Guest Column: Proof Complexity and Beyond

This essay is a highly personal and biased account of some main concepts and several research directions in modern propositional proof complexity. Special attention will be paid to connections with other disciplines.

[1]  Jan Krajícek,et al.  Exponential Lower Bounds for the Pigeonhole Principle , 1992, STOC.

[2]  Prasad Raghavendra,et al.  Lower Bounds on the Size of Semidefinite Programming Relaxations , 2014, STOC.

[3]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[4]  Toniann Pitassi,et al.  Circuit Complexity, Proof Complexity, and Polynomial Identity Testing , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[5]  Michael Alekhnovich,et al.  Pseudorandom generators in propositional proof complexity , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

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

[7]  Jan Krajícek,et al.  Some Consequences of Cryptographical Conjectures for S12 and EF , 1998, Inf. Comput..

[8]  Grant Schoenebeck,et al.  Linear Level Lasserre Lower Bounds for Certain k-CSPs , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[9]  Michael Alekhnovich,et al.  Resolution Is Not Automatizable Unless W[P] Is Tractable , 2008, SIAM J. Comput..

[10]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[11]  Alexander A. Razborov,et al.  Lower bounds for the polynomial calculus , 1998, computational complexity.

[12]  Massimo Lauria,et al.  On the Automatizability of Polynomial Calculus , 2010, Theory of Computing Systems.

[13]  Madhur Tulsiani.  Lovász‐Schrijver Reformulation , 2011 .

[14]  Rocco A. Servedio,et al.  Poly-logarithmic Frege depth lower bounds via an expander switching lemma , 2016, STOC.

[15]  Jan Krajícek,et al.  Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds , 2004, Journal of Symbolic Logic.

[16]  EDWARD A. HIRSCH,et al.  COMPLEXITY OF SEMIALGEBRAIC PROOFS , 2003 .

[17]  E. Ben-Sasson,et al.  Expansion in proof complexity. (הרחבה ומורכבות הוכחות.) , 2001 .

[18]  Samuel R. Buss,et al.  Linear gaps between degrees for the polynomial calculus modulo distinct primes , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[19]  A. Razborov.  Bounded Arithmetic and Lower Bounds in Boolean Complexity , 1995 .

[20]  Eli Ben-Sasson.  Hard examples for bounded depth frege , 2002, STOC '02.

[21]  P. Pudlák.  Sets and Proofs: On the Complexity of the Propositional Calculus , 1999 .

[22]  Stephen Cook,et al.  Corrections for "On the lengths of proofs in the propositional calculus preliminary version" , 1974, SIGA.

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

[24]  Allan Sly,et al.  Proof of the Satisfiability Conjecture for Large k , 2014, STOC.

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

[26]  Paul Beame,et al.  More on the relative strength of counting principles , 1996, Proof Complexity and Feasible Arithmetics.

[27]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

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

[29]  Jacobo Torán,et al.  Space Bounds for Resolution , 1999, STACS.

[30]  Jakob Nordström.  On the interplay between proof complexity and SAT solving , 2015, SIGL.

[31]  Russell Impagliazzo,et al.  Using the Groebner basis algorithm to find proofs of unsatisfiability , 1996, STOC '96.

[32]  Prasad Raghavendra,et al.  Graph expansion and the unique games conjecture , 2010, STOC '10.

[33]  R. O'Donnell,et al.  Hypercontractive inequalities via SOS, and the Frankl-Rödl graph , 2012 .

[34]  Pravesh Kothari,et al.  A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[35]  Jakob Nordstr.  PEBBLE GAMES, PROOF COMPLEXITY, AND TIME-SPACE TRADE-OFFS ∗ , 2013 .

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

[37]  Dmitrii V. Pasechnik,et al.  Complexity of semialgebraic proofs , 2002 .

[38]  Uriel Feige,et al.  Resolution lower bounds for the weak pigeon hole principle , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[39]  Toniann Pitassi,et al.  Hardness amplification in proof complexity , 2009, STOC '10.

[40]  Jan Krajícek,et al.  Tautologies from Pseudo-Random Generators , 2001, Bulletin of Symbolic Logic.

[41]  Neil Thapen,et al.  Total Space in Resolution , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[42]  Madhur Tulsiani.  CSP gaps and reductions in the lasserre hierarchy , 2009, STOC '09.

[43]  Alexander A. Razborov.  On the Width of Semi-Algebraic Proofs and Algorithms , 2016, Electron. Colloquium Comput. Complex..

[44]  S. Cook,et al.  Logical Foundations of Proof Complexity: INDEX , 2010 .

[45]  Toniann Pitassi,et al.  Rank Bounds and Integrality Gaps for Cutting Planes Procedures , 2006, Theory Comput..

[46]  Michael Alekhnovich,et al.  Lower bounds for polynomial calculus: non-binomial case , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

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

[48]  Alexander A. Razborov,et al.  Parameterized Bounded-Depth Frege Is Not Optimal , 2011, ICALP.

[49]  Eli Ben-Sasson,et al.  Short proofs are narrow—resolution made simple , 2001, JACM.

[50]  Michael Alekhnovich.  Lower bounds for k-DNF resolution on random 3-CNFs , 2005, STOC.

[51]  Dima Grigoriev,et al.  Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity , 2001, Theor. Comput. Sci..

[52]  David Steurer,et al.  Sum-of-squares proofs and the quest toward optimal algorithms , 2014, Electron. Colloquium Comput. Complex..

[53]  Sanjeev Arora,et al.  Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy , 2005, STOC '05.

[54]  Dima Grigoriev,et al.  Complexity of Semi-algebraic Proofs , 2002, STACS.

[55]  Alexander A. Razborov,et al.  Pseudorandom generators hard for $k$-DNF resolution and polynomial calculus resolution , 2015 .

[56]  Alexander A. Razborov,et al.  Proof Complexity of Pigeonhole Principles , 2001, Developments in Language Theory.

[57]  Jan Krajícek,et al.  Forcing with Random Variables and Proof Complexity , 2006, London Mathematical Society lecture note series.

[58]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

[59]  Michael Alekhnovich,et al.  Space Complexity in Propositional Calculus , 2002, SIAM J. Comput..