Modelling legal reasoning in a mathematical environment through model theoretic semantics

ABSTRACT

We introduce a mathematical model of legal reasoning using an underlying conditional logic semantics, to allow its tractability in some special cases. The main idea is to capture the entailment of legal consequences through a model of 0-1 programming. For such task, first we model legal reasoning with Lehmann's Lexicographic semantics and then we translate it to an instance of weighted MAXSAT problem, in order to compute the logical consequences of legal reasoning. Hence, combinatorial optimization algorithms can be used to yield the legal consequences of defeasible reasoning over legal conditional knowledge bases.

REFERENCES

• 1: A. Aarnio, R. Alexy, and A. Peczenik. The foundation of legal reasoning. Rechtstheorie, 12:133--158, 1981.
• 2: Colin Bell , Anil Nerode , Raymond T. Ng , V. S. Subrahmanian, Mixed integer programming methods for computing nonmonotonic deductive databases, Journal of the ACM (JACM), v.41 n.6, p.1178-1215, Nov. 1994 [doi>10.1145/195613.195637]
• 3: Colin Bell , Anil Nerode , Raymond T. Ng , V. S. Subrahmanian, Implementing deductive databases by mixed integer programming, ACM Transactions on Database Systems (TODS), v.21 n.2, p.238-269, June 1996 [doi>10.1145/232616.232691]
• 4: V. Chandru and J. N. Hooker. Optimization Methods for Logical Inference. Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., 1999.
• 5: Stephen A. Cook, The complexity of theorem-proving procedures, Proceedings of the third annual ACM symposium on Theory of computing, p.151-158, May 03-05, 1971, Shaker Heights, Ohio, United States [doi>10.1145/800157.805047]
• 6: Thomas Eiter , Thomas Lukasiewicz, Default reasoning from conditional knowledge bases: complexity and tractable cases, Artificial Intelligence, v.124 n.2, p.169-241, Dec. 2000 [doi>10.1016/S0004-3702(00)00073-4]
• 7: B. Garcia. New tractable classes for default reasoning from conditional knowledge bases. To appear.
• 8: Moisés Goldszmidt , Judea Pearl, On the consistency of defeasible databases, Artificial Intelligence, v.52 n.2, p.121-149, Dec. 1991 [doi>10.1016/0004-3702(91)90039-M]
• 9: J. Gu, P. W. Purdom, J. Franco, and B. W. Wah. Algorithms for the satisfiability (SAT) problem: a survey. In D. Du, J. Gu, and P. M. Pardalos, editors, Satisfiability Problem: Theory and Applications, volume 35 of DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, pages 19--152. American Mathematical Society, 1997.
• 10: Theodore Hailperin, Boole's logic and probability: a critical exposititon from the standpoint of contemporary algebra, logic, and probability theory, North-Holland Publishing Co., Amsterdam, The Netherlands, 1986
• 11: J. N. Hooker, A quantitative approach to logical inference, Decision Support Systems, v.4 n.1, p.45-69, March 1988 [doi>10.1016/0167-9236(88)90097-8]
• 12: R. G. Jeroslow. Logic-Based Decision Support. Mixed Integer Model Formulation. Elsevier, Amsterdam, 1988.
• 13: V. Kagan, A. Nerode, and V. S. Subrahmanian. Computing definite logic programs by partial instantiation. Annals of Pure and Applied Logic, 67(1--3):161--182, 1994.
• 14: D. Lehmann. Another perspective on default reasoning. Annals of Mathematics and Artificial Intelligence, 15(1):61--82, 1995.
• 15: Judea Pearl, System Z: a natural ordering of defaults with tractable applications to nonmonotonic reasoning, Proceedings of the 3rd conference on Theoretical aspects of reasoning about knowledge, March 04-07, 1990, Pacific Grove, California
• 16: P. Simons. Towards constraint satisfaction through logic programs and the stable model semantics. Research report A47, Helsinki University of Technology, 1997.
• 17: S. Toulmin. The Uses of Argument. Cambridge University Press, 1958.
• 18: H. P. Williams. Fourier-Motzkin elimination extension to integer programming problems. Journal of Combinatorial Theory (A), 21:118--123, 1976.

INDEX TERMS

Keywords:
MAX-SAT, artificial intelligence and law, legal reasoning, lexicographic semantics, nonmonotonic reasoning