Abstract not found

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately-owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago. ii

. At CADE-10 Ross Overbeek proposed a two-part contest to stimulate and reward work in automated theorem proving. The first part consists of seven theorems to be proved with resolution, and the second part of equational theorems. Our theorem provers Otter and its parallel child Roo proved all of the resolution theorems and half of the equational theorems. This paper represents a family of entries in the contest. Key Words. Automated theorem proving, resolution, paramodulation, Knuth-Bendix completion, strategy. 1 Introduction The Conference on Automated Deduction (CADE) has been for nearly twenty years a meeting where both theoreticians and practitioners present their work. Feeling perhaps that the conference was becoming dominated by the theoreticians, Ross Overbeek proposed at CADE-10 in 1990 a contest to stimulate work on the implementation and use of automated theorem-proving systems. The challenge was to prove a set of theorems, and do so with a uniform approach. That is, one w...

In one sense, this article is a personal tribute to Woody Bledsoe. As such, the style will in general be that of private correspondence. However, since this article is also a compendium of experiments with an automated reasoning program, researchers interested in automated reasoning, mathematics, and logic will find pertinent material here. The results of those experiments strongly suggest that research frequently benefits greatly from the use of an automated reasoning program. As evidence, I select from those results some proofs that are better than one can find in the literature, and focus on some theorems that, until now, had never been proved with an automated reasoning program, theorems that Hilbert, Church, and various logicians thought significant. To add spice to the article, I present challenges for reasoning programs, including questions that are still open. 1 This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Depa...

This article is the thirty-first of a series of articles discussing various open research problems - p in automated reasoning. The problem proposed for research asks one to find a strategy that can be cou led with the inference rule hyperresolution to control the behavior of an automated reasoning program K

1 1 Introduction 1 2 Results 1 3 Settings and Set of Support 2 3.1 Settings for the Basic Set : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 3.2 Settings for the Equality Set : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 3.3 Description of the Settings : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 4 Failures on Equality Theorems 6--10 5 5 Summary of Otter Outputs for the Basic Set 6 5.1 Theorem 1: x 2 = e Groups are Commutative (P-form) : : : : : : : : : : : : 6 5.2 Theorem 2: The Commutator Theorem (P-form) : : : : : : : : : : : : : : : 8 5.3 Theorem 3: x 2 = x Rings are Commutative (P-form) : : : : : : : : : : : : : 11 5.4 Theorem 4: Equivalential Calculus, XGK ! PYO : : : : : : : : : : : : : : 14 5.5 Theorem 5: Implicational Calculus Single Axiom, CD-67 (Imp-4) : : : : : : 16 5.6 Theorem 6: Many-valued Sentential Calculus, CD-57 : : : : : : : : : : : : : 19 5.7 Theorem 7: Many-valued Sentential Calculus, CD-60 : : : : : : : : : : : : : 21 6 Su...

An automated reasoning program has provided invaluable assistance in answering certain previously open questions in mathematics and in formal logic. These questions would not have been answered, at least by those who obtained the results, were it not for the program's contribution. Others have used such a program to design logic circuits, many of which proved superior (with respect to transistor count) to the existing designs, and to validate the design of other circuits. These successes establish the value of an automated reasoning program for research and suggest the value for practical applications. We thus conclude that the field of automated reasoning is on the verge of becoming one of the more significant branches of computer science. Further, we conclude that the field has already advanced from stage 1, that of potential usefulness, to stage 2, that of actual usefulness. To pass to stage 3, that of wide acceptance and use, requires, among other things, easy access to an automated reasoning program and an understanding of the various aspects of automated reasoning. In fact, an automated reasoning program is available that is portable and can be run on relatively inexpensive machines. Moreover, a system exists for producing a reasoning program tailored to given specifications. As for the requirement of understanding the aspects of automated reasoning, much research remains—research aided by access to a reasoning program. A large obstacle has thus been removed, permitting many to experiment with and find uses for a computer program that can be relied upon as a most valuable automated reasoning assistant.