Beckert, B. and R. Gore, Free variable tableaux for propositional modal logics, Studia Logica 69 (2001), pp. 59--96.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1994] Though the techniques are routine, the combination is still somewhat unusual. Research in modal logic whether the investigation is more mathematical [Gore, 1992, Massacci, 1998b, Massacci, 1998a, Gore, 1999] or primarily concerns algorithms for proof search [Otten and Kreitz, 1996, Beckert and Gore, 1997, Schmidt, 1998] is dominated by the study of the propositional logic of a single modal operator (or accessibility relation) Moreover, researchers who have investigated modal logic in a first order setting have tended to jump directly into a discussion of particular theorem proving strategies, ....

Beckert, B. and Gore, R. (1997). Free variable tableaux for propositional modal logics. In TABLEAUX'97, LNAI 1227, pages 91--106. Springer.

....on the size of candidate proofs, whose values are entered by the user. It is complete in the sense that if a formula is valid, then a proof is found, provided that the input bounds are big enough. Note that, although some theorem provers for non classical logics exist (for instance, ModLeanTap [1] deals with propositional modal logics, while IleanTap [12] deals with quantification, but only in a intuitionistic logic setting) up to our knowledge, not much has been done on QML. Section 5 concludes this work with a discussion on equality. 2 Quantified Modal Logics In Kripke semantics for ....

B. Beckert and R. Gore. Free variable tableaux for propositional modal logics. In Proc. of TABLEAUX'97, pages 91--106. Springer, 1997.

....few works address the problem of dealing with interaction axioms. For instance, in [22] a resolution for some modal and multimodal logics is presented, but, in the case of multimodal logics, the interaction axioms considered are simple and limited to axioms I and 4M . In the case of tableaux [28, 14, 39, 32, 33, 17, 10, 11], most of the proposed calculi deal with modal but not with multimodal systems, and those developed for multimodal logics, see for instance [14, 33] only deal with specific cases (see Section 5. for a detailed discussion) Perhaps the most general proposals to deal with multimodal logics are the ....

....hand, we deal with a wider class of logics. In particular, we think that it is hard to extend the unification method of prefixes so to deal with all the classes of logics that we considered for the same reasons given in Section 5. 2 when discussing the proposals in [28, 39, 32] The proposals in [33, 17, 10] address the problem of an efficient implementation of the tableau calculi for a wide class of modal logics. They generalize the prefixes by allowing occurrences of variables and they use unification to show that two prefixes are names for the same world. While a straightforward implementation of ....

B. Beckert and R. Gor e. Free Variable Tableaux for Propositional Modal Logics. In D. Galmiche, editor, Proc. of the International Conference on Automatic Reasoning with Analytic Tableaux and Related Methods, TABLEAUX'97, volume 1227 of LNAI, pages 91--106. Springer-Verlag, 1997.

.... of tableau and sequent calculi have been proposed for modal logics and surveys can be found in [15, 20] For sake of simplicity, we stick to the Gentzen like formulation here without adding extra logical symbols, although these techniques can be easily adapted to prefixed systems such as those of [15, 26, 5]. The corresponding sequent rule are shown in Fig. 4 where different logics can be obtained by changing the operation S . Table 2 lists the major modal logics of knowledge and belief [21] and other logics can be found in [15, 20] To obtain deontic versions of those logics one simply need ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. In D. Galmiche, editor, Proceedings of the International Conference on Analytic Tableaux and Related Methods (TABLEAUX-97), Lecture Notes in Artificial Intelligence. Springer-Verlag, 1997.

....algebra and to specific algorithmic procedures for manipulating labels. In the literature, it is usual to distinguish two ways to deal with labels: 1) labels are propagated using logic dependent inference rules (see, for example, Fitting s prefixed tableaux [17] and single step tableaux [33, 6]) 2) labels are matched according to logic depended label unifications. Even if our system employs the second strategy, it is worth noting that, in both cases, it is necessary to take care of the formulas that parameterize labels when inference rules or unifications are defined, and the ....

Bernhard Beckert and Rajeev Gore. Free-variable tableaux for propositional modal logic. Studia Logica, forthcoming.

....and their labels # B unify. For detailed accounts of KEM see [1, 8, 9] 2. 3 Single Step Tableaux (SST) Single Step Tableaux [12] originate from and add modularity to Fitting s prefix tableaux [6] The free variable version we shall focus on here has been proposed by Beckert and Gore [2]. The basic idea of SST is that (modal) formulas are used to move the evaluation point to the neighbourhood of the labels they are associated with, that is, each time we are allowed to move only to one step apart. In other words the information that can be extracted from a formula are ....

Bernard Beckert and Rajeev Gore. Free variable tableaux for propositional modal logics. Studia Logica, 2001.

