Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing
Tóm tắt
We propose a general way of combining background reasoners in theory reasoning. Using a restricted version of the Craig interpolation lemma, we show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures. In this case, constraint information is propagated across reasoners eexchanging residues that are, in essence, disjunctions of ground literals over a common signature. As an application of our approach, we describe a multitheory version of the semantic tableau calculus, and we prove it sound and complete.
Tài liệu tham khảo
Baalen, J. V. and Roach, S.: Using decision procedures to accelerate domain-specific deductive synthesis systems, in P. Flener (ed.), Proceedings of the 8th International Workshop on Logic Programming Synthesis and Transformation (LOPSTR’98), Manchester, UK, Lecture Notes in Comput. Sci. 1559, Springer-Verlag, 1999, pp. 61-70.
Baumgartner, P.: An ordered theory resolution calculus, in A. Voronkov (ed.), Proceedings of the 1st International Conference on Logic Programming and Automated Reasoning, Lecture Notes in Artificial Intelligence 624, Springer-Verlag, 1992, pp. 119-130.
Baumgartner, P.: Linear and unit-resulting refutations for Horn theories, J. Automated Reasoning 16(3) (1996), 241-319.
Baumgartner, P.: Personal communication, 2001.
Baumgartner, P., Furbach, U. and Petermann, U.: A unified approach to theory reasoning, Research Report 15-92, Universität Koblenz-Landau, Koblenz, Germany. Fachberichte Informatik, 1992.
Beckert, B. and Pape, C.: Incremental theory reasoning methods for semantic tableaux, in P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proceedings of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, TABLEAUX’92 Palermo (Italy), Lecture Notes in Comput. Sci. 1071, Springer-Verlag, 1996, pp. 93-109.
Bibel, W.: Automated Theorem Proving, Friedr. Vieweg & Sohn, Braunschweig, Germany, 1982.
Bürckert, H.-J.: A resolution principle for constraint logics, Artificial Intelligence 66 (1994), 235-271.
Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn, Graduate Texts in Computer Science, Springer-Verlag, Berlin, 1996.
Hodges, W.: Logical features of Horn clauses, in D. Gabbay, C. Hogger and J. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Oxford University Press, 1993a, pp. 449-503.
Hodges, W.: Model Theory, Enclyclopedia of Mathematics and Its Applications 42, Cambridge University Press, 1993b.
Jaffar, J. and Maher, M.: Constraint logic programming: A survey, J. Logic Programming 19/20 (1994), 503-581.
Käufl, T. and Zabel, N.: Cooperation of decision procedures in a tableau-based theorem prover, Revue d’Intelligence Artificielle 4(3) (1990), 99-126.
Lassez, J.-L. and McAloon, K.: A constraint sequent calculus, in Proceedings of the 5th IEEE Symposium on Logic in Computer Science, LICS’90, IEEE Computer Society Press, Philadelphia, 1990, pp. 52-61.
Malitz, J.: Universal classes in infinitary languages, Duke Math. J. 36(3) (1969), 621-630.
Nelson, G. and Oppen, D. C.: Simplification by cooperating decision procedures, ACM Trans. on Programming Languages and Systems 1(2) (1979), 245-257.
Oppen, D. C.: Complexity, convexity and combinations of theories, Theoret. Comput. Sci. 12 (1980), 291-302.
Petermann, U.: Connection calculus theorem proving with multiple built-in theories, J. Symbolic Comput. 29(2) (2000), 373-392.
Plotkin, G.: Building in equational theories, in B. Meltzer and D. Michie (eds.), Machine Intelligence, Vol. 7, Edinburgh University Press, Edinburgh, Scotland, 1972, pp. 73-90.
Socher-Ambrosius, R. and Johann, P.: Deduction Systems, Springer-Verlag, Berlin, 1997.
Stickel, M. E.: Automated deduction by theory resolution, J. Automated Reasoning 1(4) (1985), 333-355.
Tinelli, C.: Cooperation of background reasoners in theory reasoning by residue sharing, in International Workshop on First Order Theorem Proving, FTP’2000, St Andrews (Scotland), 2000.
Tinelli, C. and Harandi, M. T.: A new correctness proof of the Nelson-Oppen combination procedure, in F. Baader and K. Schulz (eds.), Frontiers of Combining Systems: Proceedings of the 1st International Workshop (Munich, Germany), Kluwer Academic Publishers, 1996, pp. 103-120.
Tinelli, C. and Harandi, M. T.: Constraint logic programming over unions of constraint theories, J. Functional and Logic Programming 6 (1998).
Tinelli, C. and Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures, Theoret. Comput. Sci. 290(1) (2002), 291-253.
Zarba, C.: A tableau calculus for combining non-disjoint theories, in C. Fermüller and U. Egly (eds.), Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, TABLEAUX 2002 (Copenhagen, Denmark), Lecture Notes in Comput. Sci. 2381, Springer, 2002, pp. 315-329.