Trade-Offs Between Size and Degree in Polynomial Calculus

Building on [Clegg et al. \u2796], [Impagliazzo et al. \u2799] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(?(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen \u2716], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexity measure that works against any resolution refutation---previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods

Space complexity in polynomial calculus

During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well understood from this point of view. For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]: 1. We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHPm n with m pigeons and n holes, and show that this is tight. 2. For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. ’02] measured in the width of the formulas. 3. We then present another encoding of the pigeonhole principle that has constant width, and prove an Ω(n) space lower bound in PCR for these formulas as well. 4. Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity

Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are the amounts of time and memory used. In proof complexity, these resources correspond to the length and space of resolution proofs. There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution. Our collection of trade-offs cover almost the whole range of values for the space complexity of formulas, and most of the trade-offs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these trade-offs in fact extend to the exponentially stronger k-DNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question whether the k-DNF resolution systems form a strict hierarchy with respect to space in the affirmative. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formula F can be transformed by simple variable substitution into a new formula F' such that if F has the right properties, F' can be proven in essentially the same length as F, whereas on the other hand the minimal number of lines one needs to keep in memory simultaneously in any proof of F' is lower-bounded by the minimal number of variables needed simultaneously in any proof of F. Applying this theorem to so-called pebbling formulas defined in terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical reports TR09-034 and TR09-047, and it hence subsumes these two report

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples

Understanding space in resolution: optimal lower bounds and exponential trade-offs

We continue the study of tradeoffs between space and length of resolution proofs and focus on two new results: begin{enumerate} item We show that length and space in resolution are uncorrelated. This is proved by exhibiting families of CNF formulas of size $O(n)$ that have proofs of length $O(n)$ but require space $Omega(n / log n)$. Our separation is the strongest possible since any proof of length $O(n)$ can always be transformed into a proof in space $O(n / log n)$, and improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and H{aa}stad 2008]. item We prove a number of trade-off results for space in the range from constant to $O(n / log n)$, most of them superpolynomial or even exponential. This is a dramatic improvement over previous results in [Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007]. end{enumerate} The key to our results is the following, somewhat surprising, theorem: Any CNF formula $F$ can be transformed by simple substitution transformation into a new formula $F\u27$ such that if $F$ has the right properties, $F\u27$ can be proven in resolution in essentially the same length as $F$ but the minimal space needed for $F\u27$ is lower-bounded by the number of variables that have to be mentioned simultaneously in any proof for $F$. Applying this theorem to so-called pebbling formulas defined in terms of pebble games over directed acyclic graphs and analyzing black-white pebbling on these graphs yields our results

Electronic Colloquium on Computational Complexity, Report No. 114 (2007) A Simplified Way of Proving Trade-off Results for Resolution

We present a greatly simplified proof of the length-space trade-off result for resolution in Hertel and Pitassi (2007), and also prove a couple of other theorems in the same vein. We point out two important ingredients needed for our proofs to work, and discuss possible conclusions to be drawn regarding proving trade-off results for resolution. Our key trick is to look at formulas of the type F = G∧H, where G and H are over disjoint sets of variables and have very different length-space properties with respect to resolution. This trick is not present in the proof of Hertel and Pitassi, and thus their techniques can likely be used to prove results not obtainable by our methods. In these notes, we present a simplification of the length-space trade-off result for resolution in [9] (soon to appear together with [8] as [10]), and show how the same ideas can be used to prove other related theorems. The simplified proof is given in Section 1. In Section 2 we prove two other tradeoff results of a similar flavour. We point out two key ingredients needed for our proofs to work in Sections 3 and 4, and discuss possible conclusions to be drawn regarding proving trade-off results for resolution. Finally, in Section 5 we mention a couple of open problems that seem both natura