This is an informal description of my ideas about using formal logic as a tool for reasoning systems using computers. Introduction The title of this paper contains both the words'mechanized' and'theory'. I want to make the point that the ideas presented here are not only of interest to theoreticians. I believe that any theory of interest to artificial intelligence must be realizable on a computer. I will not present difficult examples.

ON CLOSED WORLD DATA BASES Raymond Reiter The University of British Columbia Vancouver, British Columbia ABSTRACT Deductive question-answering systems generally evaluate queries under one of two possible assumptions which we in this paper refer to as the open and closed world assumptions. The open world assumption corresponds to the usual first order approach to query evaluation: Given a data base DB and a query Q, the only answers to Q are those which obtain from proofs of Q given DB as hypotheses. Under the closed world assumption, certain answers are admitted as a result of failure to find a proof. More specifically, if no proof of a positive ground literal exists, then the negation of that literal is assumed true. In this paper, we show that closed world evaluation of an arbitrary query may be reduced to open world evaluation of socalled atomic queries.

This paper explores the truism that people think about what they say. It proposes that, to satisfy their own goals, people often plan their speech acts to affect their listeners' beliefs, goals, and emotional states. Such language use can be modelled by viewing speech acts as operators in a planning system, thus allowing both physical and speech acts to be integrated into plans. Methodological issues of how speech acts should be defined in a planbased theory are illustrated by defining operators for requesting and informing. Plans containing those operators are presented and comparisons ore drawn with Searle's formulation.

ABSTRACT This talk reviews those efforts in automatic theorem proving, during the past few years, which have emphasized techniques other than resolution. These include: knowledge bases, natural deduction, reduction, (rewrite rules), typing, procedures, advice, controlled forward chaining, algebraic simplification, built-in associativity and commutativity, models, analogy, and man-machine systems. Examples are given and suggestions are made for future work. Earlier work by Newell, Simon, Shaw, and Gelernter in the middle and late 1950s emphasized the heuristic approach, but the weight soon shifted to various syntactic methods culminating in a large effort on resolution type systems in the last half of the 1960s. It was about 1970 when considerable interest was revived in heuristic methods and the use of human supplied, domain dependent, knowledge.

In this paper we look at some of the ingredients and processes involved in the understanding of mathematics. We analyze elements of mathematical knowledge, organize them in a coherent way and take note of certain classes of items that share noteworthy roles in understanding. We thus build a conceptual framework in which to talk about mathematical knowledge. We then use this representation to describe the acquisition of understanding. We also report on classroom experience with these ideas.

North-Holland Advances in Artificial Intelligence, T. O'Shea (Ed.) Elsevier Science Publishers By. (North-I lolland) 0 ECCAI, 1985 367 THE UBIQUITOUS DIALECTIC Edwina L. Rissland Department of Computer and Information Science University of Massachusetts Amherst, MA 01003 In this paper, we discuss the central role played by examples in reasoning in various fields including mathematics, law, linguistics, and computer science. In particular, we consider the dialectic between proposing a concept, conjecture, or proposition and testing and refining it with examples. We provide several examples of this ubiquitous process. In mathematics, examples can be said to be as important to understanding as the traditionally exalted definitions, theorems, and proofs (Rissland 1978). In fact, some mathematical areas developed in response to troublesome counterexamples like modern real function theory which has been called'the branch of mathematics which deals with counterexamples" (Munroe 1953).

A new predicate calculus deduction system based on production rules is proposed. The system combines several developments in Artificial Intelligence and Automatic Theorem Proving research including the use of domain-specific inference rules and separate mechanisms for forward and backward reasoning. It has a clean separation between the data base, the production rules, and the control system. Goals and subgoals are maintained in an AND/OR tree structure. Logical deduction is a basic activity in many artificial intelligence (Al) systems.

A program called "AM" is described which carries on simple mathematics research, defining and studying new concepts under the guidance of a large body of heuristic rules. The 250 heuristics communicate via an agenda mechanism, a global priority queue of small tasks for the program to perform, and reasons why each task is plausible (for example, "Find generalizations of'primes', because'primes' turned out to be so useful a concept"). Each concept is represented as an active, structured knowledge module. One hundred very incomplete modules are initially supplied, each one corresponding to an elementary set-theoretic concept (for example, union). This provides a definite but immense space which AM begins to explore.

A programming language needs simple and well defined semantics. The two favoured theoretical bases for languages have been lambda calculus as advocated by Landin and others, and predicate calculus as advocated by Kowalski (see Landin (1966) and Kowalski (1973)). In this paper I adopt an approach based on predicate calculus, but in a manner that differs from the existing PROLOG language (Warren 1975 and Battani & Meloni 1973) in that I adopt a "forward inference" approach -- inferring conclusions from premises, rather than the "backward inference" approach of PROLOG, which starts with a desired conclusion and tries to find ways of inferring it. This difference is reflected in the internal structure of the associated implementations, that of PROLOG being a "backtrack search" kind of implementation, while the most obvious implementation of the system proposed here involves a kind of mass operation on tables of data, reminiscent of APL (Iverson 1962) but in fact identical in many respects with the work of Codd (Codd 1970) on relational data bases. Indeed, from one perspective this paper can be seen as an extension of Codd's work into the realm of general purpose computing.

Machine intelligence problems are sometimes defined as those problems which (i) computers can't yet do, and (ii) humans can. We shall further consider how much "knowledge" about a finite mathematical function can, on certain assumptions, be credited to a computer program. Although our approach is quite general, we are really only interested in programs which evaluate "semihard" functions, believing that the evaluation of such functions constitutes the defining aspiration of machine intelligence work. If a function is less hard than "semihard," then we can evaluate it by pure algorithm (trading space for time) or by pure lookup (making the opposite trade), with no need to talk of knowledge, advice, machine intelligence, or any of those things. We call such problems "standard."