Final algebras, cosemicomputable algebras, and degrees of unsolvability

Abstract

This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many-sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature Σ and a set V of visible sorts, for every Σ-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension Σ′ of Σ (without new sorts) and a finite set E of Σ′-equations such that A is isomorphic to a reduct of the final (Σ′, E)-algebra relative to V. This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial (Σ′, E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature Σ, there are either countably many Σ-congruences on the free Σ-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions which separate these two cases. We introduce the notion of the Turing degree of a minimal algebra. Using the results above prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of Σ-equations such the initial (Σ, E)-algebra has degree d. There is a two-sorted signature Σ0 and a single visible sort such that for every r.e. degree d there is a finite set E of Σ-equations such that the initial (Σ, E, V)-algebra is computable and the final (Σ, E, V)-algebra is cosemicomputable and has degree d.KeywordsFunction SymbolRecursive FunctionCritical PairConstant SymbolAbstract Data TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.