Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In Pierangelo Miglioli, Ugo Moscato, Daniele Mundici, and Mario Ornaghi, editors, Proceedings, 5th International Workshop on Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Arti cial Intelligence, pages 93-109. Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....previous section can be used to integrate multiple background reasoners into a theory reasoning calculus. To be specific we will define a multi reasoner extension of the partial theory version of (free variable) semantic tableaux. Our exposition of the calculus will follow closely the one given in [3]. There, a tableau is defined as a multiset of tableau branches, where a branch is a multiset of first order formulas. As usual, branches are interpreted logically as the conjunction of their elements, and tableaux as the disjunction of their branches. Tableaux are manipulated according to the ....

B. Beckert and Ch. Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli et al, editors, Proc. of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX'92, volume 1071 of Lecture Notes in Computer Science, pages 93--109. Springer, 1996.

....key sets e#ectively and on the background reasoner side how to generate residues e#ciently. These same issues remain crucial in our approach as well. We did show that, under the A noteworthy approach partially addressing these issues and based on incremental methods is described in [BP96] 27 right conditions, it is enough to consider only certain types of key sets and residues. But even within these restrictions the number of possible choices is still large enough to make actual applications impractical without further optimizations. As we mentioned in the introduction, the ....

....possible choices is still large enough to make actual applications impractical without further optimizations. As we mentioned in the introduction, the research challenge is now to identify specific combinations of theories and more or less general implementation techniques like those described in [BP96] for which our cooperation approach is feasible. Focusing on the specialized results from Section 5, one potentially interesting application could come in conjunction with Baumgartner s results on linearizing completion [Bau96] a technique for producing background reasoners automatically for ....

Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Proceedings of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX'92 Palermo, (Italy), volume 1071 of Lecture Notes in Computer Science, pages 93--109. Springer, 1996.

....previous section can be used to integrate multiple background reasoners into a theory reasoning calculus. To be speci c we will de ne a multi reasoner extension of the partial theory version of (free variable) semantic tableaux. Our exposition of the calculus will follow closely the one given in [3]. There, a tableau is de ned as a multiset of tableau branches, where a branch is a multiset of rst order formulas. As usual, branches are interpreted logically as the conjunction of their elements, and tableaux as the disjunction of their branches. Tableaux are manipulated according to the ....

B. Beckert and Ch. Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli et al, editors, Proc. of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX'92, volume 1071 of Lecture Notes in Computer Science, pages 93-109. Springer, 1996.

....possible choices is still large enough to make actual applications impractical without further optimizations. As we mentioned in the introduction, the research challenge is now to identify speci c combinations of theories and more or less general implementation techniques like those described in [BP96] for which our cooperation approach is feasible. Focusing on the specialized results from Section 5, one potentially interesting application could come in conjunction with Baumgartner s results on linearizing completion [Bau96] a technique for producing background reasoners automatically for ....

....would like to thank Melving Fitting for pointing out Malitz s interpolation theorem to me, and Peter Baumgartner for his clari cations on some computational aspects of theory reasoning. 18 A noteworthy approach partially addressing these issues and based on incremental methods is described in [BP96] 19 Strictly speaking we have done that only for the semantic tableaux calculus. But, again, the same is is possible for other calculi. 27 A Proofs In the following, we will use the standard model theoretic notions of embedding, isomorphism, substructure, generators, reducts and so on. The ....

Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Proceedings of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX'92 Palermo, (Italy), volume 1071 of Lecture Notes in Computer Science, pages 93-109. Springer, 1996. 32

....restricted. Notice that the satis ability problem for pure rst order 98 sentences with equality has been rst solved by Bernays and Sch on nkel (cf. 9] A more general approach to theory reasoning has been rst described in [12] in the context of resolution) and, more recently, in [1] (in the context of semantic tableaux) In particular, 1] proposes a general incremental approach to apply theory reasoning to the framework of free variable semantic tableaux. Though our approach is more restricted in scope, in some favorable cases it yields decidability results. 2 First order ....

.... for pure rst order 98 sentences with equality has been rst solved by Bernays and Sch on nkel (cf. 9] A more general approach to theory reasoning has been rst described in [12] in the context of resolution) and, more recently, in [1] in the context of semantic tableaux) In particular, [1] proposes a general incremental approach to apply theory reasoning to the framework of free variable semantic tableaux. Though our approach is more restricted in scope, in some favorable cases it yields decidability results. 2 First order Theories We closely follow the notation and terminology ....

[Article contains additional citation context not shown here]

B. Beckert and C. Pape. Incremental theory reasoning methods for semantic tableaux. In Proc. of 5th International Workshop TABLEAUX '96, volume 1071 of Lecture Notes in Articial Intelligence, pages 93-109. Springer-Verlag, 1996.

....previous section can be used to integrate multiple background reasoners into a theory reasoning calculus. To be specific we will define a multi reasoner extension of the partial theory version of (free variable) semantic tableaux. Our exposition of the calculus will follow closely the one given in [3]. There, a tableau is defined as a multiset of tableau branches, where a branch is a multiset of first order formulas. As usual, branches are interpreted logically as the conjunction of their elements, and tableaux as the disjunction of their branches. Tableaux are manipulated according to the ....

B. Beckert and Ch. Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli et al, editors, Proc. of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX'92, volume 1071 of Lecture Notes in Computer Science, pages 93--109. Springer, 1996.

....by the fact that most equalities present on a branch are actually not needed to close it, such that computing a completion of all available equalities not only is expensive, but quite useless. These diculties can (at least partially) be avoided by using incremental methods for equality reasoning [8]. These are algorithms that after a futile try to solve an E uni cation problem allow to store the results of the equality reasoner s computations and to reuse them for a later call (with additional equalities) Then, in case of doubt, the equality reasoner can be called early without running the ....

Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Proc. 5th TABLEAUX, Terrasini/Palermo, Italy, LNCS 1071, pages 93{

....during the construction of a single tableau branch. After a futile try to find a unifier, the data computed by the background reasoner is reused for later calls, in particular it is reused for different extensions of the branch. We call this feature of 3 T A P incremental equality reasoning [4]. Search Space Restrictions One possibility to restrict the search space is to avoid putting redundant axioms on the tableau in the first place. This is particularly useful with huge axiomatizations where only a small subset of the axioms is actually needed to prove a given theorem. Only formulae ....

Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In Proceedings, 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, Palermo, Italy, LNCS. Springer, 1996.

....rule for equality. We will use the completion based method for mixed universal and rigid E unification of [Bec94] This procedure was developed to be used within free variable semantic tableaux, and its application within the modified hyper tableaux calculus is very natural. Its recent version [BP96] is in particular attractive because it improves the interaction between the foreground reasoner (i.e. hyper tableaux) and the equality reasoner by preserving the results of the completion procedure to subsequent calls to it. ....

B. Beckert and C. Pape. Incremental Theory Reasoning Methods for Semantic Tableaux. In P. Moscato, U. Moscato, D. Mundici, and M. Ornaghi, editors, Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Artificial Intelligence, pages 93--109. Springer, 1996.

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Bernhard Beckert and Christian Pape. Incremental theory reasoning methods for semantic tableaux. In Pierangelo Miglioli, Ugo Moscato, Daniele Mundici, and Mario Ornaghi, editors, Proceedings, 5th International Workshop on Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Arti cial Intelligence, pages 93-109. Springer, 1996.

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