The functional dimension of inductive definitions

Abstract

If R( x,y) is an inductively defined relation, we say that x and y are dependent, if there exists a function ⨍ such that ∀ x, y R( x, y) ⇔ R( x, f( x)). The functional dimension of an inductive query is the number of independent recursion variables, and is an important logical property of R, related to its efficient evaluation. In this paper, we compare the expressive power of various languages: the language of iterated inductive definitions, the stratified logic programs and second-order logic. The functional dimension is an important parameter in this comparison. We prove that on the class of finite-valued graphs with a successor relation, the query SP, where SP( a, b, i) if the shortest path between a and b is of cost, i, is not monadic second-order definable, i.e. it cannot be defined by a second-order formula with existential and universal second-order quantifiers on monadic relations. We then conclude that it can neither be defined by iterated inductions with one recursion variable, nor by stratified logic programs with one recursion variable. We prove that in the general case it is undecidable to determine if an inductive query is of functional dimension 1, although it can be decidable in certain cases.