characterization but our target rst order language is more restricted. Moreover, we do not use MSO de nable built in relations, just plain GF . This has also advantages from the complexity point of view. Indeed, GF ....

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H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations (extended abstract). In LICS'99, pages 24-34. IEEE Computer Society Press, 1999.

....Of theoretical interest is that a variety of inference techniques have been developed for the guarded fragment. The fragment and its extensions have been shown decidable using ordered resolution [3, 5] alternating automata [9] tableaux methods [12] or embedding into monadic second order logic [6]. In fact, in [4] it is shown how any ALC(#,#, concept translates into a guarded formula. One important property of the guarded fragment is that the guards, that is the atoms we obtain from the translation of roles, are always positive. Therefore, and its extensions contain concepts ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pp. 24--34. IEEE Computer Society Press, 1999.

....1 the question of its decidability enumerability remains open. 2 Minsky machines The (two counter) Minsky machine represents a universal model of computation [Min61, Min67] Being of very simple structure, Minsky machines are very convenient for proving undecidability results (see for example [Hut94, KR95, CZ97, GMV99]) A Minsky machine M is a simple imperative program consisting of a sequence of instructions labelled by natural numbers from 1 to some L. It starts from an instruction labelled 1 and operates with two counters S 1 and S 2 each containing a nonnegative integer. Any instruction is one of the ....

.... nor the new formula given above are guarded. So, the question about decidability enumerability of the monodic and guarded fragment with equality is open as before. Related papers dealing with undecidable guarded fragments of non temporal firstorder logic with added transitive relations are [GMV99] and [Gra99] In [Gra99] it is shown that the three variable guarded fragment equipped with two transitive binary relations is not recursively enumerable, while in [GMV99] the authors have shown that the two variable guarded fragment without equality, but equipped with five transitive relations ....

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H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive realtions. In Proceedings of 14th Annual IEEE Symposium on Logic in Computer Science(LICS'99), pages 24--34, 1999.

....and by extending the notion of a clash. Please note that this is not true for the full Guarded Fragment because it becomes undecidable when extended with number restrictions [ 5 ] Another interesting extension is the one with transitive roles (binary relations) and or axioms. It was shown in [ 4 ] that the Guarded Fragment extended with transitive relations, even when restricted to two variables, becomes undecidable. The investigation of these extensions is ongoing work. ....

H. Ganzinger, Chr. Meyer, and M. Veanes. The twovariable guarded fragment with transitive relations. In Proc. 14th IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, 1999. To appear in LICS'99.

....work. Classi cation. Is it the case that any regular grammar logic has a satis ability problem that is either EXPTIME hard or in PSPACE If the answer is positive, then is it a decidable problem to check whether a regular grammar logic is EXPTIME hard or in PSPACE First order fragments. In [GMV99], it is shown that the guarded fragment without equality, with ve built in transitive relations, and two variables is undecidable (see also [Gr a99a] Is there a simple rst order extension of either FO2 or the guarded fragment that is decidable and such that GSP(REG) can be naturally translated ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations (extended abstract). In LICS'99, pages 24-34, 1999. Available via http://www.mpi-sb.mpg.de/hg on WWW. 47

....under submodels. They show that the guarded fragment (GF) shares all these properties with basic modal logic. Decidability was also shown for various extensions of the guarded fragment, like the loosely guarded fragment [15, 19] guarded xpoint logic [20] or monadic GF 2 with transitive guards [16]. The various decision procedures exploit the nite model property, use ordered resolution, alternating automata, or embeddings into monadic second order logic. This is an interesting contrast to the literature on decidable modal logics and description logics, where tableaux based decision ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pages 24-34. IEEE Computer Society, 1999.

....knowledge bases of n ary description logics and contains many modal logics. It was introduced by Andr eka, van Benthem and Nem eti [1, 2] The fragment as well as its several extensions like the loosely guarded fragment [6, 11] guarded xpoint logic [12] or monadic GF 2 with transitive guards [7] have been shown decidable. The various decision procedures have double exponential worst case time and space complexity and exploit the nite model property, use ordered resolution, alternating automata, or embedding into monadic second order logic. They are an interesting contrast to the ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pages 24-34. IEEE Computer Society, 1999.

