Complexity of algebraic specifications

Abstract

In the field of algebraic specification, the semantics of an equationally specified datatype is given by the initial algebra of the specifications. We show in this paper that in general the theory of the initial algebra of a given set of equations is II2◦-complete. The impossibility of complete finite axiomatization of equations as well as inequations true in the initial algebra is therefore established. We, further, establish that the decision problem is, in general, II1◦-complete if the equational theory is decidable. We extend the investigation to a certain semantics of parameterized specifications (the so-called free extension functor semantics) that has been proposed in the literature. We present some results characterizing the recursion-theoretic properties of (a) the theory of the free extension algebras of parameter algebras relative to the the theory of the parameter algebras, and (b) the theory of the models of a parameterized specification relative to the theory axiomatized by the specification itself.