Terminating Turing Machine Computations and the Complexity and/or decidability of Correspondence Problems, Grammars, and Program Schemes

Reviewer: Keith Harrow

This is a unifying paper which brings together ideas and techniques from a number of fields. The author includes material from the study of logical theories, complexity theory, classical automata and formal language theory, and program schemes. The paper begins by noting the difference between the complexity of computing a predicate and the complexity of the language which is the domain of the predicate. The author gives some new definitions for the decision problem of a language with respect to a domain, as well as for the complexity of this decision problem. These definitions enable the complexity difference noted above to be studied more carefully than in the past. One of the main ideas of the paper is the use of a technique from logic (the idea of two sets being recursively inseparable) to show that a decision problem is of nonrecursive complexity (i.e., it has no recursive time bound). More precisely, a decision problem is decidable if and only if certain sets are r.e. separable; the decision problem is recursively time-bounded if and only if the sets are recursively separable. Three natural classes of decision problems are discussed. Each of these is shown to have decision problems which are of nonrecursive complexity. One of these classes is related to the well-known Post Correspondence Problem and linear context-free grammars; another deals with arbitrary context-free grammars; and the third is concerned with program schemes. A series of undecidability and related results are derived for these and similar structures. An earlier paper by the author [1] used these ideas to show a variety of results for context-free grammars. The present paper extends many of the earlier results to more general situations. Although a number of the results in the present paper were previously known, the paper does present many new and interesting conclusions relating to correspondence problems, ambiguous grammars, and equivalence problems for program schemes. In addition, the paper presents a unified framework for viewing many of the results that are not new. The paper is well written and relatively self-contained, although there are numerous references to earlier work in a number of fields. As noted above, it uses many of the ideas and methods from one area of theory to study another area. This cross-fertilization will probably prove to be as useful as the direct results presented here.