Both the analytical philosopher and the designer of intelligent software are doing what might be called'mental engineering': constructing precise, formal models of some aspects of intelligent thought or behaviour. There is a wide gap between them, but it is narrowing.

The solution to this simple problem would then guide the solution of the original problem. Minsky (1961) has discussed complex planning of this'homomorphic model' type, and has stressed the potential reduction in total search effort to be won. In the same paper he has also considered the use of semantic models' as a form of complex planning in a mathematical context. The successful geometry theorem-proving program of Gelernter (1959), which used a diagram' to test the validity of propositions, is a wellknown example of this form of planning. Recently Sandewall (1969) has defined a Planning Problem Solver (P P This is an attempt to explore in detail complex planning of the homomorphic model' type as applied to the

Rather, the program exists in a state of continual revision and extension, and that part of it which is running and yielding results at the moment is intended to form the nucleus of a much larger and more extensive automatic deduction system in the future. The program is being developed with a number of different questions and applications in mind. First of all, some of the present generation of deduction programs are already capable of proving the sort of theorem of an elementary mathematical theory that appears in the textbooks either as a basic theorem or standard exercise (sometimes as a'starred' exercise), and might reasonably be classified as'somewhat' tricky. We have in mind the theorems of algebra and number theory obtained by programs discussed by Wos, Robinson and Carson (1965), Wos et al. (1967), and Luckham (1968). In addition, it is reported by Guard et al. (1969) that an open problem in modular lattice theory was solved with the aid of an on-line program (and it is interesting to note that, with reference to the initial set of axioms and hypotheses, this seems not to be the most difficult theorem that has been proved by a program so far). We are thus motivated to ask if, by adding a flexible interactive facility to a good deduction program, it is possible to construct a system that would be useful in investigating some fairly basic mathematics.

A formal description is given of GT 4, a revised and extended version of the Graph Traverser. Methods are described whereby GT4 can improve its performance at run time (a) by automatic optimization of parameters used by the evaluation function and (b) by dynamic re-ordering of operators. Neither method depends upon there being any successful searches in the program's past experience of a given problem. The essential feasibility of both approaches has been validated in experimental tests using sliding block puzzles. Two planned extensions, 'local smoothing' and'regionalization' are described. INTRODUCTION The Graph Traverser (Doran and Michie 1966), and subsequent work based on it, represents an attempt to adapt game-playing methods, particularly those of Samuel (1959), to automatic problem-solving. The design objective is not the simulation of human problem-solving as a study in psychology, but rather to provide an efficient general-purpose search procedure appropriate to non-numerical problem domains. There is a parallel with the development of direct search techniques for numerical function minimization, for example pattern search (Hooke and Jeeves 1961), simplex (Spendley, Hext and Himsworth 1962, Nelder and Mead 1965).

In particular its task domain is the analysis of mass spectra, chemical data gathered routinely from a relatively new analytical instrument, the mass spectrometer. This collaboration of chemists and computer scientists has produced what appears to be an interesting program from the viewpoint of artificial intelligence and a useful tool from the viewpoint of chemistry. For this discussion it is sufficient to say that a mass spectrometer is an instrument into which is put a minute sample of some chemical compound and out of which comes data usually represented as a bar graph. This is what is referred to here as the mass spectrum. The x-points of the bar graph represent the masses of ions produced and the y-points represent the relative abundances of ions of these masses. The first, preliminary inference (or planning), obtains clues from the data as to which classes of chemical compounds are suggested or forbidden by the data.

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.

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.