**DIMACS - Series in Discrete Mathematics and Theoretical Computer Science**

Volume: 39;
1998;
320 pp;
Hardcover

MSC: Primary 03; 68;

Print ISBN: 978-0-8218-0577-0

Product Code: DIMACS/39

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-3997-2
Product Code: DIMACS/39.E**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

# Proof Complexity and Feasible Arithmetics

Share this page *Edited by *
*Paul W. Beame; Samuel R. Buss*

A co-publication of the AMS and DIMACS

Questions of mathematical proof and logical inference have been a significant thread in modern mathematics and have played a formative role in the development of computer science and artificial intelligence. Research in proof complexity and feasible theories of arithmetic aims at understanding not only whether logical inferences can be made, but also what resources are required to carry them out. Understanding the resources required for logical inferences has major implications for some of the most important problems in computational complexity, particularly the problem of whether NP is equal to co-NP. In addition, these have important implications for the efficiency of automated reasoning systems.

The last dozen years have seen several breakthroughs in the study of these resource requirements. Papers in this volume represent the proceedings of the DIMACS workshop on
“Feasible Arithmetics and Proof Complexity” held in April 1996 at Rutgers University in New Jersey as part of the DIMACS Institute's Special Year on Logic and Algorithms.

This book brings together some of the most recent work of leading researchers in proof complexity and feasible arithmetic reflecting many of these advances. It covers a number of aspects of the field, including lower bounds in proof complexity, witnessing theorems and proof systems for feasible arithmetic, algebraic and combinatorial proof systems, interpolation theorems, and the relationship between proof complexity and Boolean circuit complexity.

Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1–7 were co-published with the Association for Computer Machinery (ACM).

#### Readership

Graduate students and research mathematicians interested in mathematical logic and foundations.

#### Reviews & Endorsements

The volume adequately reflects current interests and trends in the area of feasible proofs, and many papers in it simply define the current state of the art there. Its importance for everybody interested in this beautiful theory, be it the experienced researcher, a beginner, or just a curious outsider, can hardly be overstated.

-- Journal of Symbolic Logic

# Table of Contents

## Proof Complexity and Feasible Arithmetics

- Cover Cover11
- Title page v6
- Contents vii8
- Foreword ix10
- Preface xi12
- Plausibly hard combinatorial tautologies 114
- More on the relative strength of counting principles 1326
- Ranking arithmetic proofs by implicit ramification 3750
- Lower bounds on Nullstellensatz proofs via designs 5972
- Relating the provable collapse of π to ππΒΉ and the power of logical theories 7386
- On ππ»π, π π‘-connectivity, and odd charged graphs 93106
- Descriptive complexity and the π hierarchy 119132
- Lower bounds on sizes of cutting plane proofs for modular coloring principles 135148
- Equational calculi and constant depth propositional proofs 149162
- Exponential lower bounds for semantic resolution 163176
- Bounded arithmetic: Comparison of Bussβ witnessing method and Siegβs Herbrand analysis 173186
- Towards lower bounds for bounded-depth Frege proofs with modular connectives 195208
- A quantifier-free theory based on a string algebra for ππΆΒΉ 229242
- A propositional proof system for π β±β 253266
- Algebraic models of computation and interpolation for algebraic proof systems 279292
- Self-reflection principles and NP-hardness 297310
- Back Cover Back Cover1335