A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov ’03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can “cluster” clauses and variables in a way that “respects the structure” of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov ’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. 1998 ACM Subject Classification F.2.2[Analysis of Algorithms and Problem Complexity]: [Nonnumerical Algorithms and Problems] —Complexity of proof procedures; F.1.3[Computation by Abstract Devices]: Complexity Measures and Classes; I.2.3[Artificial Intelligence]: Deduction and Theorem Proving; F.4.1[Mathematical Logic and Formal Languages]: Mathematical Logic —computational logic