Deriving Focused Calculi for Transitive Relations
We propose a new method for deriving focused ordered resolution calculi, exemplified by chaining calculi for transitive relations. Previously, inference rules were postulated and a posteriori verified in semantic completeness proofs. We derive them from the theory axioms. Completeness of our calculi then follows from correctness of this synthesis. Our method clearly separates deductive and procedural aspects: relating ordered chaining to Knuth-Bendix completion for transitive relations provides the semantic background that drives the synthesis towards its goal. This yields a more restrictive and transparent chaining calculus. The method also supports the development of approximate focused calculi and a modular approach to theory hierarchies.
Unable to display preview. Download preview PDF.
- 2.L. Bachmair and H. Ganzinger. Rewrite techniques for transitive relations. In Ninth Annual IEEE Symposium on Logic in Computer Science, pages 384–393. IEEE Computer Society Press, 1994.Google Scholar
- 4.L. Bachmair and H. Ganzinger. Strict basic superposition. In 15th International Conference on Automated Deduction, volume 1421 of LNAI, pages 160–174. Springer-Verlag, 1998.Google Scholar
- 6.W. W. Bledsoe and L. M. Hines. Variable elimination and chaining in a resolution-based prover for inequalities. In W. Bibel and R. Kowalski, editors, 5th Conference on Automated Deduction, volume 87 of LNCS, pages 70–87. Springer-Verlag, 1980.Google Scholar
- 9.U. Martin and T. Nipkow. Ordered rewriting and con uence. In M. Stickel, editor, 10th International Conference on Automated Deduction, volume 449 of LNCS, pages 366–380. Springer-Verlag, 1990.Google Scholar
- 10.M. M. Richter. Some reordering properties for inequality proof trees. In E. Börger, G. Hasenjaeger, and D. Rödding, editors, Logic and Machines: Decision Problems and Complexity, Proc. Symposium “Rekursive Kombinatorik”, volume 171 of LNCS, pages 183–197. Springer-Verlag, 1983.Google Scholar
- 12.M. Stickel. Automated deduction by theory resolution. In A. Joshi, editor, 9th International Joint Conference on Artificial Intelligence, pages 1181–1186. Morgan Kaufmann, 1985.Google Scholar
- 13.G. Struth. Deriving focused calculi for transitive relations (extended version). http://www.informatik.uni-freiburg.de/~struth/papers/focus.ps.gz.
- 14.G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut für Informatik, Universität des Saarlandes, 1998.Google Scholar