Search

We use mosaics to provide a simple, sound, complete and terminating tableau reasoning procedure for the temporal logic of until and since over general linear time. [ABSTRACT FROM AUTHOR]

We present a sound, complete and relatively straightforward tableau method for deciding valid formulas in the propositional version of the bundled (or suffix and fusion closed) computation tree logic BCTL*. This proves that BCTL* is decidable. It is also moderately useful to have a tableau available for a reasonably expressive branchingtime temporal logic. However, the main interest in this should be that it leads us closer to being able to devise a tableau-based technique for theorem-proving in the important full computational tree logic CTL*. [ABSTRACT FROM AUTHOR]

In this paper, three refinements on I-SATCHMO are presented. I-SATCHMO is an application of the backjumping strategy for SATCHMO, which is a model generation theorem prover and is formalized as positive unit hyperresolution tableaux, to pruning away redundant branches. However, I-SATCHMO does not completely implement backjumping. Since marking an atom as useful whenever it contributes to a forward chaining reasoning, I-SATCHMO suffers from two problems. One is that the atoms that only contribute to irrelevant forward chaining reasoning are marked as ‘useful’. The other is that the atoms that are only useful for the sibling branches of the branch under consideration are taken as ‘useful’ for the branch. In both cases, I-SATCHMO might take some irrelevant violated clauses as ‘relevant’ and therefore use them to expand the search space. Moreover, not utilizing the partial reasoning results derivable during reasoning, I-SATCHMO might make repeated reasoning. Addressing these problems, our extension, called R-SATCHMO, refines I-SATCHMO in three ways: (1) only making the atoms that contribute to relevant forward chaining reasoning as useful; (2) it distinguishes between the set of useful atoms in different branches of a proof tree; (3) it summarizes the derived refutations as nogoods and utilizes them to avoid repeated reasoning. Hence, R-SATCHMO always searches a subspace of that which I-SATCHMO does. R-SATCHMO is optimal in the sense that during reasoning no same complete subtree is constructed in a proof tree. Moreover, R-SATCHMO can be used to generate minimal unsatisfiable subsets of an unsatisfiable set of clauses, and the strategies proposed in R-SATCHMO are applicable to minimal model generation procedures. We describe our refinements, present our implementation, prove the correctness, provide examples to demonstrate the power of our refinements, and discuss the application to minimal model generation. [ABSTRACT FROM PUBLISHER]

This paper presents a complete axiomatization for propositional interval temporal logic (PITL) with projection. The axiomatization is based on a tableau decision procedure for the logic, which in turn is founded upon a normal form for PITL formluae. The construction of the axiomatization provides a general mechanism for generating axiomatizations thus: given a normal form for a new connective, axioms can be generated for the connective from the tableau construction using that normal form. The paper concludes with a discussion of aspects of compositionality for PITL with projection. [ABSTRACT FROM PUBLISHER]

A sequent calculus of a new sort is extracted from the Prolog program lean7AP. This calculus is sound and complete, even though it lacks almost all structural rules. Thinking of lean7AP as a sequent calculus provides a new perspective on it and, in some ways, makes it easier to understand. It is also easier to verify correctness and completeness of the Prolog implementation. In addition, it suggests extensions to other logics, some of which are considered here. [ABSTRACT FROM PUBLISHER]