Bounded Arithmetic and Propositional Proof Complexity

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories and of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses.

[1]  Jacques Herbrand.  Recherches sur la théorie de la démonstration , 1930 .

[2]  G. Gentzen.  Die Widerspruchsfreiheit der reinen Zahlentheorie , 1936 .

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

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

[5]  Stephen A. Cook,et al.  Review: Alan Cobham, Yehoshua Bar-Hillel, The Intrinsic Computational Difficulty of Functions , 1969 .

[6]  Rohit Parikh,et al.  Existence and feasibility in arithmetic , 1971, Journal of Symbolic Logic.

[7]  S. Cook,et al.  On the lengths of proofs in the propositional calculus (Preliminary Version) , 1974, STOC '74.

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

[9]  Stephen A. Cook,et al.  Feasibly constructive proofs and the propositional calculus (Preliminary Version) , 1975, STOC.

[10]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[11]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

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

[13]  Martin Dowd.  Propositional representation of arithmetic proofs (Preliminary Version) , 1978, STOC '78.

[14]  M. Dowd,et al.  Propositional representation of arithmetic proofs , 1979 .

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

[16]  Bernard R. Hodgson,et al.  An Arithmetical Characterization of NP , 1982, Theor. Comput. Sci..

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

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

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

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

[21]  Jeff B. Paris,et al.  On the scheme of induction for bounded arithmetic formulas , 1987, Ann. Pure Appl. Log..

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

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

[24]  S. Cook.  Computational complexity of higher type functions , 1990 .

[25]  Jan Krajícek,et al.  Quantified propositional calculi and fragments of bounded arithmetic , 1990, Math. Log. Q..

[26]  Christos H. Papadimitriou,et al.  ON GRAPH-THEORETIC LEMMATA AND COMPLEXITY CLASSES (Extended Abstract ) , 1990, FOCS 1990.

[27]  S. Buss.  Axiomatizations and Conservation Results for Fragments of Bounded Arithmetic ∗ , 1990 .

[28]  Samuel R. Buss,et al.  Propositional Consistency Proofs , 1991, Ann. Pure Appl. Log..

[29]  Jan Krajícek,et al.  Bounded Arithmetic and the Polynomial Hierarchy , 1991, Ann. Pure Appl. Log..

[30]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[31]  Petr Hájek,et al.  Metamathematics of First-Order Arithmetic , 1993, Perspectives in mathematical logic.

[32]  S. Buss,et al.  An Application of Boolean Complexity to Separation Problems in Bounded Arithmetic , 1994 .

[33]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[34]  Samuel R. Buss,et al.  On Herbrand's Theorem , 1994, LCC.

[35]  Alexander A. Razborov,et al.  On provably disjoint NP-pairs , 1994, Electron. Colloquium Comput. Complex..

[36]  Samuel R. Buss,et al.  Are there Hard Examples for Frege Systems , 1995 .

[37]  Samuel R. Buss,et al.  Unprovability of Consistency Statements in Fragments of Bounded Arithmetic , 1995, Ann. Pure Appl. Log..

[38]  Samuel R. Buss.  Relating the Bounded Arithmetic and Polynomial Time Hierarchies , 1995, Ann. Pure Appl. Log..

[39]  Domenico Zambella.  Notes on Polynomially Bounded Arithmetic , 1996, J. Symb. Log..

[40]  外史 竹内.  Bounded Arithmetic と計算量の根本問題 , 1996 .

[41]  Astrophysics,et al.  The Characteristics of Millisecond Pulsar Emission. I. Spectra, Pulse Shapes, and the Beaming Fraction , 1998, astro-ph/9801177.