INTRODUCTION The evidence is already in favour of the computer becoming an aid to research in branches of mathematics which do not involve numerical computation. Programs have now been constructed for performing certain symbolic operations in mathematics, for example algebraic simplification of equations and indefinite integration, and for solving some of the problems (which are certainly not in the class of numerical calculations) now occurring in theoretical physics and other areas. One may reasonably expect that sooner or later programs of this type will be incorporated in a'questionanswering' package. As the construction of multi-programming systems progresses, and'on-line' use of the computer is accepted as normal, it will become more and more practicable for the mathematician to rely on the computer for answers to routine non-numerical questions. It is, therefore, of interest from a practical as well as a theoretical point of view to ask if this development can be taken a stage further by mechanising the processes of mathematical proof. In this way the computer could be used as a tool for establishing true mathematical statements or checking proofs for correctness. The problem is essentially to produce practicable programs for proving theorems which will answer some reasonably complicated questions before overstepping the bounds of available time and memory space. This is potentially one of the most rewarding areas of computer applications, but as many people already know, it is also one of the most frustrating.

The author's main source was from personal discussion.)

In the'condition omitted' column: denotes that the full complement of heuristic devices was in use denotes that the coefficient of the feature L was set to zero denotes that the coefficient of the feature B was set to zero de,notes that the coefficient of the feature E was set to zero denotes that the coefficient of the feature T was set to zero PD denotes that all neighbours of a node were produced at one development M denotes that the example group was not in use 44 POPPLESTONE F denotes that the equality was proved in reverse, i.e., that the RHS was converted into the LHS. Note that the LHS is longer than the RHS in every problem where the lengths are different. In the'result' column: S denotes that the problem was solved F denotes that the problem was unsolved after 100 distinct nodes had been produced - FF denotes that the problem was unsolved after 100 distinct nodes had been produced but that the problem was solved after restarting one or more times from the most promising node produced so far (see Doran's paper in this volume). FL denotes that the problem was unsolved because a term too large to hold in the array assigned to hold it was produced. The nodes listed' column gives the total number of distinct nodes produced.

RELATION TO PHYSIOLOGY It is possible to compare the organisation of this system with the organisation of the visual system, as revealed by microelectrode studies in the cat (Hubel & Wiesel 1962, 1965) and the frog (Lettvin, Maturana, McCulloch & Pitts 1959). Briefly the following points emerge: (1) Cells in the visual cortex only respond to local properties of the visual scene, e.g., edges, line segments. This mirrors the immediate constituent constraint imposed for economic reasons in the picture grammar.

Psychological theory attempts to explain how thinking--the subject matter of psychology--is possible by a brain composed of single mechanistic elements--the basic assumption of psychology. The problem of programming digital computers to behave in complex fashions is equivalent to this aspect of the psychological problem. Today automata theorists agree that no fundamental barrier blocks the development of machines which can think, by any reasonable definition of the term. However, the precise techniques for implementing general thinking proceGseE-J have been only partially developed. An example of a high-level, general thinking process is comprehension: the understanding of passages of a natural language.

The ability of the higher animals to accept and interpret information from distant objects confers enormous advantages for creatures (or machines) which respond only to immediate stimulation and have no opportunity to anticipate the future. Distance receptors, especially the eyes, serve as early warning systems by giving information of distance events, making it possible to gauge the probable future. The classical biological notion of stimulusresponse applies to creatures limited to touch information. The development of distance-receptors evidently allowed brains to develop to give strategic behaviour. It is unfortunate that the early, now classical, studies of reflexes involving touch and the internal regulation of the body have been so largely taken over to describe brain function, for these concepts are inadequate for describing the central nervous system. They tell nothing about how brains handle information from the eyes, to allow animals and man to see. They tell us nothing about decision-making: how present experience is related to the stored past to predict the immediate future.

Might computers one day be able to think every thought human brains can think? Might robots be able to behave in all the ways humans behave? Would such artifacts be fully human even though they had different structure and used different means? Since human evolution proceeds at a much slower pace than technological advance, might artifacts someday do everything better than people and become super-human beings? While computers and robots have made a promising start at overtaking humans at many thinking tasks and skilled behaviors, from chess to violin playing, these achievements have been made possible primarily by an exponential increase in computing power since the 1950s.

Using the Neomycin rule base, and inputting Neomycin's own actions to the action justification generator, the average size of J(.4,) was ten and the maximum size was approximately one hundred. When an Odysseus-generated rule base for the Neomycin domain was used, these set sizes increased by a factor of four to five. After the set J(Ai) is generated, the action justification ranking subsystem of Odysseus determines the likelihood that J(Ai) contains ji, the action justification of the specialist. This involves, first, ranking ji,„ in order of likelihood of being equal to the unknown An example of ranking rule is: given two elements of a J(.4,), where,4, occurs early in the problem solving session, the

LOGIC CROUP KNOWLEDGE SYSTEMS LABORATORY Department of Computer Science Stanford tIniversity Stanford, California 91305 Decision Precedures Abstract Distributed artificial intelligence is the study of how a group of individual intelligent agents can combine to solve a difficult global problem. This paper discusses in very general terms the problems of achieving this global goal by considering simpler, local subproblems; we drop the usual requirement that the agents working on the subproblems do not interact. We are led to a single assumption. An example of a distributed computation using these ideas is presented. Introduction The thrust of research in distributed artificial intelligence (DAI) is the investigation of the possibility of solving a difficult problem by presenting each of a variety of machines with simpler parts of it. The approach that has been taken has been to consider the problem of dividing tho original problem: what:,libtasks should be pursued at any given time? To which available machine should a..iven subtask be assigned? The question of how the individual machines should g9 about solving their subproblems has been left to the non-distributed Al community (or perhaps to a recursive application of DAI techniques). The assumption underlying this approach--that each of the agents involved in the solution of the subproblems can proceed independently of the others--has recently been called into question 12,3,6,7,101. It has been realized that, in a world of limited resources, it is inappropriate to dedicate a substantial fraction of those resources to each processor. The increasing attract:witless of parallel architectures in which processors share memory is an example of this: memory is a scarce resource. Automated factories must inevitably encounter similar difficulties. Are the robots working in such factories to be given distinct bins of component parts, and non-overlapping regions in which to work or to travel from one area of the factory to another?

Some of the more popular alternativo used to build knowledge systems are production systems, backward-chained reasoning, logic programming, heuristic search, and the Blackboard framework. Many of the applications implemented in production systems have been written in the OPS language [8]. In this framework, knowledge is represented as a set of homogeneous rules that are scanned for applicability in a data base that contains the current state of solution. Backward chaining also has a homogeneous set of rules, but the search for applicable rules is driven by a hierarchy of goals and sub-goals. The best known system for implementing this type of program is EMYCIN [4].