COMPUTATIONAL EQUIVALENCE

Abstract

The notion of effectively calculable function has been approached and studied from several directions. General recursiveness, λ-defineability, Post Cannonical Systems, Combinatory definability, and Turing Computability are the best known notions and they have all been shown formally to be equivalent one to the other (in the sense that they all define the same class of effective computations). These formal proofs taken together with “Church's Thesis” have lead to the general acceptance of all these notions as indeed capturing the idea of “effective computability”. Characteristic of all these approaches is a mechanistic procedure which acts with a finite amount of information to compute the desired functions. The modern stored program electronic computer is a mechanization of a “reasonable approximation” to a mechanistic “clerk” who carries out the procedures described in any of the systems mentioned above. In that sense, each of the representations of a function in any of those systems has an analogue in the computer program, and the power and the limitations embodied in each of those systems carries over to the corresponding computer programs. The subject of this research is a study of the properties of a particular notion of algorithm which is much closer in form and appearance to the commonly accepted notions of computer programs than any of the formalisms mentioned above. These algorithms are called “programs” to reflect this bias, and we shall be concerned particularly with relations of computational equivalence between programs. Deciding computational equivalence is the core of any attempts at mechanical simplification or verification of computer programs and in this study we will assess the relative difficulty involved in trying to decide several types of “equivalence relation”.