Hidden Structure in Unsatisfiable Random 3-SAT: an Empirical Study

Recent advances in propositional satisfiability (SAT) include studying the hidden structure of unsatisfiable formulas, i.e. explaining why a given formula is unsatisfiable. Although theoretical work on the topic has been developed in the past, only recently two empirical successful approaches have been proposed: extracting unsatisfiable cores and identifying strong backdoors. An unsatisfiable core is a subset of clauses that defines a sub-formula that is also unsatisfiable, whereas a strong backdoor defines a subset of variables which assigned with all values allow concluding that the formula is unsatisfiable. The contribution of this paper is two-fold. First, we study the relation between the search complexity of unsatisfiable random 3-SAT formulas and the sizes of unsatisfiable cores and strong backdoors. For this purpose, we use an existing algorithm which uses an approximated approach for calculating these values. Second, we introduce a new algorithm that optimally reduces the size of unsatisfiable cores and strong backdoors, thus giving more accurate results. Experimental results indicate that the search complexity of unsatisfiable random 3-SAT formulas is related with the size of unsatisfiable cores and strong backdoors. 1

A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories

Finding small unsatisfiable cores for SAT problems has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Surprisingly, the problem of finding unsatisfiable cores in SMT has received very little attention in the literature; in particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. In this paper we present a novel approach to this problem. The main idea is to combine an SMT solver with an external propositional core extractor: the SMT solver produces the theory lemmas found during the search; the core extractor is then called on the boolean abstraction of the original SMT problem and of the theory lemmas. This results in an unsatisfiable core for the original SMT problem, once the remaining theory lemmas have been removed. The approach is conceptually interesting, since the SMT solver is used to dynamically lift the suitable amount of theory information to the boolean level, and it also has several advantages in practice. In fact, it is extremely simple to implement and to update, and it can be interfaced with every propositional core extractor in a plug-and-play manner, so that to benefit for free of all unsat-core reduction techniques which have been or will be made available. We have evaluated our approach by an extensive empirical test on SMT-LIB benchmarks, which confirms the validity and potential of this approach

Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least a^a^...^a clauses, where a is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between title of the paper and title of the arxiv submissio

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

SAT-based Explicit LTL Reasoning

We present here a new explicit reasoning framework for linear temporal logic (LTL), which is built on top of propositional satisfiability (SAT) solving. As a proof-of-concept of this framework, we describe a new LTL satisfiability tool, Aalta\_v2.0, which is built on top of the MiniSAT SAT solver. We test the effectiveness of this approach by demonnstrating that Aalta\_v2.0 significantly outperforms all existing LTL satisfiability solvers. Furthermore, we show that the framework can be extended from propositional LTL to assertional LTL (where we allow theory atoms), by replacing MiniSAT with the Z3 SMT solver, and demonstrating that this can yield an exponential improvement in performance