A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the size-width relationship is tight.

[1]  Alasdair Urquhart,et al.  Formal Languages]: Mathematical Logic--mechanical theorem proving , 2022 .

[2]  Phokion G. Kolaitis,et al.  On the Expressive Power of Datalog: Tools and a Case Study , 1995, J. Comput. Syst. Sci..

[3]  Albert Atserias.  Unsatisfiable random formulas are hard to certify , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[4]  Pavel Pudlák.  Proofs as Games , 2000, Am. Math. Mon..

[5]  Alexander A. Razborov.  Resolution lower bounds for perfect matching principles , 2004, J. Comput. Syst. Sci..

[6]  Alexander A. Razborov,et al.  Electronic Colloquium on Computational Complexity, Report No. 75 (2001) Resolution Lower Bounds for the Weak Functional Pigeonhole Principle , 2001 .

[7]  Phokion G. Kolaitis,et al.  A Game-Theoretic Approach to Constraint Satisfaction , 2000, AAAI/IAAI.

[8]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[9]  Phokion G. Kolaitis,et al.  Conjunctive-Query Containment and Constraint Satisfaction , 2000, J. Comput. Syst. Sci..

[10]  Martin Grohe,et al.  Finite Variable Logics in Descriptive Complexity Theory , 1998, Bulletin of Symbolic Logic.

[11]  Maria Luisa Bonet,et al.  Optimality of size-width tradeoffs for resolution , 2001, computational complexity.

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

[13]  Phokion G. Kolaitis,et al.  Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics , 2002, CP.

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

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

[16]  Toniann Pitassi,et al.  Simplified and improved resolution lower bounds , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

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

[18]  Jacobo Torán.  Lower Bounds for Space in Resolution , 1999, CSL.

[19]  Eli Ben-Sasson,et al.  Space complexity of random formulae in resolution , 2003, Random Struct. Algorithms.

[20]  Søren Riis.  A complexity gap for tree resolution , 2001, computational complexity.

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

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

[23]  Albert Atserias,et al.  On sufficient conditions for unsatisfiability of random formulas , 2004, JACM.

[24]  Samuel R. Buss,et al.  How to Lie Without Being (Easily) Convicted and the Length of Proofs in Propositional Calculus , 1994, CSL.

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

[26]  Maria Luisa Bonet,et al.  On the automatizability of resolution and related propositional proof systems , 2002, Inf. Comput..

[27]  Michael Alekhnovich,et al.  Pseudorandom Generators in Propositional Proof Complexity , 2004, SIAM J. Comput..

[28]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..