Algebraic answer set programming

Non-monotonic reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via answer set programming. Describing complex objects as the composition of elementary ones is a common strategy in computer science and science in general. This paper contributes to the foundations of answer set programming and artificial intelligence by introducing and studying the sequential composition of answer set programs. Specifically, we show that the notion of composition gives rise to a family of finite monoids and seminearrings, baptized ASP monoids and ASP seminearrings in this paper. Particularly, we show that the combination of composition and union yields the structure of a finite idempotent seminearring. We also show that the restricted class of proper Krom-Horn programs, which only contain rules with exactly one body atom, yields a finite idempotent semiring. On the semantic side, we show that the van Emden-Kowalski immediate consequence operator of a program can be represented via composition, which allows us to compute the least model semantics of Horn programs without any explicit reference to operators. As a result, we characterize answer sets algebraically, which bridges the conceptual gap between the syntax and semantics of an answer set program in a mathematically satisfactory way, and which provides an algebraic characterization of strong and uniform equivalence. Moreover, it gives rise to an algebraic meta-calculus for answer set programs. In a broader sense, this paper is a further step towards an algebra of rule-based logical theories and in the future we plan to adapt and generalize the methods of this paper to wider classes of formalisms, most importantly to first-order and disjunctive answer set programs and extensions thereof.