All solutions of a system of recursion equations in infinite trees and other contraction theories

Abstract

A system Σ of recursion equations is a finite set of equations ϕ_i = t_i i = 1,…, n, where the expressions ti denote terms built from individual variables, “function constants,” and the “function variables” ϕ₁,…,ϕ_n. An interpretation of such a system assigns a meaning to each function constant and a possible meaning to each individual and function variable. Standard interpretations have imposed an ordering on the class of possible “solutions” of Σ and only the least (or greatest) solution is found. In this paper contraction theories are defined and used as interpretations for Σ. Several known kinds of interpretations are shown to be contraction theories, including the collection of rooted labeled trees. No ordering is imposed on solutions in a contraction theory, but a metric is involved. All solutions of Σ in any contraction theory are described.