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.

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.

ABSTRACT This paper shows how a question-answering system can be constructed using first-order logic as its language and a resolution-type theorem-prover as its deductive mechanism. A working computer-program, Q A3, based on these ideas is described. The performance of the program compares favorably with several other general question-answering systems. The type of questionanswering system considered in this paper is ideally one having the following features: 1. A language general enough to describe any reasonable questionanswering subjects and express desired questions and answers. This paper argues the case for formal methods to achieve such a system and presents one particular approach in detail. The name'question-answering system' requires clarification. The system described above might be named an'advice taker' or a'multi-purpose problem-solving system' or'general problem-solving system'. McCarthy (1958) proposed using formal languages and deduction to construct such a system, and suggested allowing the user to give hints or advice on how to answer a question; he referred to the proposed system as an'advice taker'. Research on'multi-purpose' or'general problem-solving' tends to differ from questionanswering as described above by placing more emphasis on solving deeper, more difficult problems and less emphasis on user interaction, formality, and efficient retrieval of relevant facts from a large data base. The situation is further confused by the use of'question-answering' to refer sometimes to natural language systems, sometimes to information retrieval systems having little deductive ability, and sometimes to systems with deductive ability limited to the propositional calculus. It is important to emphasize the distinction between general versus specialpurpose question-answering. If the class of questions asked of a system is small, completely specified in advance, and concerned with a particular subject area, such as the question-answering system of Green, Wolf, Chomsky, and Laughery (1963) concerned with baseball, or that of Lindsay (1963) concerned with family relations, then we shall call such a system'specialpurpose'. Frequently the goal in designing a special-purpose system is to achieve good performance, measured in terms of running speed and memory utilization. In this case the best approach may be first to construct a special data base or memory that is optimized for that subject area and question class, and then to write special question-answering subroutines that are optimized for the particular data base and question class.

R.M.BURSTALL 373 23 Assertions: programs written without specifying unnecessary order.

It is a natural generalization of the various versions and extensions of the resolution principle, each of which it includes as special cases; but in addition it supplies all of the inferential machinery which is needed in order to be able to treat the intended interpretation of the equality symbol as'built in', and obviates the need to include special axioms of equality in the formulation of every theorem-proving problem which makes use of that notion. The completeness theory of the generalized resolution principle exploits the very intuitive and natural idea of attempting to construct counterexamples to the theorems for which proofs are wanted, and makes this the central concept. It is shown how a proof of a theorem is generated automatically by the breakdown of a sustained attempt to construct a counterexample for it. The kind of proof one gets depends entirely on the way in which the attempt to construct a counterexample is organized, and the theory places virtually no restrictions on how this shall be done. Consequently there is a very wide freedom in the form which proofs may take: the individual inferences in a proof may be very'small' or very'large' (in a scale of measurement which, roughly speaking, weighs the amount of computing necessary to check that the inference is correct). It is even correct to infer the truth of a true proposition in just one step, but, presumably, to offer such a proof to someone who wishes to be convinced of the proposition's truth would not be helpful epistemologically. His conviction would come, not from contemplating the proof itself, but rather from examining the computation which shows the correctness of its single inference step.

The general problem of re-Presentation is concerned with the relationship between different ways of formulating a problem to a problem solving system and the efficiency with which the system can be expected to find a solution to the problem. An understanding of the relationship between problem formulation and problem solving efficiency is a prerequisite for the design of procedures that can automatically choose the most appropriate' representation of a problem (they can find a point of view' of the problem that maximally simplifies the process of finding a solution). Many problems of practical importance are problems of reasoning about actions. In these problems, a course of action has to be found that satisfies a number of specified conditions. A formal definition of this class of problems is given in the next section, in the context of a general conceptual framework for formulating these problems for computers. Everyday examples of reasoning about actions include planning an airplane trip, organizing a dinner party, etc. There are many examples of industrial and military problems in this category, such as scheduling assembly and transportation processes, designing a program for a computer, planning a military operation, etc. The research presented in this paper was sponsored in part by the Air Force Office of Scientific Research, under Contract Number A F49(638)-1184. Part of this work was done while the author was on a visiting appointment at the Computer Science Department of the Carnegie Institute of Technology, Pittsburgh, Pa.

Networks as models of word storage: G. R. Kiss