Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ ( √ n) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordström 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.

Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and trade-offs between proof size and proof space.

We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω ( √ m). Trade-offs optimality follows from our result and from the short refutations of Graph Ordering Principle in [Bonet and Galesi 1999; 2001]. We then introduce the algebraic proof system PCRk which combines together Polynomial Calculus (PC) and k-DNF Resolution (RESk). We show a size hierarchy theorem for PCRk: PCRk is exponentially separated from PCRk+1. This follows from the previous degree lower bound and from techniques developed for RESk. Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in PCRk.

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Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [16] (FOCS’07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles. We broadly investigate Parameterized Resolution obtaining the following main results: ∙ We introduce a purely combinatorial approach to obtain lower bounds to the proof size in tree-like Parameterized Resolution. For this we devise a new asymmetric Prover-Delayer game which characterizes proofs in (parameterized) tree-like Resolution. By exhibiting good Delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle. ∙ Interpreting a well-known FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution, we devise a proof search algorithm for Parameterized Resolution and show that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF’s. ∙ We answer a question posed by Dantchev, Martin, and Szeider in [16] showing that dag-like Parameterized Resolution is not fpt-bounded. We obtain this result by proving that the pigeonhole principle requires proofs of size

In this note we show that the asymmetric Prover-Delayer game developed in [3] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, Ω(

This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP.

We examine the proof-theoretic strength of parameterized tree-like resolution—a proof system for the coW[2]-complete set of parameterized tautologies. Parameterized resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS’07). In that paper, Dantchev et al. show a complexity gap in parameterized tree-like resolution for propositional formulas arising from translations of firstorder principles. Here we pursue a purely combinatorial approach to obtain lower bounds to the proof size in parameterized treelike resolution. For this we devise a prover-delayer game suitable for parameterized resolution. By exhibiting good delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle. On the other hand, we demonstrate that parameterized tree-like resolution is a very powerful system, as it allows short refutations of all parameterized contradictions given as bounded-width CNF’s. Thus, a number of principles such as Tseitin tautologies, pebbling contradictions, or random 3-CNF’s which serve as hard examples for classical resolution become easy in the parameterized setting.

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