Final algebras, cosemicomputable algebras and degrees of unsolvability

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 of 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, we 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.