.... search procedure (see [15] Besides the original leanTAP implementation for classical rst order logic, lean tableau provers are also available for various non classical logics, i.e. rstorder intuitionistic logic (ileanTAP, 21] and the propositional modal logics K, KD, KT, and S4 (ModLeanTAP, [4]) The linTAP implementation lls the gap for the multiplicative linear logic. The source code of linTAP can be obtained at http: www.intellektik.informatik.tu darmstadt.de jeotten linTAP . ....

B. Beckert, R. Gore. Free Variable Tableaux for Propositional Modal Logics. Proc. 6 th TABLEAUX Conference, LNAI 1227, pp. 91-106, Springer 1997.

....logic programming and knowledge representation remains a novel one. In fact, even today, research in modal logic whether the investigation is more mathematical [Gore, 1992, Massacci, 1998b, Massacci, 1998a, Gore, 1999] or primarily concerns algorithms for proof search [Otten and Kreitz, 1996, Beckert and Gore, 1997, Schmidt, 1998] is dominated by the study of the propositional logic of a single modal operator (or accessibility relation) When multiple modal operators are considered, their interpretations and interactions are often predefined. In PDL and terminological logics we have combinations of ....

Beckert, B. and Gore, R. (1997). Free variable tableaux for propositional modal logics. In TABLEAUX'97, LNAI 1227, pages 91--106. Springer.

....in order to conclude with success the proof. Moreover, adding exsplicitly the subprefixes, as the one above, is not enough to solve the problem, since all prefixes representing the same world have to be identified. Similar consideration can be done for the proposals in [18, 32] The proposals in [18, 32, 5] address the problem of an efficient implementation of the tableau calculi for a wide class of modal logics. They generalize the prefixes by allowing occurrences of variables and they use unification to show that two prefixes are names for the same world. While a straightforward implementation of ....

B. Beckert and R. Gor'e. Free Variable Tableaux for Propositional Modal Logics. In Proc. of TABLEAUX'97, volume 1227 of LNAI, pages 91--106. Springer-Verlag, 1997.

....satisfiability of formulae in propositional modal logics are becoming available. These systems, including DLP [ 30 ] FaCT [ 23 ] KSatC [ 13 ] SAT [ 35 ] andTA [ 26 ] have more optimisations and are much faster than the previous generation of modal decision procedures, such as leanK [ 3 ] Logics Workbench [ 21 ] and#KE [ 32 ] As with most theorem proving problems, neither computational complexity nor algorithm complexity is useful in determining the e#ectiveness of optimisations. The worst case complexity of the problem, of course, remains unchanged. For many propositional ....

....results for DLP and di#er considerably for TA because they were performed on a di#erent version of the systems. 300 An Analysis of Empirical Testing for Modal Decision Procedures branch d4 dum grz lin path ph poly t4p K p n p n p n p n p n p n p n p n p n leanK 2. 0 1 0 1 1 0 0 0 4 2 0 3 1 2 0 0 0 #KE 13 3 13 3 4 4 3 1 2 17 5 4 3 17 0 0 3 LWB 1 0 6 7 8 6 13 19 7 13 11 8 12 10 4 8 8 11 8 7 TA 9 9 18 20 20 6 9 16 17 19 KSAT 8 8 8 5 11 17 3 4 8 5 5 13 12 10 18 SAT 1.2 12 8 12 Crack 1.0 2 1 2 3 3 1 5 2 2 6 2 3 1 1 ....

[Article contains additional citation context not shown here]

B. Beckert and R. Gore. Free variable tableaux for propositional modal logics. In Automated Reasoning with Analytic Tableaux and Related Methods: International Conference Tableaux'97, number 1227 in Lecture Notes in Artificial Intelligence, pages 91--106, Berlin, 1997. SpringerVerlag.

....also called labels, used in various proof systems for modal logics (see e.g. Fit83, Wal90] no proof systems for nominal tense logics using this conceptual similarity exist. This is all the more surprising because during the last years, prefixed calculi have regained some interest (see e.g. [Ogn94, Mas94, Gov95, Rus96, Gab96, BMV97, BG97]) Although designing general frameworks defining proof systems for modal logics is a fundamental task, other works deal with the difficult problem of improving significantly the mechanization of logics by finding refined properties, mostly proof theoretical, that provide better complexity bounds ....

....length of proof steps. Since the calculi involved in the paper use a very restricted cut rule (the size of the cut formula is linear in the size of the conclusion) our definition is sufficient for our needs. As no subexponential algorithm for any NP hard problem is known, such a cubic overhead 4 [BG97] is one of the rare papers where such a relationship is explicitly recognized. 5 Since and have the same cardinality, without any loss of generality, we can assume that the individual constants and the nominals are respectively of the form a oe and i oe where oe 2 . jOE X j 2 O(ji 1 : ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. In TABLEAUX'97, pages 91--106. LNAI 1227, Springer, 1997.

