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  1. Restarts are a widely-used class of techniques integral to the efficiency of Conflict-Driven Clause Learning (CDCL) Boolean SAT solvers. While the utility of such policies has been well-established empirically, a theoretical understanding of whether restarts are indeed crucial to the power of CDCL solvers is missing. In this paper, we prove a series of theoretical results that characterize the power of restarts for various models of SAT solvers. More precisely, we make the following contributions. First, we prove an exponential separation between a drunk randomized CDCL solver model with restarts and the same model without restarts using a family ofmore »satisfiable instances. Second, we show that the configuration of CDCL solver with VSIDS branching and restarts (with activities erased after restarts) is exponentially more powerful than the same configuration without restarts for a family of unsatisfiable instances. To the best of our knowledge, these are the first separation results involving restarts in the context of SAT solvers. Third, we show that restarts do not add any proof complexity-theoretic power vis-a-vis a number of models of CDCL and DPLL solvers with non-deterministic static variable and value selection.« less
  2. We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: • We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there aremore »no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. • We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.« less