2005/01/13 by Marc Denecker, Denecker, Marc, Eugenia Ternovska +1 · 1 citation
Computer Science · #Advanced Algebra and Logic #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Semantic Web and Ontologies #cs.AI #cs.LO
paper · pdf · doi:10.48550/arxiv.cs/0501025
50 pages, TOCL submission
arxiv created 2005/01/13 · openalex publication_date 2005/01/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Well-known principles of induction include monotone induction and different sorts of non-monotone induction such as inflationary induction, induction over well-founded sets and iterated induction. In this work, we define a logic formalizing induction over well-founded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for Non-Monotone Inductive Definitions (ID-logic). The semantics of the logic is strongly influenced by the well-founded semantics of logic programming. Our main result concerns the modularity properties of inductive definitions in ID-logic. Specifically, we formulate conditions under which a simultaneous definition \D of several relations is logically equivalent to a conjunction of smaller definitions \D1 ∧ ... ∧ \Dn with disjoint sets of defined predicates. The difficulty of the result comes from the fact that predicates Pi and Pj defined in \Di and \Dj, respectively, may be mutually connected by simultaneous induction. Since logic programming and abductive logic programming under well-founded semantics are proper fragments of our logic, our modularity results are applicable there as well.