Finite Horn Monoids and Near-Semirings

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 knowledge representation and database theory by introducing and studying the sequential composition of propositional Horn theories. Specifically, we show that the notion of composition gives rise to a family of monoids and near-semirings, which we will call {\em Horn monoids} and {\em Horn near-semirings} in this paper. Particularly, we show that the combination of sequential composition and union yields the structure of a finite idempotent near-semiring. We also show that the restricted class of proper propositional Krom-Horn theories, which only contain rules with exactly one body atom, yields a finite idempotent semiring. On the semantic side, we show that the immediate consequence or van Emden-Kowalski operator of a theory can be represented via composition, which allows us to compute its least model semantics without any explicit reference to operators. This bridges the conceptual gap between the syntax and semantics of a propositional Horn theory in a mathematically satisfactory way. Moreover, it gives rise to an algebraic meta-calculus for propositional Horn theories. In a broader sense, this paper is a first 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 theories, most importantly to first-, and higher-order logic programs, and non-monotonic logic programs under the stable model or answer set semantics and extensions thereof.