Inductive definitions over finite structures

Abstract

Inductive definitions have played a central role in the foundations of mathematics for over a century. They were used in the 1970s as the backbone of one major generalization of Recursive Function Theory (Moschovakis, 1974; Aczel, 1977). In recent years the relevance of inductive definitions (in particular over finite structures) to Database Theory, to Descriptive Computational Complexity, and to Logics of programs has been recognized. A seminal paper on inductive definitions in Database Theory is Chandra and Hare1 (1982), where they define a hierarchy of queries over finite structures, within which minor steps (successor ordinals) correspond to first-order quantifier alternations, and major steps (limit ordinals) correspond to uses of lixpoints. They left open the question of whether the hierarchy remains strict above the first major step (level w). This problem was answered in the negative by Immerman (1986). Since the collection of first-order lixpoint queries over finite structures is closed under composition and first-order operations other than negation (Moschovakis, 1974), the main component of Immerman’s solution was