....di erent levels of guardedness . In this paper we consider guarded fragments of rst order logic and least xed point logic with two notions of guardedness. While model theoretic properties and satis ability algorithms for guarded logics have already been studied rather extensively (see, e.g. [1, 8, 11, 12, 15]) the model checking problem has not yet received as much attention. In [9] a guarded variant of Datalog, called Datalog LITE, has been introduced which is shown to admit ecient query evaluation (linear time in the query length and the size of the database) Datalog LITE is equivalent, via ecient ....

H. Ganzinger, C. Meyer, and M. Veanes, The two-variable guarded fragment with transitive relations, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, 24-34.

....they exhibit at least double exponential worst case time and space complexity [4, 9] which is in contrast to the low complexity of the satis ability problem of basic modal logic. Moreover, extensions of the guarded fragment with transitivity or number restrictions lead to undecidability [7], even though modal logics extended with transitivity or number restrictions are decidable. This shows the guarded fragment as a whole is too general and expressive and cannot thoroughly explain the good computational properties of modal logics and related description logics. A natural question ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pp. 24-34. IEEE Computer Society, 1999.

....thereofore is undecidable by [23] Example 3.6 If instead of QCL we take the theory of transitive guards, the Guarded Fragment even with two individual variables becomes undecidable. However, if it is also required that the set A should contain only monadic atoms, the decidability is regained [14]. Example 3.7 On the other hand, consider the extension of guarded formulas, in which the contexts of quantification are #x (#(x, y) # #(x, y) 4. CUBIC FRAGMENTS 177 where # is a conjunction of atoms, such that every two variables from x, y co occur in some conjunct of #. Then we obtain ....

....is a very complex fragment of classical logic with a simple description. 2. Is the logic K4 2 decidable Note that K4 2 n is equivalent to SF with transitive guards. This logic seems to be close to undecidability; it is known that a similar fragment GF with transitive guards is undecidable [14]; also, due to a recent result by A.Kurucz and M.Zakharyaschev, K4 3 is undecidable. On the other hand, its sublogic K K4 is decidable, but lacks the product f.m.p. 46] 3. Is the logic K 3 (or the cubic fragment CF 3 ) decidable From [12] we know that K 3 has the f.m.p. and is RE. ....

H. Ganzinger, C. Meyer, M. Veanes. The two-variable Guarded Fragment with transitive relations. In: Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, Trento, pp. 24-34, 1999.

....we may also want to make a distinction between action predicates and state predicates. This makes a substantial di#erence for logical properties, but it remains to be determined what its influence is on computational properties. As a first step in this direction, Ganzinger, Meyer, and Veanes [12] show that GF 2 (GF with only two variables) where, in addition, binary relations may be specified as transitive, is undecidable, but that making the strict separation between action and state predicates and allowing transitivity statements only for action predicates, restores decidability. 6 ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. Unpublished.

.... symbols [27, 5] the fragment with only two individual variables [36, 31] 1 the guarded fragment with quanti cation of the form 9y(G(x; y) x; y) where the guard G(x; y) is atomic 2 [1] The current state of art in this eld is presented in the recent monograph [6] see also [1, 43, 44, 13, 14, 18, 29, 32]. For modal logicians the decision problem in rst order modal logics seemed almost hopeless. The following list covers basically all known results and leaves not too much space for maneuver: the monadic fragment of practically all modal predicate logics is undecidable [25] see also [24] ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proceedings of the 14th IEEE Symposium, pages 24-34, 1999.

....under submodels. They show that the guarded fragment (GF) shares all these properties with basic modal logic. Decidability was also shown for various extensions of the guarded fragment, like the loosely guarded fragment [12, 15] guarded xpoint logic [16] or monadic GF 2 with transitive guards [13]. The various decision procedures exploit the nite model property, use ordered resolution, alternating automata, or embeddings into monadic secondorder logic. This is an interesting contrast to the literature on decidable modal logics and description logics, where tableaux based decision ....

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pp. 24-34. IEEE Computer Society, 1999.

