Automated Theorem Proving: After 25 Years

AMERICAII MATHEMATICAL SOCIETY VOLUME 29

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COnTEMPORARY MATHEMATICS

Volume21

Automated Theorem Proving: After 25 Years

W. W. Bledsoe and D. W. Loveland. Editors

AMERICAII MATHEMATICAL SOCIETY Providence • RhOde 111•

EDITORIAL BOARD R. 0. Wells, Jr.,

managing editor Jeff Cheeger Adriano M. Garsia

Kenneth Kunen James I. Lepowsky Johannes C. C. Nitsche Irving Reiner

PROCEEDINGS OF THE SPECIAL SESSION ON AUTOMATIC THEOREM PROVING

89TH ANNUAL MEETING OF THE AMERICAN MATHEMATICAL SOCIETY

HELD IN DENVER, COLORADO

JANUARY 5-9, 1983

1980 Mathematics Subject Classification. Primary 68G15; Secondary 03835.

Llbruy of Congr• Cataloging in Publia~tion D• Special Seaion on Automatic Theorem Proving (1983: Denver, Colo.)

Automated theorem proving. (Contemporary mathematics, 0271-4132; v. 29) "Proceedings of the Special Session on Automatic Theorem Proving, 89th Annual Meeting

of the Ameria~n MathematiCIII Society, held in Denver. Colorado, January 5-9, 1983"-T. p. wrso. Includes bibliographies. 1. Automatic theorem proving-Congresses. I. Bledsoe, W.W. II. Loveland, Donald W.

Ill. AmeriCIIn Mathematical Society. Meeting (89th: 1983: Denver, Colo.l IV. Title. V. Series: Contemporary mathematics (American Mathematical Society; v. 291 QA76.9.A96S64 1983 511.3 84-9226 ISBN 0-8218-6027-X

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Table of Contents

Preface ..................................................................................................... vii

Acknowledgments ... .. ..... ...... ...... .... ...... ..... ................. ..... ..... ...... ..... ....... ix

D. W. Loveland. Automated Theorem Proving: a Quarter Century Review............................................................... 1

Citation to Hao Wang ............................................................................. 47

Hao Wang. Computer Theorem Proving and Artificial Intelligence ....................................................................... 49

Citation to Lawrence Was and Steven Winker .................................... 71

L. Was and S. Winker. Open Questions Solved with the Assistance of AURA .... ~ ...................................................... 73

W. W. Bledsoe. Some Automatic Proofs in Analysis .......................... 89

R. S. Boyer and J. S. Moore. Proof-Checking, Theorem-Proving, and Program Verification ............................... 119

R. S. Boyer and J. S. Moore. A Mechanical Proof of the Turing Completeness of Pure LISP ..................................... 133

P. B. Andrews, D. A. Miller, E. L. Cohen and F. Pfenning. Automating Higher-order Logic ..................................................... 169

D. Lankford, G. Butler and B. Brady. Abelian Group Unification Algorithms for Elementary Terms ............................. 193

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vi Table of Contents

G. Nelson. Combining Satisfiability Procedures by Equality Sharing ......................................................................... 201

Wu Wen-Tsun. On the Decision Problem and the Mechanization of Theorem-Proving in Elementary Geometry ................................................................. 213

Wu Wen-Tsun. Some Recent Advances in Mechanical Theo:-em-Proving of Geometries ................................ 235

Shang-Ching Chou. Proving Elementary Geometry Theorems Using Wu's Algorithm ................................... 243

D. B. Lenat. Automated Theory Formation in Mathematics ................................................................................. 287

J. McDonald and P. Suppes. Student Use of an Interactive Theorem Prover ...................................................... 315

Preface

The genesis of the collection of papers presented in this volume is a special session on automated theorem proving (ATP), held in conjunction with the annual meeting of the American Mathematical Society in Denver, January, 1983, to honor several research leaders in the field of ATP. Hao Wang received the first Milestone Prize for his pioneering work in the early years of research in ATP and Lawrence Wos and Steven Winker received the Current Research Prize for recent and ongoing work. The cita-tions appear in this volume with papers by the recipients. It is anticipated that both awards will be given periodically at a yet undetermined interval.

A secondary purpose for the special session was to bring to the attention of the mathematics research community the status of work in ATP. It is clear from the work of several research groups that the state has been reached where serious thought can be given by mathematicians to the possibilities of computer-aided proof discovery. (Fully automated proof, and theorem, discovery remains a goal but current success, in some ways quite impressive, is still very limited.) It is our hope that some active mathemati-cians will choose to accept the challenge this field extends and work, probably with an existing research group in ATP, to extend our capabilities towards frontier mathematics.

The papers in this volume derive primarily from the presenta-tions by invited speakers at the special sessions, although invita-tions were extended to Wu Wen-Tsun and Patrick Suppes to submit

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viii Preface

papers for the volume. It was our feeling that the presentation in the special session collectively portrayed the state-of-the-art, with a sense of progress and high activity that exists in the field today. We also feel that the papers presented in this volume are even more successful at presenting the position and the momentum of the field of ATP in 1983.

The first paper in the volume serves as the introduction and overview to this volume as well as to the field of ATP. The papers of Bledsoe, Boyer and Moore, and Wos and Winker have a tutorial aspect and might be read next by those with little prior acquain-tance with the field. A non-technical personal perspective of the endeavor to automate theorem proving is offered by Hao Wang. The papers by Lenat and Suppes outline substantial projects of considerable interest to our community but whose published reports have generally appeared in other settings. The remaining papers are more technical but report substantial contributions to the field.

We wish to thank the American Mathematics Society for their interest in this developing field of Automated Theorem Proving, both for the staging of the special session at the AMS Annual Meet-ing in Denver, January 1983, and for their interest in publishing this volume within the Contemporary Mathematics series. We thank Tony Palermino for competent stewardship at AMS in bring-ing this volume to press. Also, we thank David Mutchler at Duke University for his time and expertise in guiding several of these papers· through a balking text-processor and output device, we thank Barbara Smith at the University of Texas, Austin, for typing several papers under time pressure, and indirectly we thank those who typed and supervised camera-ready copy preparation for indi-vidual papers.

W.W. Bledsoe D. W. Loveland

February 15, 1984

Acknowledgments

Wu Wen-Tsun. On the decision problem and the mechanization of theorem proving in elementary geometry. Reprinted from .SCientia.

Si.nica 21 (2), 1978, with permission of the author.

Douglas Lenat. Automated theory formation in mathematics. Adopted from the article of the same title in the Proceedings of the

International Joint Conference on Artificial Intelligence, August, 1977, with the permission of the International Joint Conferences on Artificial Intelligence, Inc. Copies of the proceedings are avail-able from William Kaufmann, Inc., 95 First Street, Los Altos, CA 94022, USA.

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