Propositional Proof Complexity An Introduction

This article is an abridged and revised version of a 1996 McGill University technical report [15]. The technical report was based on lectures delivered by the author at a workshop in Holetown, Barbados and on the authors prepared overhead transparencies. The audience at this workshop wrote scribe notes which then formed the technical report [15]. The material selected for the present article corresponds roughly to the content of the author’s lectures at the NATO summer school held in Marktoberdorf, Germany in July-August 1997.

[1]  A. Slisenko Studies in constructive mathematics and mathematical logic , 1969 .

[2]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

[3]  Samuel R. Buss Polynomial Size Proofs of the Propositional Pigeonhole Principle , 1987, J. Symb. Log..

[4]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

[5]  Jan Kra,et al.  Lower Bounds to the Size of Constant-depth Propositional Proofs , 1994 .

[6]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[7]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[8]  Samuel R. Buss,et al.  An Optimal Parallel Algorithm for Formula Evaluation , 1992, SIAM J. Comput..

[9]  Samuel R. Buss,et al.  Size-Depth Tradeoffs for Boolean Fomulae , 1994, Inf. Process. Lett..

[10]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[11]  Daniele Mundici Tautologies with a unique craig interpolant, uniform vs. nonuniform complexity , 1984, Ann. Pure Appl. Log..

[12]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, STOC '95.

[13]  Russell Impagliazzo,et al.  Exponential lower bounds for the pigeonhole principle , 1992, STOC '92.

[14]  R. Statman Complexity of Derivations from Quantifier-Free Horn Formulae, Mechanical Introduction of Explicit Definitions, and Refinement of Completeness Theorems , 1977 .

[15]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[16]  Miklós Ajtai,et al.  The complexity of the Pigeonhole Principle , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[17]  Jan Krajícek,et al.  An Exponenetioal Lower Bound to the Size of Bounded Depth Frege Proofs of the Pigeonhole Principle , 1995, Random Struct. Algorithms.

[18]  P. Clote,et al.  Arithmetic, proof theory, and computational complexity , 1993 .

[19]  Andreas Goerdt Cuting Plane Versus Frege Proof Systems , 1990, CSL.

[20]  William J. Cook,et al.  On the complexity of cutting-plane proofs , 1987, Discret. Appl. Math..

[21]  S. Rao Kosaraju,et al.  A tree-partitioning technique with applications to expression evaluation and term matching , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[22]  Samuel R. Buss,et al.  Chapter I - An Introduction to Proof Theory , 1998 .

[23]  Jan Krajícek,et al.  Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic , 1997, Journal of Symbolic Logic.

[24]  William Craig,et al.  Linear reasoning. A new form of the Herbrand-Gentzen theorem , 1957, Journal of Symbolic Logic.

[25]  C. Burdet,et al.  On cutting planes , 1973 .

[26]  Peter Clote,et al.  Cutting planes, connectivity, and threshold logic , 1996, Arch. Math. Log..

[27]  Jan Krajícek,et al.  Propositional proof systems, the consistency of first order theories and the complexity of computations , 1989, Journal of Symbolic Logic.

[28]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[29]  Samuel R. Buss,et al.  Some remarks on lengths of propositional proofs , 1995, Arch. Math. Log..

[30]  R. Ladner The circuit value problem is log space complete for P , 1975, SIGA.

[31]  Richard P. Brent,et al.  The Parallel Evaluation of General Arithmetic Expressions , 1974, JACM.

[32]  Sloman,et al.  Automation of Reasoning , 1983, Symbolic Computation.

[33]  Jan Krajícek,et al.  The number of proof lines and the size of proofs in first order logic , 1988, Arch. Math. Log..

[34]  Samuel R. Buss,et al.  The Boolean formula value problem is in ALOGTIME , 1987, STOC.

[35]  Allan Borodin,et al.  Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines , 1984, Inf. Control..

[36]  Nader H. Bshouty,et al.  Size-depth tradeoffs for algebraic formulae , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.