A Logic of Non-monotone Inductive Definitions and Its Modularity Properties

Abstract

Well-known principles of induction include monotone induction and different sorts of non-monotone induction such as inflationary induction, induction over well-ordered sets and iterated induction. In this work, we define a logic formalizing induction over well-ordered 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 (NMID-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 NMID-logic. Specifically, we formulate conditions under which a simultaneous definition Δ of several relations is logically equivalent to a conjunction of smaller definitions Δ₁ Λ... ΛΔ _n with disjoint sets of defined predicates. The difficulty of the result comes from the fact that predicates P _i and P _j defined in Δ _i and Δ _j , 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. As an example of application of our logic and theorems presented in this paper, we describe a temporal formalism, the inductive situation calculus, where causal dependencies are naturally represented as rules of inductive definitions.