Theorem Proving Modulo

Journal of Automated Reasoning

Volumn 31, Issue 1, 2003, Pages 33-72

  Dowek, Gilles ^a   Hardin, Thérèse ^b   Kirchner, Claude ^c  

^a ÉCOLE POLYTECHNIQUE   (France)

^b INRIA ROCQUENCOURT   (France)

^c LORIA   (France)

Author keywords

Automated theorem proving; Cut elimination; Deduction modulo; Higher order logic; Narrowing; Resolution; Rewriting; Sequent calculus modulo; Skolemization

Indexed keywords

COMPUTATIONAL COMPLEXITY; COMPUTER SIMULATION; FORMAL LOGIC; PROBLEM SOLVING;

DEDUCTION MODULO; SEQUENCE CALCULUS MODULO;

THEOREM PROVING;

EID: 0344927106     PISSN: 01687433     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1027357912519     Document Type: Article

References (52)

1. Abadi, M., Cardelli, L., Curien, P.-L. and Levy, J.-J.: Explicit substitutions, J. Funct. Programming 1(4) (1991), 375-416.
2. Andrews, P. B.: Resolution in type theory, J. Symbolic Logic 36 (1971), 414-432.
3. Baader, F. and Nipkow, T.: Term Rewriting and All That, Cambridge University Press, 1998.
4. Bachmair, L.: Proof methods for equational theories, Ph.D. thesis, University of Illinois, Urbana-Champaign, IL, 1987, Revised version, August 1988.
5. Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W.: Basic paramodulation, Inform. and Comput. 121(2) (1995), 172-192.
6. Barendregt, H. and Barendsen, E.: Autartik computations in formal proofs, J. Automated Reasoning 28(3) (2002), 321-336.
7. Bürckert, H.-J.: A resolution principle for clauses with constraints, in M. E. Stickel (ed.), Proceedings 10th International Conference on Automated Deduction, Kaiserslautern (Germany), Lecture Notes in Comput. Sci. 449, Springer-Verlag, New York, 1990, pp. 178-192.
8. Bürckert, H.-J.: A Resolution Principle for a Logic with Restricted Quantifiers, Lecture Notes in Artificial Intelligence 568, Springer-Verlag, 1991.
9. Curien, P.-L., Hardin, T. and Levy, J.-J.: Confluence properties of weak and strong calculi of explicit substitutions, J. ACM 43(2) (1996), 362-397.
10. Degtyarev, A. and Voronkov, A.: Equality reasoning in sequent-based calculi, in A. Robinson and A. Voronkov (eds.), Handbook of Automated Reasoning, Vol. 1, Elsevier, 2001, Chapt. 10, pp. 611-706.
11. Dershowitz, N. and Jouannaud, J.-P: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Elsevier (North-Holland), 1990, Chapt. 6, pp. 244-320.
12. Dowek, G.: Lambda-calculus, combinators and the comprehension scheme, in M. Dezani-Ciancaglini and G. Plotkin (eds.), Typed Lambda Calculi and Applications, Lecture Notes in Comput. Sci. 902, 1995, pp. 154-170.
13. Dowek, G.: Proof normalization for a first-order formulation of higher-order logic, in E. Gunter and A. Felty (eds.), Theorem Proving in Higher Order Logics, Lecture Notes in Comput. Sci. 1275, 1997, pp. 105-119.
14. Dowek, G.: La Part du Calcul, Université de Paris 7, Mémoire d'habilitation, 1999.
15. Dowek, G.: Axioms vs. rewrite rules: From completeness to cut elimination, in H. Kirchner and C. Ringeissen (eds.), Frontiers of Combining Systems, Lecture Notes in Artificial Intelligence 1794, Springer-Verlag, 2000, pp. 62-72.
16. Dowek, G., Hardin, T. and Kirchner, C.: Theorem proving modulo, Rapport de Recherche 3400, Institut National de Recherche en Informatique et en Automatique, 1998. http://www.inria.fr/rrrt/rr-3400.html.
17. Dowek, G., Hardin, T. and Kirchner, C.: Higher-order unification via explicit substitutions, Inform. and Comput. 157 (2000), 183-235.
18. Dowek, G., Hardin, T. and Kirchner, C.: HOL-λσ, an intentional first-order expression of higher-order logic, Math. Structures Comput. Sci. 11(1) (2001), 21-45.
19. Dowek, G., Hardin, T. and Kirchner, C.: Theorem proving modulo, revised version, Rapport de Recherche 4861, Institut National de Recherche en Informatique et en Automatique, 2003. http://www.inria.fr/rrrt/rr-4861.html.
20. Dowek, G. and Werner, B.: Proof normalization modulo, in Types for Proofs and Programs, Lecture Notes in Comput. Sci. 1657, 1999, pp. 62-77.
21. Gallier, J. H.: Logic for Computer Science: Foundations of Automatic Theorem Proving, Computer Science and Technology Series 5, Harper & Row, New York, 1986.
22. Gallier, J., Raatz, S. and Snyder, W.: Rigid E-unification and its applications to equational matings, in H. Aït-Kaci and M. Nivat (eds.), Resolution of Equations in Algebraic Structures, Vol. 1, Academic Press, New York, 1989, pp. 151-216.
23. Girard, J.-Y., Lafont, Y. and Taylor, P.: Proofs and Types, Cambridge Tracts Theoret. Comput. Sci. 7, Cambridge University Press, 1989.
24. Huet, G.: Constrained resolution: A complete method for type theory, Ph.D. thesis, Case Western Reserve University, 1972.
