A logic of nonmonotone inductive definitions

ACM Transactions on Computational Logic

Volumn 9, Issue 2, 2008, Pages

  Denecker, Marc ^a   Ternovska, Eugenia ^b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Classical logic; Inductive definitions; Logic programming

Indexed keywords

DATABASE SYSTEMS; ITERATIVE METHODS; MATHEMATICAL MODELS; SEMANTICS;

CLASSICAL LOGIC; INDUCTIVE DEFINITIONS; MONOTONE INDUCTION; NONMONOTONE INDUCTION;

LOGIC PROGRAMMING;

EID: 42149166294     PISSN: 15293785     EISSN: 1557945X     Source Type: Journal    
DOI: 10.1145/1342991.1342998     Document Type: Article

References (56)

1. ACZEL, P. 1977. An introduction to inductive definitions. In Handbook of Mathematical Logic, J. Barwise, Ed. North-Holland Publishing Company, Amsterdam, The Netherlands, 739-782.
2. APT, K., BLAIR, H., AND WALKER, A. 1988. Towards a theory of declarative knowledge. In Proceedings of the Simposium Foundations of Deductive Databases and Logic Programming, J. Minker, Ed. Morgan Kaufmann, San Francisco, CA. 89-148.
3. BAADER, F., CALVANESE, D., MCGUINNESS, D., NARDI, D., AND PATEL-SCHNEIDER, P., EDS. 2002. The Description Logic Handbook. Theory, Implementation and Applications. Cambridge University Press, Cambridge, MA.
4. BRACHMAN, R. J. AND LEVESQUE, H. J. 1982. Competence in Knowledge Representation. In National Conference on Artificial Intelligence (AAAI'82). 189-192.
5. BUCHHOLZ, W., FEFERMAN, S., AND SIEG, W. P. W. 1981. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics, vol. 897. Springer-Verlag, New York.
6. CLARK, K. L. 1978. Negation as failure. In Logic and Databases, H. Gallaire and J. Minker, Eds. Plenum Press, New York, 293-322.
7. COMPTON, K. 1993. A deductive system for existential least fixpoint logic. J. Logic Comput. 3, 2, 197-213.
8. DENECKER, M. 1998. The well-founded semantics is the principle of inductive definition. In Proceedings of the Logics in Artificial Intelligence (JELIA'98), J. Dix, L. Fariñas del Cerro, and U. Furbach, Eds. Lecture Notes in Artificial Intelligence, vol. 1489. Springer-Verlag, New York, 1-16.
9. DENECKER, M. 2000. Extending classical logic with inductive definitions. In Proceedings of the 1st International Conference on Computational Logic (CL'2000), J. Lloyd et al., Ed. Lecture Notes in Artificial Intelligence, vol. 1861. Springer-Verlag, New York, 703-717.
10. DENECKER, M. 2004. Whatś in a model? Epistemological analysis of Logic Programming. In Ninth International Conference on Principles of Knowledge Representation and Reasoning (KR'04). 106-113. URL=http://www.cs.kuleuven.ac.be/cgibin-dtai/publ_info.pl?id=41086.
11. DENECKER, M., BRUYNOOGHE, M., AND MAREK, V. 2001. Logic programming revisited: Logic programs as inductive definitions. ACM Trans. Comput. Logic 2, 4 (Oct.), 623-654.
12. DENECKER, M., MAREK, V., AND TRUSZCZYŃSKI, M. 2000. Approximating operators, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In Logic-based Artificial Intelligence, J. Minker, Ed. Kluwer Academic Publishers, Chapter 6, 127-144.
13. DENECKER, M. AND TERNOVSKA, E. 2004a. Inductive Situation Calculus. In Proceedings of the 9th International Conference on Principles of Knowledge Representation and Reasoning (KR'04). 545-553.
