Sequential composition of answer set programs
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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. Recently, the author introduced the sequential composition of Horn logic programs for syntactic program composition and decomposition in the context of logic-based analogical reasoning and learning. This paper contributes to the foundations of answer set programming and artificial intelligence by generalizing the construction of composition from Horn to (propositional) answer set programs containing negation as failure. This task turns out to be non-trivial due to the intricate algebraic properties of composing negation as failure occurring in rule bodies. Specifically, we show that the notion of composition gives rise to a family of finite magmas and algebras, baptized ASP magmas and ASP algebras in this paper. 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 can 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. In a broader sense, this paper is a further step towards an algebra of rule-based logical theories with applications to logic-based analogical reasoning and learning, and in the future we plan to adapt and generalize the methods of this paper to wider classes of formalisms, most importantly to higher-order and disjunctive logic programs and extensions thereof.
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