Lower bounds for cutting planes proofs with small coefficients

Abstract We consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

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

[2]  Peter Clote,et al.  Cutting planes, connectivity, and threshold logic , 1996, Arch. Math. Log..

[3]  A. Razborov.  Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[4]  A A Razborov.  Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[5]  Jan Krajícek,et al.  Some Consequences of Cryptographical Conjectures for S_2^1 and EF , 1994, LCC.

[6]  Russell Impagliazzo,et al.  Upper and lower bounds for tree-like cutting planes proofs , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

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

[8]  Jan Kra,et al.  Lower Bounds to the Size of Constant-depth Propositional Proofs , 1994 .

[9]  Paul Beame,et al.  Randomized versus nondeterministic communication complexity , 1992, STOC '92.

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

[11]  Peter Clote,et al.  Cutting planes and constant depth Frege proofs , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[12]  Andreas Goerdt.  Cuting Plane Versus Frege Proof Systems , 1990, CSL.

[13]  Ran Raz,et al.  Monotone circuits for matching require linear depth , 1990, STOC '90.

[14]  Ran Raz,et al.  Probabilistic communication complexity of Boolean relations , 1989, 30th Annual Symposium on Foundations of Computer Science.

[15]  Mauricio Karchmer,et al.  Communication complexity - a new approach to circuit depth , 1989 .

[16]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

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

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

[19]  Samuel R. Buss,et al.  Polynomial size proofs of the propositional pigeonhole principle , 1987, Journal of Symbolic Logic.

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

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

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

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

[24]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[25]  S. Vajda.  Recent Advances in Mathematical Programming , 1964 .

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

[27]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .