James R. Slagle

Automated Theorem-Proving for Theories with Simplifiers Commutativity, and Associativity

To prove really difficult theorems, resolution principle programs need to make better inferences and to make them faster. An approach is presented for taking advantage of the structure of some special theories. These are theories with simplifiers, commutativity, and associativity, which are valuable concepts to build in, since they so frequently occur in important theories, for example, number theory (plus and times) and set theory (union and intersection). The object of the approach is to build in such concepts in a (refutation) complete, valid, efficient (in time) manner by means of a “natural” notation and/or new inference rules. Some of the many simplifiers that can be built in are axioms for (left and right) identities, inverses, and multiplication by zero.nAs for results, commutativity is built in by a straightforward modification to the unification (matching) algorithm. The results for simplifiers and associativity are more complicated. These theoretical results can be combined with one another and/or extended to either C-linear refutation completeness or theories with partial ordering, total ordering, or sets. How these results can serve as the basis of practical computer programs is discussed.

Automatic Theorem Proving With Renamable and Semantic Resolution

The theory of J. A. Robinsons resolution principle, an inference rule for first-order predicate calculus, is unified and extended. A theorem-proving computer program based on the new theory is proposed and the proposed semantic resolution program is compared with hyper-resolution and set-of-support resolution programs. Renamable and semantic resolution are defined and shown to be identical. Given a model <italic>M</italic>, semantic resolution is the resolution of a latent clash in which each “electron” is at least sometimes false under <italic>M</italic>; the nucleus is at least sometimes true under <italic>M</italic>.nThe completeness theorem for semantic resolution and all previous completeness theorems for resolution (including ordinary, hyper-, and set-of-support resolution) can be derived from a slightly more general form of the following theorem. If <italic>U</italic> is a finite, truth-functionally unsatisfiable set of nonempty clauses and if <italic>M</italic> is a ground model, then there exists an unresolved maximal semantic clash [<italic>E</italic><subscrpt>1</subscrpt>, <italic>E</italic><subscrpt>2</subscrpt>, · · ·, <italic>E</italic><subscrpt>q</subscrpt>, <italic>C</italic>] with nucleus <italic>C</italic> such that any set containing <italic>C</italic> and one or more of the electrons <italic>E</italic><subscrpt>1</subscrpt>, <italic>E</italic><subscrpt>2</subscrpt>, · · ·, <italic>E</italic><subscrpt>q</subscrpt> is an unresolved semantic clash in <italic>U</italic>.

A New Algorithm for Generating Prime Implicants

This paper describes an algorithm which will generate all the prime implicants of a Boolean function. The algorithm is different from those previously given in the literature, and in many cases it is more efficient. It is proved that the algorithm will find all the prime implicants. The algorithm may possibly generate some nonprime implicants. However, using frequency orderings on literals, the experiments with the algorithm show that it usually generates very few ( possibly none) nonprime implicants. Furthermore, the algorithm may be used to find the minimal sums of a Boolean function. The algorithm is implemented by a computer program in the LISP language.

A Heuristic Program that Solves Symbolic Integration Problems in Freshman Calculus

A large high-speed general-purpose digital computer (IBM 7090) was programmed to solve elementary symbolic integration problems at approximately the level of a good college freshman. The program is called SAINT, an acronym for Symbolic Automatic INTegrator. This paper discusses the SAINT program and its performance. SAINT performs indefinite integration. I t also performs definite and multiple integration when these are trivial extensions of indefinite integration. I t uses many of the methods and heuristics of students attacking the same problems. SAINT took an average of two minutes each to solve 52 of the 54 attempted problems taken from the Massachusetts Institute of Technology freshman calculus final examinations. Based on this and other experiments with SAINT, some conclusions concerning computer solution of such problems are: (1) Pattern recognition is of fundamental importance. (2) Great benefit would have been derived from a larger memory and more convenient symbol manipulating facilities. (3) The solution of a symbolic integration problem by a commercially available computer is far cheaper and faster than by man.

Experiments With Some Programs That Search Game Trees

Many problems in artificial intelligence involve the searching of large trees of alternative possibilities—for example, game-playing and theorem-proving. The problem of efficiently searching large trees is discussed. A new method called “dynamic ordering” is described, and the older minimax and Alpha-Beta procedures are described for comparison purposes. Performance figures are given for six variations of the game of kalah. A quantity called “depth ratio” is derived which is a measure of the efficiency of a search procedure. A theoretical limit of efficiency is calculated and it is shown experimentally that the dynamic ordering procedure approaches that limit.

An admissible and optimal algorithm for searching AND/OR graphs

Abstract An AND/OR graph is a graph which represents a problem-solving process. A solution graph is a subgraph of the AND/OR graph which represents a derivation for a solution of the problem. Therefore, solving a problem can be viewed as searching for a solution graph in an AND/OR graph. A “cost” is associated with every solution graph. A minimal solution graph in a solution graph with minimal cost. In this paper, an algorithm for searching for a minimal solution graph in an AND/OR graph is described. If the “lower bound” condition is satisfied, the algorithm is guaranteed to find a minimal solution graph when one exists. Furthermore, the “optimality” of the algorithm is also proved.

Experiments with a deductive question-answering program

As an investigation in artificial intelligence, computer experiments on deductive question-answering were run with a LISP program called DEDUCOM, an acronym for DEDUctive COMmunicator. When given 68 facts, DEDUCOM answered 10 questions answerable from the facts. A fact tells DEDUCOM either some specific information or a method of answering a general kind of question. Some conclusions drawn in the article are: (1) DEDUCOM can answer a wide variety of questions. (2) A human can increase the deductive power of DEDUCOM by telling it more facts. (3) DEDUCOM can write very simple programs (it is hoped that this ability is the forerunner of an ability to self-program, which is a way to learn). (4) DEDUCOM is very slow in answering questions. (5) DEDUCOMs search procedure at present has two bad defects: some questions answerable from the given facts cannot be answered and some other answerable questions can be answered only if the relevant facts are given in the right order. (6) At present, DEDUCOMs method of making logical deductions in predicate calculus has two bad defects: some facts have to be changed to logically equivalent ones before being given to DEDUCOM, and some redundant facts have to be given to DEDUCOM.

Experiments With a Multipurpose, Theorem-Proving Heuristic Program

The heuristic program discussed searches for a constructive proof or disproof of a given proposition. It uses a search procedure which efficiently selects the seemingly best proposition to work on next. This program is multipurpose in that the domains it can handle are varied.nAs an initial experiment, the program was given the task of searching for proofs and disproofs of propositions about kalah end games. Kalah is a two-person game. In another experiment the program, after some modifications, played the game of kalah. This program was compared with another tree-searching procedure, the Alpha-Beta minimax procedure; the results have been encouraging since the program is fast and efficient. Its greatest usefulness is in solving large problems. It is hoped that this program has added one more step toward the goal of eventually obtaining computer programs which can solve intellectually difficult problems.

Automatic Theorem Proving with Built-in Theories Including Equality, Partial Ordering, and Sets

Oxidizing butane to acetic acid which comprises contacting a sufficient concentration of an oxygen-containing gas with normal butane in the presence of catalyst consisting essentially of bromine and cobalt to initiate a self-sustaining exothermic reaction.

Application of game tree searching techniques to sequential pattern recognition

A sequential pattern recognition (SPR) procedure does not test all the features of a pattern at once. Instead, it selects a feature to be tested. After receiving the result of that test, the procedure either classifies the unknown pattern or selects another feature to be tested, etc. Medical diagnosis is an example of SPR. In this paper the authors suggest that SPR be viewed as a one-person game played against nature (chance). Virtually all the powerful techniques developed for searching two-person, strictly competitive game trees can easily be incorporated either directly or by analogy into SPR procedures. In particular, one can incorporate the “miniaverage backing-up procedure” and the “gamma procedure,” which are the analogues of the “minimax backing-up procedure” and the “alpha-beta procedure,” respectively. Some computer simulated experiments in character recognition are presented. The results indicate that the approach is promising.