25. Huet, G.: A mechanization of type theory, in Proceeding of the Third International Joint Conference on Artificial Intelligence, 1973, pp. 139-146.
26. Huet, G.: A unification algorithm for typed lambda calculus, Theoret. Comput. Sci. 1(1) (1975), 27-57.
27. Huet, G.: Résolution d'equations dans les langages d'ordre 1, 2...., ω, Thèse de Doctorat d'Etat, Université de Paris 7, France, 1976.
28. Hullot, J.-M,: Canonical forms and unification, in Proceedings 5th International Conference on Automated Deduction, Les Arcs (France), 1980, pp. 318-334.
29. Jaffar, J. and Lassez, J.-L.: Constraint logic programming, in Proceedings of the 14th Annual ACM Symposium on Principles Of Programming Languages, Munich (Germany), 1987, pp. 111-119.
30. Jouannaud, J.-P. and Kirchner, H.: Completion of a set of rules modulo a set of equations, SIAM J. Comput. 15(4) (1986), 1155-1194. Preliminary version in Proceedings 11th ACM Symposium on Principles of Programming Languages, Salt Lake City (USA), 1984.
31. Jouannaud, J.-P. and Kirchner, C.: Solving equations in abstract algebras: A rule-based survey of unification, in J.-L. Lassez and G. Plotkin (eds.), Computational Logic. Essays in honor of Alan Robinson, MIT Press, Cambridge, MA, 1991, Chapt. 8, pp. 257-321.
32. Kirchner, C.: Méthodes et outils de conception systématique d'algorithmes d'unification dans les théories équationnelles, Thèse de Doctorat d'Etat, Université Henri Poincaré - Nancy 1, 1985.
33. Kirchner, H.: Orderings in automated theorem proving, in F. Huffman (ed.), Mathematical Aspects of Artificial Intelligence, Proc. Symposia Appl. Math. 55, Amer. Math. Soc., 1998, pp. 55-95.
34. Kirchner, C., Kirchner, H. and Rusinowitch, M.: Deduction with symbolic constraints, Revue d'Intelligence Artificielle 4(3) (1990), 9-52. Special issue on Automatic Deduction.
35. Kirchner, C. and Ringeissen, C.: Higher-order equational unification via explicit substitutions, in Proceedings 6th International Joint Conference ALP'97-HOA'97, Southampton (UK), Lecture Notes in Comput. Sci. 1298, Springer-Verlag, 1997, pp. 61-75.
36. Klop, J., van Oostrom, V. and van Raamsdonk, F.: Combinatory reduction systems: Introduction and survey, Theoret. Comput. Sci. 121 (1993), 279-308.
37. Knuth, D. E. and Bendix, P. B.: Simple word problems in universal algebras, in J. Leech (ed.), Computational Problems in Abstract Algebra, Pergamon Press, Oxford, 1970, pp. 263-297.
38. Kolata, G.: With major math proof, brute computers show flash of reasoning power, New York Times, Tuesday, December 10, 1996.
39. Lee, S.-J. and Plaisted, D.: Use of replace rules in theorem proving, Methods of Logic in Computer Science 1(2) (1994), 217-240.
40. Marché, C.: Normalised rewriting and normalised completion, in S. Abramsky (ed.), Proceedings 9th IEEE Symposium on Logic in Computer Science, Paris, 1994, pp. 394-403.
41. McCune, W.: Solution of the Robbins problem, J. Automated Reasoning 19(3) (1997), 263-276.
42. Nieuwenhuis, R. and Rubio, A.: AC-superposition with constraints: No AC-unifiers needed, in A. Bundy (ed.), Proceedings 12th International Conference on Automated Deduction, Nancy (France), Lecture Notes in Artificial Intelligence 814, Springer-Verlag, 1994, pp. 545-559.
43. Nutt, W., Réty, P. and Smolka, G.: Basic narrowing revisited, J. Symbolic Comput. 7(3-4) (1989), 295-318. Special issue on unification. Part one.
44. Peterson, G.: A technique for establishing completeness results in theorem proving with equality, SIAM J. Comput. 12(1) (1983), 82-100.
45. Peterson, G. and Stickel, M. E.: Complete sets of reductions for some equational theories, J. ACM 28 (1981), 233-264.
46. Plaisted, D. A. and Potter, R. C.: Term rewriting: Some experimental results, J. Symbolic Comput. 11(1-2) (1991), 149-180.
47. Plotkin, G.: Building-in equational theories, Machine Intelligence 7 (1972), 73-90.
48. Stickel, M.: Automated deduction by theory resolution, J. Automated Reasoning 1(4) (1985), 285-289.
49. Stuber, J.: A model-based completeness proof of extended narrowing and resolution, in 1st Int. Joint Conf. on Automated Reasoning (IJCAR-2001), Siena, Italy, Lecture Notes in Comput. Sci. 2083, Springer-Verlag, 2001, pp. 195-210.
50. Terese (M. Bezem, J. W. Klop and R. de Vrijer, eds.): Term Rewriting Systems, Cambridge University Press, 2002.
51. Vigneron, L.: Positive deduction modulo regular theories, in H. K. Büning (ed.), Annual Conference of the European Association for Computer Science Logic, Lecture Notes in Comput. Sci. 1092, Springer-Verlag, 1995, pp. 468-485. Selected papers.
52. Viry, P.: Equational rules for rewriting logic, Theoret. Comput. Sci. 285(2) (2002), 487-517.