INTRODUCTION The problem discussed in this paper, namely that of finding a function to satisfy a given argument-value table, is by no means new to computing science, or to mathematics. Thus, for example, the problem of fitting a curve to a set of points is a part of numerical analysis. However, I am concerned with finding a function over a non-metric space, and so my work is closer to that of Feldman et al. (1969) in what they call, 'grammatical inference' or to the automaton-synthesizing programs described by Fogel, Owens and Walsh (1966). There have been some applications of learning devices. Perhaps the best known is Samuel's checkers program (Samuel 1967), but Murray and Elcock (1968) have a system for describing generalized board states in Go-Moku that employs a much richer language to describe the concepts learnt.

We will define the notions of abstract theorem-proving graph, abstract theorem-proving problem g and search strategy E for g. These concepts generalize the usual tree (or graph) searching problem and admit Hart, Nilsson and Raphael (1968) and Pohl (1969) theories of heuristic search. In particular the admissibility and optimality theorems of Hart, Nilsson and Raphael generalize for the classes 0 and 0" of diagonal search strategies for abstract theorem-proving problems. In addition the subclass au of 0 is shown to be optimal for 2. Implementation of diagonal search is treated in some detail for theorem-proving by resolution rules (Robinson 1965). SEARCH STRATEGIES, COMPLETENESS AND EFFICIENCY Completeness and efficiency of proof procedures can be studied only in the context of search strategies. A system T of inference rules and axioms can be complete or incomplete for a given class of intended interpretations. Similarly a search strategy E for T may or may not be complete for ...

Engeler, E., Proof theory and the accuracy of computations (to appear).

The equivalence problem for program schemes, or for programs, is reduced to the proving of a theorem in second-order logic. This work extends Manna's first-order logic reductions. Some examples of the technique are given together with a suggested method for obtaining proofs in special cases by firstorder methods. INTRODUCTION Several workers in recent years have considered using techniques and ideas of various mathematical theories of computation for proving interesting results about computer programs. This paper is concerned with two of these approaches.

COGNITIVE PROCESSES: METHODS AND MODELS 17 Steps towards a model of word selection.

Konig's infinity lemma 61,90, 161 spanning tree of, see tree Kowalski 176-8, 181 Graph Traverser program 450,456-7 Kripke 493-5, 501 Greanias 376, 381 Green, B. F. jnr.

In reviewing the first three Machine Intelligence volumes, Nature (14 December 1968) wrote: 'The enterprise of Edinburgh University in fostering many of the developments reported has been amply rewarded'. It also noted that several of the contributions were speculative, and seemed'to grope in the darkness for some signs of a breakthrough towards a true Machine Intelligence'. A Vice-Chancellor may be forgiven these days for seizing every chance to stress the obvious. The practice of research is proper and vital to a university. Such pursuits -- speculative or experimental -- often lead, penultimately, to a blind alley; and such negative results are themselves positive contributions to knowledge.

Robinson, J.A. (1965) A machine-oriented logic based on the resolution principle.

The necessary logical apparatus can be kept remarkably simple. We use a formulation of predicate logic containing individual constants and function symbols. To simplify the description of the method, it is convenient to restrict the formulae F to which the method is applicable. Firstly, it is supposed that F is closed and in prenex normal form. Secondly, it is supposed that all existential quantifiers are eliminated. To see how this can The main part of this paper was also presented in lectures at the University of Stockholm and the Technische Hochschule of Hanover in the spring of 1967.

F(n,$) as introduced here corresponds to A.N.Prior's (1957, 1968) expression Fir.