We present the best known separation between tree-like and general resolution, improving on the recent exp(n ) separation of [BEGJ98].

An exponential lower bound for the size of tree-like Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between tree-like resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13].

Dedicated to the memory of Misha Alekhnovich Abstract: This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial. ACM Classification: F.2.2, F.2.3 AMS Classification: 03F20, 68Q17 Key words and phrases: resolution, proof complexity, lower bounds

We prove an exponential lower bound for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution from (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Putnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...

We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their kind, have implications on the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof

We consider the problem of establishing jam-resistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.

Karchmer, Raz, and Wigderson, 1991, discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires dk 2 \Omega\Gamma119 n= log log n) depth. This would separate AC from NC . They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk \Gamma O(d ) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits k) circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds.

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.

In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the runtime of the Grobner basis algorithm for instances of satisfiability, and also on the degree of Polynomial Calculus (PC) and Nullstellensatz refutations of unsatisfiable formulas. We show that the Nullstellensatz degree is equivalent to the homogenized PC degree. Using this relationship, we prove nearly linear separations between Nullstellensatz and PC degree, for 3CNF formulas. Research partially supported by NSF grant CCR-9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation. + Research supported by NSF CCR-9734911, Sloan Research Fellowship BR-3311, and by a cooperative research grant INT9600919 /ME-103 from NSF and the M SMT (Czech Republic), and USA-Israel-BSF Grant 97-00188 # Research supported by NSF grant CCR-9457782 and US-I...

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.