Decision problems in computational models

Abstract

Some of the assertions about programs which we might be interested in proving are concerned with correctness, equivalence, accessibility of subroutines and guarantees of termination. We should like to develop techniques for determining such properties efficiently and intelligently wherever possible. Though theory tells us that for a realistic programming language almost any interesting property of the behaviour is effectively undecidable, this situation may not be intolerable in practice. An unsolvability result just gives us warning that we may not be able to solve all of the problems we are presented with, and that some of the ones we can solve will be very hard. In such circumstances it is very reasonable to try and determine necessary or sufficient conditions on programs for our techniques to be assured of success; however, in this paper we shall discuss a more qualitative, indirect, approach. We consider a range of more or less simplified computer models, chosen judiciously to exemplify some particular feature or features of computation. A demonstration of unsolvability in such a model reveals more accurately those sources which can contribute to unsolvability in a more complicated structure. On the other hand a decision procedure may illustrate a technique of practical use. It is our thesis that this kind of strategy of exploration can and will yield insight and practical advances in the theory of computation. Provided that the model retains some practical relevance, the dividends are the greater the nearer the decision problem lies to the frontier between solvability and unsolvability.