Representation results for defeasible logic

Reviewer: Aida Pliuskeviciene

Defeasible Logic is a simple but efficient formalism for nonmonotonic reasoning presented, for example, by D.Nute in [1]. Defeasible Logic considered in the paper follows the presentation of D.Billington in [2]. Defeasible Logic is based on the idea of logic programming without negation as failure. A defeasible theory (a knowledge base in Defeasible Logic, or a defeasible logic program) consists of five different kinds of knowledge: facts, strict rules, defeasible rules, defeaters, and a superiority relation. In the first two sections of the paper Defeasible Logic is introduced, and proof theory of this logic is considered. The main results of the paper are transformations of the original theory to the equivalent theory which has an empty set of superiority relations, and no defeaters nor facts. One main consequence of this results is that, without loss of generality, simpler form of Defeasible Logic can be studied. A defeasible theory, received after transformation process, is called by normal form of the theory. All three transformations (for elimination of facts, superiority relation, and defeaters) are described explicitly and it is proved that the transformations are correct, i.e. transformed theory has the same meaning as the original theory, or, more formally, both theories have the same conclusions. The transformations described in the paper are important for two main reasons: they can be used as a theoretical tool that leads to a deeper understanding of the formalism and they have been used in the development of an efficient implementation of Defeasible Logic. Two key properties of transformations, modularity and incrementality, are considered. Both properties are important for implementations. It is proved that if a transformation is modular then it is correct and incremental, but in general the inverse proposition does not hold. It is proved that some transformation considered are not modular but they are correct. Conditions, under which the transformation to eliminate superiority relation is modular, are formulated. It is proved that neither strict rules nor defeasible rules can be eliminated while maintaining the set of conclusions. It is mentioned that main transformations presented are utilized in the implementation of Defeasible Logic and the implementation relies on a linear time algorithm. The paper is theoretical self-content investigation containing the proofs of propositions and theorems, and helpful examples. References [1] Nute, D. 1994. Defeasible logic. In Handbook of Logic in Artificial Intelligence and Logic Programming, D.Gabbay, C.Hogger, and J.Robinson, Eds. Vol. 3. Oxford University Press, Oxford, 353-395. [2] Billington, D. 1993. Defeasible logic is stable. Journal of Logic and Computation 3, 370-400.