A complexity gap for tree resolution

Abstract. This paper shows that any sequence $ \psi_ n $ of tautologies which expresses the validity of a fixed combinatorial principle either is “easy”, i.e. has polynomial size tree-resolution proofs, or is “difficult”, i.e. requires exponential size tree-resolution proofs. It is shown that the class of tautologies which are hard (for tree resolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Further it is shown that the gap phenomenon is valid for tautologies based on infinite mathematical theories (i.e. not just based on a single proposition).¶A corollary to this classification is that it is undecidable whether a sequence $ \psi_ n $ has polynomial size tree-resolution proofs or requires exponential size tree-resolution proofs. It also follows that the degree of the polynomial in the polynomial size (in case it exists) is non-recursive, but semi-decidable.

[1]  Jesper G. Henriksen.  An Expressive Extension of TLC , 1999, ASIAN.

[2]  Alasdair Urquhart,et al.  The Complexity of Propositional Proofs , 1995, Bulletin of Symbolic Logic.

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

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

[5]  Alexander Leitsch,et al.  On Skolemization and Proof Complexity , 1994, Fundam. Informaticae.

[6]  R. Impagliazzo,et al.  Lower bounds on Hilbert's Nullstellensatz and propositional proofs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

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

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

[9]  Moni Naor,et al.  Search Problems in the Decision Tree Model , 1995, SIAM J. Discret. Math..

[10]  Søren Riis,et al.  Count(q) Does Not Imply Count(p) , 1994, Ann. Pure Appl. Log..

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

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

[13]  M. Sitharam,et al.  Uniformly generated submodules of permutation modules , 1998 .

[14]  Miklós Ajtai,et al.  The independence of the modulo p counting principles , 1994, STOC '94.

[15]  Meera Sitharam,et al.  Generating hard tautologies using predicate logic and the symmetric group , 1998, Log. J. IGPL.

[16]  H. L. Hartley,et al.  Manuscript Preparation , 2022 .

[17]  P. Pudlák.  On the length of proofs of finitistic consistency statements in first order theories , 1986 .

[18]  Meera Sitharam,et al.  Non-constant Degree Lower Bounds imply linear Degree Lower Bounds , 1997, Electron. Colloquium Comput. Complex..

[19]  Søren Riis.  A complexity gap for tree resolution , 1999 .

[20]  James M. Crawford,et al.  Experimental Results on the Crossover Point in Random 3-SAT , 1996, Artif. Intell..

[21]  Umesh V. Vazirani,et al.  An Introduction to Computational Learning Theory , 1994 .

[22]  Samuel R. Buss,et al.  Resolution and the Weak Pigeonhole Principle , 1997, CSL.

[23]  Zvi Galil,et al.  On the Complexity of Regular Resolution and the Davis-Putnam Procedure , 1977, Theor. Comput. Sci..

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

[25]  Luca Aceto,et al.  Conservative Extension in Structural Operational Semantics , 1999, Bull. EATCS.

[26]  J. Krajícek.  On the degree of ideal membership proofs from uniform families of polynomials over a finite field , 2001 .

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

[28]  Olivier Danvy,et al.  On proving syntactic properties of CPS programs , 1999, Electron. Notes Theor. Comput. Sci..

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

[30]  Alexander Leitsch,et al.  The Resolution Calculus , 1997, Texts in Theoretical Computer Science An EATCS Series.

[31]  Olivier Danvy,et al.  Lambda-dropping: transforming recursive equations into programs with block structure , 1997, Theor. Comput. Sci..

[32]  Christian N. S. Pedersen,et al.  Finding Maximal Quasiperiodicities in Strings , 1999, CPM.

[33]  Olivier Danvy,et al.  An Extensional Characterization of Lambda-Lifting and Lambda-Dropping , 1998, Fuji International Symposium on Functional and Logic Programming.

[34]  Luca Aceto,et al.  On the Two-Variable Fragment of the Equational Theory of the Max-Sum Algebra of the Natural Numbers , 1999, STACS.

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

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

[37]  Søren Riis.  Count( $q$) versus the pigeon-hole principle , 1997, Arch. Math. Log..

[38]  Michael E. Saks,et al.  On the complexity of unsatisfiability proofs for random k-CNF formulas , 1998, STOC '98.