LIPIcs

We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko ’15] as a tool for obtaining results in cryptography. We consider instead the non- deterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10–15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström ’08, ’11] we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure

Resolution and the binary encoding of combinatorial principles.

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn for m > n. Using the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn requires proofs of size 2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2 p n log n there are quasipolynomial size proofs of Bin-PHPmn in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form and with no finite model

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity

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 are 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

A framework for space complexity in algebraic proof systems

Algebraic proof systems, such as Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR), refute contradictions using polynomials. Space complexity for such systems measures the number of distinct monomials to be kept in memory while verifying a proof. We introduce a new combinatorial framework for proving space lower bounds in algebraic proof systems. As an immediate application, we obtain the space lower bounds previously provided for PC/PCR [Alekhnovich et al. 2002; Filmus et al. 2012]. More importantly, using our approach in its full potential, we prove Ω(n) space lower bounds in PC/PCR for random k-CNFs (k ≥ 4) in n variables, thus solving an open problem posed in Alekhnovich et al. [2002] and Filmus et al. [2012]. Our method also applies to the Graph Pigeonhole Principle, which is a variant of the Pigeonhole Principle defined over a constant (left) degree expander graph