ECCC-Report TR15-078https://eccc.weizmann.ac.il/report/2015/078Comments and Revisions published for TR15-078en-usTue, 05 May 2015 02:13:33 +0300
Paper TR15-078
| A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds |
Jakob Nordström,
Mladen Mikša
https://eccc.weizmann.ac.il/report/2015/078We 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. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.Tue, 05 May 2015 02:13:33 +0300https://eccc.weizmann.ac.il/report/2015/078