IBERAMIA 2006, SBIA 2006: Advances in Artificial Intelligence - IBERAMIA-SBIA 2006 pp 582-591

The Predicate-Minimizing Logic MIN

• Francicleber Martins Ferreira
• Ana Teresa Martins
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4140)

Abstract

The concept of minimization is widely used in several areas of Computer Science. Although this notion is not properly formalized in first-order logic, it is so with the logic MIN(FO) [13] where a minimal predicate P is defined as satisfying a given first-order description φ(P). We propose the MIN logic as a generalization of MIN(FO) since the extent of a minimal predicate P is not necessarily unique in MIN as it is in MIN(FO). We will explore two different possibilities of extending MIN(FO) by creating a new predicate defined as the union, the U-MIN logic, or intersection, the I-MIN logic, of the extent of all minimal P that satisfies φ(P). We will show that U-MIN and I-MIN are interdefinable. Thereafter, U-MIN will be just MIN. Finally, we will prove that simultaneous minimizations does not increase the expressiveness of MIN, and that MIN and second-order logic are equivalent in expressive power.

Keywords

Minimal Model Expressive Power Predicate Symbol Denotational Semantic Nonmonotonic Reasoning
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Davis, M.D., Sigal, R., Weyuker, E.J.: Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science, 2nd edn. Morgan Kaufmann, San Diego (1994)Google Scholar
2. 2.
de Kleer, J., Konolige, K.: Eliminating the Fixed Predicates from a Circumscription. Artificial Intelligence 39, 391–398 (1989)
3. 3.
Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical Logic. Springer, New York (1994)
4. 4.
Grädel, E.: Guarded fixed point logics and the monadic theory of countable trees. Theoretical Computer Science 288(1), 129–152 (2002)
5. 5.
Hrbacek, K., Jech, T.: Introduction to Set Theory, 2nd edn. Marcel Dekker, INC., New York (1984)
6. 6.
Libkin, L.: Elements of Finite Model Theory. Springer, Berlin (2004)
7. 7.
Lifschitz, V.: Circumscription. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming – Nonmonotonic Reasoning and Uncertain Reasoning, vol. 3, pp. 297–352. Clarendon Press, Oxford (1994)Google Scholar
8. 8.
Lloyd, J.W.: Foundations of Logic Programming, 2nd edn. Springer, New York (1987)
9. 9.
McCarthy, J.L.: Circumscription – a form of non-monotonic reasoning. Artif. Intell. 13(1-2), 27–39 (1980)
10. 10.
McCarthy, J.L.: Applications of circumscription to formalizing common-sense knowledge. Artif. Intell. 28(1) (1986)Google Scholar
11. 11.
Schmidt, D.A.: Denotational Semantics: a methodology for language development. Allyn and Bacon, INC, Boston (1986)Google Scholar
12. 12.
Scott, D.: Domains for Denotational Semantics. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 577–613. Springer, Berlin (1982)
13. 13.
van Benthem, J.: Minimal Predicates, Fixed-Points, and Definability. J. Symbolic Logic 70(3), 696–712 (2005)