....logic. The first uniform tableau calculi for various modal logics and intuitionistic logic are presented in [Fitting, 1983] Like in our approach prefixes are used to describe the peculiarities of each logic. Uniform tableau methods for all 15 classes of modal logics are described in [Beckert and Gor e, 1997]. These methods are limited to propositional logic and do not use a specialized string unification which is necessary for a more efficient proof search. There are also approaches for systematizing calculi for finite valued logics [Carnielli, 1987, Hahnle, 1993, Surma,1984] These approaches ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. 6 th International Conference on Analytic Tableaux and Related Methods, LNAI 1227, pp. 91--106, 1997.

.... procedure (see [15] Besides the original leanTAP implementation for classical first order logic, lean tableau provers are also available for various non classical logics, i.e. firstorder intuitionistic logic (ileanTAP, 21] and the propositional modal logics K, KD, KT, and S4 (ModLeanTAP, [4]) The linTAP implementation fills the gap for the multiplicative linear logic. The source code of linTAP can be obtained at http: www.intellektik.informatik.tu darmstadt.de jeotten linTAP . ....

B. Beckert, R. Gor'e. Free Variable Tableaux for Propositional Modal Logics. Proc. 6 th TABLEAUX Conference, LNAI 1227, pp. 91--106, Springer 1997.

....none elsewhere. In Figure 9 (right) the median CPU time and the other Qth percentile plots reveal a typical easy hard easy pattern centred in the satis ability transition area, growing up to the 50 cross over point, and then decreasing gently. Now compare the plots with those of the analogous 3CNF2m test set in Figure 2 (the parameters values and the system tested are the same) Firstly, the satis ability transition here is relatively shifted to the right. Secondly, the New 3CNF2m CPU time plots are slightly harder than the 3CNF2m plots in the satis able area, becoming dramatically ....

....up to the 50 cross over point, and then decreasing gently. Now compare the plots with those of the analogous 3CNF2m test set in Figure 2 (the parameters values and the system tested are the same) Firstly, the satis ability transition here is relatively shifted to the right. Secondly, the New 3CNF2m CPU time plots are slightly harder than the 3CNF2m plots in the satis able area, becoming dramatically harder as the fraction of trivially unsatis able 3CNF2m formulae increases. In fact, the e ect is due to the Qth percentile values for the 3CNF2m test set becoming dramatically lowered ....

[Article contains additional citation context not shown here]

B. Beckert and R. Gore. Free variable tableaux for propositional modal logics. In Automated Reasoning with Analytic Tableaux and Related Methods: International Conference Tableaux'97, number 1227 in Lecture Notes in Articial Intelligence, pages 91-106, Berlin, 1997. SpringerVerlag.

.... it is cumulative as Hilbert style axiomatizations 1 ; Gamma can be used in a modular way for a wide range of modal logics; Gamma has good computational characteristics; Gamma can be used to derive result about other systems; Gamma can be easily implemented using free variables tableaux [1]; Gamma and last but not least, it can still be used for doing proofs by hand. 1.1. Results presented in this paper In this paper we focus mainly on two aspects: 1. the methodology underlying the construction of the calculus; 2. its computational properties in comparison with other formalisms. ....

....of [24] corresponds to a visit of the tree of prefixes directly in the pruned branch B 0 . We have replaced loop checking for SST with a simple and local check. 8.4. Decidability for translation methods This result can be easily extended to the free variables SST proposed by Beckert and Gor e [1] and to other methods. It gives a direct decidability check for functional [40, 41] and mixed [38] translation methods. Observe that there is a 1 1 mapping between prefixes in (single step) prefixed tableaux and ground terms with functional or mixed translation methods [46, 49] for the logics K, ....

[Article contains additional citation context not shown here]

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. In [18], pages 91--108, 1997.

....unify check, and mini rele. The first uniform tableau calculi for various modal logics and intuitionistic logic are presented in [13] Like in our approach prefixes are used to describe the peculiarities of each logic. Uniform tableau methods for all 15 classes of modal logics are described in [4]. These methods are limited to propositional logic and do not use a specialized string unification which is necessary for a more efficient proof search. There are also approaches for systematizing calculi for finite valued logics [10, 18, 43] These approaches however, do not aim at efficient ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. 6 th International Conference on Analytic Tableaux and Related Methods, LNAI 1227, pp. 91--106, 1997.