.... obtained by suitably instantiating parameters in existing theorem proving frameworks (Bachmair and Ganzinger, 1997) For the optimised functional translation this has been demonstrated in (Schmidt, 1997; Schmidt, 1999) while the relational and semi functional translation have been considered in (Ganzinger et al. 1999a; Hustadt, 1999; Hustadt and Schmidt, 1999b; Hustadt and Schmidt, 1998) Third, there are general guidelines for choosing these parameters in the right way to enforce termination and the hard part is to prove that termination is guaranteed. The most common approach uses ordering refinements of ....

....complete, and terminating decision procedures for combinations of interacting modal logics without any additional effort. It is even possible to obtain general characterisations of the boundaries of decidability of combinations of interacting modal logics in this way. For example, results from (Ganzinger et al. 1999b) imply that if we have two modal logics which satisfy the 4 axiom, that is the accessibility relations are transitive, and the modal logics are interacting, for example, by the axiom schema (1) then the combined logic is undecidable. 6. General Frameworks The considerations of previous ....

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H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science (LICS'99). IEEE Computer Society Press, 1999.

....properties of modal logics (and hence also description logics) Guarded logics have received considerable attention in the last years and it seems that the notion of guarded quanti cation is indeed very important for the design of logics that are both expressive and algorithmically manageable. See [1, 2, 3, 5, 6, 10, 9, 8, 12, 13] for background and further results on guarded logics and [11] for an informal discussion. 2 A general criterion for decidability Guarded second order logic, abbreviated GSO, has the same syntax as ordinary second order logic, but second order quanti ers are restricted semantically to range over ....

H. Ganzinger, C. Meyer, and M. Veanes, The two-variable guarded fragment with transitive relations, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 24-34.

....has been extended by transitivity on an extra logical level (since transitivity is not expressible within the logic itself) and the following results have been obtained: 3 . GF 3 with transitive relations is undecidable (see [9] LGF with one transitive relation is undecidable (see [7]) Even GF 2 with transitive relations is undecidable (see [7] Monadic GF 2 with binary transitive, symmetric and or reflexive relations is decidable (see [7] None of these results is applicable in the case of ALCRA # . The most important result concerning ALCRA # is the ....

....transitivity is not expressible within the logic itself) and the following results have been obtained: 3 . GF 3 with transitive relations is undecidable (see [9] LGF with one transitive relation is undecidable (see [7] Even GF 2 with transitive relations is undecidable (see [7]) Monadic GF 2 with binary transitive, symmetric and or reflexive relations is decidable (see [7] None of these results is applicable in the case of ALCRA # . The most important result concerning ALCRA # is the last one, since ALC is in monadic GF 2 , and the role box allows ....

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H. Ganzinger, Chr. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. 14th IEEE Symposium on Logic in Computer Science, pages 24--34. IEEE Computer Society Press, 1999. To appear in LICS'99.

No context found.

Harald Ganzinger, Christoph Meyer, and Margus Veanes. The twovariable guarded fragment with transitive relations. In Proc. 14th IEEE Symposium on Logic in Computer Science, pages 24--34. IEEE Computer Society Press, 1999.

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Ganzinger, H., C. Meyer and M. Veanes, The two-variable guarded fragment with transitive relations, in: Logic in Computer Science, 1999, pp. 24--34.

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H. Ganzinger, C. Meyer, and H. de Nivelle. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pp. 24--34. IEEE Computer Society, 1999.

No context found.

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. LICS'99, pp. 24--34. IEEE Computer Society Press, 1999.

No context found.

H. Ganzinger, Chr. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proc. 14th IEEE Symposium on Logic in Computer Science, pages 24--34. IEEE Computer Society Press, 1999. To appear in LICS'99.

No context found.

H. Ganzinger, C. Meyer, and M. Veanes. The two-variable guarded fragment with transitive relations. In Proceedings of the Fourteenth Annual IEEE Symposium on Logic in Computer Science, pages 24-34. IEEE Computer Society Press, 1999.

No context found.

Harald Ganzinger, Christoph Meyer, and Margus Veanes. The two-variable guarded fragment with transitive relations. In 14th IEEE Symposium on Logic in Computer Science (LICS), pages 24--34, Trento, Italy, July 2--5, 1999.

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H. Ganzinger, C. Meyer, and M. Veanes, The two-variable guarded fragment with transitive relations, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 24-34.

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