A Logic for Non-Monotone Inductive Definitions

TL;DR

This paper introduces a new logic, ID-logic, to formalize various forms of non-monotone induction, including iterated induction, and explores its modularity properties, with implications for logic programming.

Contribution

It formalizes the principle of iterated induction within a new logic, extending the formalization of induction principles in logical frameworks.

Findings

ID-logic captures well-founded, monotone, and iterated induction.

Modularity conditions allow decomposing complex definitions into smaller parts.

Results apply to logic programming and abductive reasoning under well-founded semantics.

Abstract

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 $\D_1 \land ... \land \D_n$ with disjoint sets of defined predicates. The difficulty of the result comes from the fact that predicates $P_i$ and $P_j$ defined in $\D_i$ and $\D_j$, 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.