....we can introduce a number of optimizations as in modal and dynamic logic. For instance we may apply the following observation: Observation 9 Rule hai must be applied to a prefixed statement oe : a says s) only if there is no prefixed statement oe:a:n : s already present in the branch. See [3, 9, 21] for further references and techniques. Other optimizations are linked to this particular calculus. For instance the statement P ) A B) is logically equivalent to P ) A P ) B. A rule transforming statements of the former type into the latter type can be useful since it may lead to ....

....novel techniques such as passing from a tree like tableau to a forest of prefixes, a run time update of global assumptions and a modification of the notion of branch. Future research is in the direction of providing a fully automated verifier, possibly using the results of Beckert and Gor e [3], and extending the formalism to first order logic for expressing quantified properties over objects. An important side effect of these deductive techniques (discussed in [30] is an EXPTIME tableau for multi modal logics with the universal modality [19, 20] Acknowledgments Most of this work was ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. In Proceedings of the International Conference on Analytic Tableaux and Related Methods (TABLEAUX-97), volume 1227 of Lecture Notes in Artificial Intelligence, pages 91--108. Springer-Verlag, 1997.

.... modal logics [14] and in particular on Simple Step Tableaux [23,9] Tableaux have many advantages: they can be used for both logical consequence and satisfiability; deduction steps follows the semantics of the logic and the calculus can be easily automated along the lines of lean theorem proving [3]. We view our calculus as a building block of a more general framework such as authentication logics for cryptographic protocols [6,30] where the issue of jurisdiction of agents is a key problem. In this perspective access control can be seen as a problem of jurisdiction in complex and ....

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. In Proc. of TABLEAUX-97, LNAI. Springer-Verlag, 1997. To appear.

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Bernhard Beckert and Rajeev Gor. Free variable tableaux for propositional modal logics. In Proceedings, International Conference on Theorem Proving with Analytic Tableaux and Related Methods, Pont--Mousson, France, LNCS 1227, pages 91--106. Springer, 1997.

....is essentially the same as that of the Prolog first order logic theorem prover leanT A P . With greater available resources on smart cards, an extension of cardT A P to first order logic is straightforward. Additionally the tableaux method is easily extended to incorporate modalities [9], allowing modal and in particular temporal notions in the logic [10] These extensions are of interest with respect to authentication logics and the use of verification to enhance the security of smart cards since authentication logics are based on modal logics [2] Our experiments yielded very ....

Bernhard Beckert and Rajeev Gor'e. Free variable tableaux for propositional modal logics. In D Galmich, editor, Proceedings of the International Conference on Theorem Proving with Analytic Tableaux and Related Methods, volume Lecture Notes in Artificial Intelligence of LNCS, pages 91--106. Springer, May 1997.

....labelled) occurrences of 3A generate the same unique label. The paper is structured as follows: In Sections 2 and 3 we introduce the syntax and semantics of labelled modal tableaux. In Section 4 we introduce our calculus, formalise its soundness and completeness (full proofs can be found in [1]) and present an example. In Section 5 we describe our implementation and present experimental results; and in Section 6 we present our conclusions and discuss future work. 2 Syntax The formulae of modal logics are built in the usual way; see [11] To reduce the number of tableau rules and the ....

....an L expansion rule (Def. 15) the substitution rule (Def. 16) or the L closure rule (Def. 18) and (c) all branches in T r are marked as closed. Theorems 20 and 22 state soundness and completeness for our calculus with respect to the Kripke semantics for logic L; the proofs can be found in [1]. Theorem 20 (Soundness) Let A be a formula in NNF. If there is an L proof T 0 ; T r for the L unsatisfiability of A (Def. 19) then A is L unsatisfiable. We prove completeness for the non deterministic and unrestricted version of the calculus, and also for all tableau procedures ....

[Article contains additional citation context not shown here]

B. Beckert and R. Gor'e. Free variable tableaux for propositional modal logics. Interner Bericht 41/96, Universitat Karlsruhe, Fakultat fur Informatik, 1996.

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Beckert, B. and R. Gore, Free variable tableaux for propositional modal logics, Studia Logica 69 (2001), pp. 59--96.

No context found.

B. Beckert and R. Gore. Free variable tableaux for propositional modal logics. In Automated Reasoning with Analytic Tableaux and Related Methods: International Conference Tableaux'97, number 1227 in Lecture Notes in Artificial Intelligence, pages 91--106, Berlin, 1997. Springer-Verlag.

No context found.

B. Beckert, R. Gore. Free Variable Tableaux for Propositional Modal Logics. : Studia Logica, 69:59--96, 2001.

No context found.

Bernhard Beckert and Rajeev Gor#e. Free variable tableaux for propositional modal logics. In D Galmich, editor, Proceedings of the International Conference on Theorem Proving with Analytic Tableaux and Related Methods,volume Lecture Notes in Arti#cial Intelligence of LNCS, pages 91#106. Springer, May 1997.

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