14. DENECKER, M. AND TERNOVSKA, E. 2004b. A logic of non-monotone inductive definitions and its modularity properties. In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'7), V. Lifschitz and I. Niemelä, Eds.
15. DENECKER, M. AND TERNOVSKA, E. 2007. Inductive situation calculus. Artif. Intell.
16. DENECKER, M., THESEIDER DUPRE, D., AND VAN BELLEGHEM, K. 1998. An inductive definition approach to ramifications. Linkaping Electronic Articles in Computer and Information Science 3, 7, 1-43. URL: http://www.ep.liu.se/ea/cis/1998/007/.
17. EBBINGHAUS, H. AND FLUM, J. 1999. Finite Model Theory, 2nd ed. Springer-Verlag, New York.
18. FEFERMAN, S. 1970. Formal theories for transfinite iterations of generalised inductive definitions and some subsystems of analysis. In Intuitionism and Proof theory, A. Kino, J. Myhill, and R. Vesley, Eds. North Holland, Amsterdam, The Netherlands, 303-326.
19. FITTING, M. 1985. A Kripke-Kleene semantics for logic programs. J. Logic Program. 2, 4, 295-312.
20. FITTING, M. 2002. Fixpoint semantics for logic programming - A survey. Theoret. Comput. Sci. 278, 25-51.
21. GELFOND, M. AND LIFSCHITZ, V. 1988. The stable model semantics for logic programming. In Proceedings of the International Joint Conference and Symposium on Logic Programming (JIC-SLP'88). MIT Press, Cambridge, MA, 1070-1080.
22. GRAEDEL, E. AND KREUTZER, S. 2003. Will deflation lead to depletion? on non-monotone fixed point inductions. In IEEE Symposium on Logic in Computer Science (LICS'03). 158-.
23. GUREVICH, Y. AND SHELAH, S. 1986. Fixed-point extensions of first-order logic. Ann. Pure Appl. Logic 32, 265-280.
24. IMMERMAN, N. 1986. Relational queries computable in polynomial time. Info. Cont. 68, 86-104.
25. KAKAS, A. C., VAN NUFFELEN, B., AND DENECKER, M. 2001. A-system : Problem solving through abduction. In Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI'01), B. Nebel, Ed. Vol. 1. Morgan Kaufmann, San Francisco, CA, Inc., 591-596.
26. KREISEL, G. 1963. Generalized inductive definitions. Tech. rep., Section III in the Stanford University report on the Foundations of Analysis.
27. LIFSCHITZ, V. 1994. Circumscription. In Handbook of Logic in AI and Logic Programming. Vol. 3. Oxford University Press, Oxford, UK, 298-352.
28. LIFSCHITZ, V. AND TURNER, H. 1994. Splitting a logic program. In Proceedings of the International Conference on Logic Programming (ICLP'94). 23-37.
29. LIVCHAK, A. 1983. The relational model for process control. Automat. Document. Math. Ling. 4, 27-29.
30. MAREK, V. W. AND TRUSZCZYŃSKI, M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm,: A 25 Years Perspective, K. Apt, V. Marek, M. Truszczyński, and D. Warren, Eds. Springer-Verlag, New York, 375-398.
31. MAREK, W. 1989. Stable theories in autoepistemic logic. Fund. Inf. 12, 2, 243-254.
32. MARIEN, M., WITTOCX, J., AND DENECKER, M. 2006. The IDP framework for declarative problem solving. In Search and Logic: Answer Set Programming and SAT. 19-34. URL= http://www.es .kuleuven.ac.be/cgi-bin-dtai/publ_info.pl?id=42379.
33. MARTIN-LÖF, P. 1971. Hauptsatz for the intuitionistic theory of iterated inductive definitions. In Proceedings of the 2nd Scandinavian Logic Symposium, J. Fenstad, Ed. 179-216.
34. MITCHELL, D. AND TERNOVSKA, E. 2005. A framework for representing and solving np search problems. In Proceedings of AAAI'05. AAAI Press/MIT Press, Cambridge, MA, 430-435.
