Different Approaches to Proof Systems

The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper

On the Relative Strength of Pebbling and Resolution

The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven 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. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic black-white pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This paper contributes to both these approaches. First, we design a restricted form of black-white pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat black-only pebbling, and in particular that the space lower bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC '10), June 201

A game characterisation of tree-like Q-Resolution size

We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover–Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from Beyersdorff et al. (2015) for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) for tree-like QU-Resolution

Complexity measures for resolution

Esta obra es una contribución al campo de la Complejidad de la Demostración, que estudia la complejidad de los sistemas de demostración en términos de los recursos necesarios para demostrar o refutar fórmulas proposicionales. La Complejidad de la Demostración es un interesante campo relacionado con otros campos de la Informática como la Complejidad Computacional o la Demostración Automática entre otros. Esta obra se centra en medidas de complejidad para sistemas de demostración refutacionales para fórmulas en FNC. Consideramos varios sistemas de demostración, concretamente Resolución, R(k) y Planos Secantes y nuestros resultados hacen referencia a las medidas de complejidad de tamaño y espacio.Mejoramos separaciones de tamaño anteriores entre las versiones generales y arbóreas de Resolución y Planos Secantes. Para hacerlo, extendemos una cota inferior de tamaño para circuitos monótonos booleanos de Ran y McKenzie a circuitos monótonos reales. Este tipo de separaciones es interesante porque algunos demostradores automáticos se basan en la versión arbórea de sistemas de demostración, por tanto la separación indica que no es siempre una buena idea restringirnos a la versión arbórea.Tras la reciente aparición de R(k), que es un sistema de demostración entre Resolución y Frege con profundidad acotada, era importante estudiar cuan potente es y su relación con otros sistemas de demostración. Resolvemos un problema abierto propuesto por Krajícek, concretamente mostramos que R(2) no tiene la propiedad de la interpolación monónota factible. Para hacerlo, mostramos que R(2) es estrictamente más potente que Resolución.Una pregunta natural es averiguar si se pueden separar sucesivos niveles de R(k) o R(k) arbóreo. Mostramos separaciones exponenciales entre niveles sucesivos de lo que podemos llamar la jerarquía R(k) arbórea. Esto significa que hay formulas que requieren refutaciones de tamaño exponencial en R(k) arbóreo, pero tienen refutaciones de tamaño polinómico en R(k+1) arbóreo. Propusimos una nueva definición de espacio para Resolución mejorando la anterior de Kleine-Büning y Lettmann. Dimos resultados generales sobre el espacio para Resolución y Resolución arbórea y también una caracterización combinatoria del espacio para Resolución arbórea usando un juego con dos adversarios para fórmulas en FNC. La caracterización permite demostrar cotas inferiores de espacio para la Resolución arbórea sin necesidad de usar el concepto de Resolución o Resolución arbórea. Durante mucho tiempo no se supo si el espacio para Resolución y Resolución arbórea coincidían o no. Hemos demostrado que no coinciden al haber dado la primera separación entre el espacio para Resolución y Resolución arbórea.También hemos estudiado el espacio para R(k). Demostramos que al igual que pasaba con el tamaño, R(k) arbóreo también forma una jerarquía respecto alespacio. Por tanto, hay fórmulas que necesitan espacio casi lineal en R(k) arbóreo mientras que tienen refutaciones en R(k+1) arbóreo con espacio contante. Extendemos todas las cotas inferiores de espacio para Resolución conocidas a R(k) de una forma sencilla y unificada, que también sirve para Resolución, usando el concepto de satisfactibilidad dinámica presentado en esta obra.This work is a contribution to the field of Proof Complexity, which studies the complexity of proof systems in terms of the resources needed to prove or refute propositional formulas. Proof Complexity is an interesting field which has several connections to other fields of Computer Science like Computational Complexity or Automatic Theorem Proving among others. This work focuses in complexity measures for refutational proof systems for CNF formulas. We consider several proof systems, namely Resolution, R(k) and Cutting Planes and our results concern mainly to the size and space complexity measures. We improve previous size separations between treelike and general versions of Resolution and Cutting Planes. To do so we extend a size lower bound for monotone boolean circuits by Raz and McKenzie, to monotone real circuits. This kind of separations is interesting because some automated theorem provers rely on the treelike version of proof systems, so the separations show that is not always a good idea to restrict to the treelike version. After the recent apparition of R(k) which is a proof system lying between Resolution and bounded-depth Frege it was important to study how powerful it is and its relation with other proof systems. We solve an open problem posed by Krajícek, namely we show that R(2) does not have the feasible monotone interpolation property. To do so, we show that R(2) is strictly more powerful than Resolution. A natural question is to find out whether we can separate successive levels of R(k) or treelike R(k). We show exponential separations between successive levels of what we can call now the treelike R(k) hierarchy. That means that there are formulas that require exponential size treelike R(k) refutations whereas they have polynomial size treelike R(k+1) refutations. We have proposed a new definition for Resolution space improving a previous one from Kleine-Büning and Lettmann. We give general results for Resolution and treelike Resolution space and also a combinatorial characterization of treelike Resolution space via a Player-Adversary game over CNF formulas. The characterization allows to prove lower bounds for treelike Resolution space with no need to use the concept of Resolution or Resolution refutations at all. For a long time it was not known whether Resolution space and treelike Resolution space coincided or not. We have answered this question in the negative because we give the first space separation from Resolution to treelike Resolution. We have also studied space for R(k). We show that, as happened with respect to size, treelike R(k) forms a hierarchy respect to space. So, there are formulas that require nearly linear space for treelike R(k) whereas they have constant space treelike R(k+1) refutations. We extend all known Resolution space lower bounds to R(k) in an easier and unified way, that also holds for Resolution, using the concept of dynamical satisfiability introduced in this work