Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We prove a size-depth trade-oo upper bound for monotone proofs of the standard encoding of the PHP as a monotone sequent. At one extreme of the trade-oo we get quasipolynomial-size monotone proofs, and at the other extreme we get subexponential-size bounded-depth monotone proofs. This result is a consequence of deriving the basic properties of certain monotone formulas computing the boolean threshold functions. We also consider the monotone sequent expressing the Clique-Coclique Principle (CLIQUE) deened by Bonet, Pitassi and Raz (1997). We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one-to-one and onto PHP, and so CLIQUE also has quasipolynomial-size monotone proofs. As a consequence of our results, Resolution , Cutting Planes with polynomially bounded coeecients, and Bounded-Depth Frege are exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudll ak.

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

[2]  E. Szemerédi,et al.  O(n LOG n) SORTING NETWORK. , 1983 .

[3]  Russell Impagliazzo,et al.  Lower bounds for the polynomial calculus and the Gröbner basis algorithm , 1999, computational complexity.

[4]  A RazborovAlexander.  Lower bounds for the polynomial calculus , 1998 .

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

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

[7]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[8]  Toniann Pitassi,et al.  Propositional Proof Complexity: Past, Present and Future , 2001, Bull. EATCS.

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

[10]  Heribert Vollmer,et al.  Introduction to Circuit Complexity , 1999, Texts in Theoretical Computer Science An EATCS Series.

[11]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

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

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

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

[15]  Peter Clote,et al.  On PHP st-connectivity, and odd charged graphs , 1996, Proof Complexity and Feasible Arithmetics.

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

[17]  Toniann Pitassi,et al.  Approximation and Small-Depth Frege Proofs , 1992, SIAM J. Comput..

[18]  Jeff B. Paris,et al.  Provability of the Pigeonhole Principle and the Existence of Infinitely Many Primes , 1988, J. Symb. Log..

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

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

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

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

[23]  Alexander A. Razborov,et al.  Read-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus , 1997, STOC '97.

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

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

[26]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, Symposium on the Theory of Computing.

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

[28]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

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

[31]  Toniann Pitassi,et al.  A new proof of the weak pigeonhole principle , 2000, STOC '00.

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

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

[34]  Samuel R. Buss,et al.  Resolution Proofs of Generalized Pigeonhole Principles , 1988, Theor. Comput. Sci..