35. MOORE, R. 1983. Semantical Considerations on non-monotonic logic. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI'83). 272-279.
36. MOSCHOVAKIS, Y. N. 1974a. Elementary Induction on Abstract Structures. North-Holland Publishing Company, Amsterdam-New, York.
37. MOSCHOVAKIS, Y. N. 1974b. On non-monotone inductive definability. Fund. Math. 82, 39-83.
38. NLEMELÄ, I. 1999. Logic programs with stable model semantics as a constraint programming paradigm. Ann. Math. Artif. Intell. 25, 3,4, 241-273.
39. PELOV, N. AND TERNOVSKA, E. 2005. Reducing inductive definitions to prepositional satisfiability. In Proceedings of the International Conference on Logic Programming (ICLP'05). 221-234.
40. POHLERS, W. 1989. Proof Theory: An Introduction. Springer-Verlag.
41. POST, E. 1943. Formal reduction of the general combinatorial decision problem. Amer. J. Math. 65, 197-215.
42. RAO, P., RAMSKRISHNAN, I., SAGONAS, K., SWIFT, T., WARREN, D., AND FREIRE, J. 1997. XSB: A system for efficiently computing well-founded semantics. In Proceedings of the International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'97). Lecture Notes in Computer Science, 1265. Springer-Verlag, New York, 430-440.
43. REITER, R. 1978. On closed world data bases. In Logic and Data Bases, H. Gallaire and J. Minker, Eds. Plenum Press, New York, 55-76.
44. REITER, R. 1980. Equality and domain closure in first-order databases. J. ACM 27, 235-249.
45. SCHLIPF, J. 1995a. Complexity and undecidability results in logic programming. Ann. Math. Artif. Intell. 15, 257-288.
46. SCHLIPF, J. 1995b. The expressive powers of the logic programming semantics. J. Comput. Syst. Sci. 51, 64-86.
47. SPECTOR, C. 1961. Inductively defined sets of natural numbers. In Infinitistic Methods (Proceedings of the 1959 Symposium on Foundation of Mathematis in Warsaw). Pergamon Press, Oxford, UK, 97-102.
48. TARSKI, A. 1955. Lattice-theoretic fixpoint theorem and its applications. Pacif. J. Math. 5, 285-309.
49. TERNOVSKAIA, E. 1998a. Causality via inductive definitions. In Working Notes of "Prospects for a Commonsense Theory of Causation", AAAI Spring Symposium Series, March 23-28.
50. TERNOVSKAIA, E. 1998b. Inductive definability and the situation calculus. In Transaction and Change in Logic Databases. Lecture Notes in Computer Science, vol. 1472. Springer-Verlag, New York.
51. VAN GELDER, A. 1993. The alternating fixpoint of logic programs with negation. J. Comput. Syst. Sci. 47, 1, 185-221.
52. VAN GELDER, A., ROSS, K. A., AND SCHLIPF, J. S. 1991. The well-founded semantics for general logic programs. J. ACM 38, 3, 620-650.
53. VARDI, M. 1982. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on the Theory of Computing. ACM, New York, 137-146.
54. VENNEKENS, J., GILIS, D., AND DENECKER, M. 2005. Splitting an operator: Algebraic modularity results for logics with fixpoints semantics. ACM Trans. Computat. Logic. URL = http://www.cs.kuleuven.ac.be/cgi-bin-dtai/publ Jnfo.pl?id=41545.
55. VERBAETEN, S., DENECKER, M., AND DE SCHREYE, D. 2000. Compositionality of normal open logic programs. J. Logic Program. 41, 3 (Mar.), 151-183.
56. YOU, J.-H., WANG, X., AND YUAN, L. Y. 1999. Compiling defeasible inheritance networks to general logic programs. Artif. Intell. 113, 1-2 (Sept.), 247-268.