Lower Bounds for QBFs of Bounded Treewidth

The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen in parameterized complexity is that QSAT when parameterized by the treewidth of the primal graph of the input formula together with the quantifier depth of the formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier depth. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth (and quantifier depth). More formally, we establish lower bounds for QSAT and treewidth, namely, that under ETH there cannot be an algorithm that solves QSAT of quantifier depth i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a versatile reduction technique to compress treewidth that encodes the essence of dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more fine-grained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy

Generalized Craig Interpolation for Stochastic Boolean Satisfiability Problems with Applications to Probabilistic State Reachability and Region Stability

The stochastic Boolean satisfiability (SSAT) problem has been introduced by Papadimitriou in 1985 when adding a probabilistic model of uncertainty to propositional satisfiability through randomized quantification. SSAT has many applications, among them probabilistic bounded model checking (PBMC) of symbolically represented Markov decision processes. This article identifies a notion of Craig interpolant for the SSAT framework and develops an algorithm for computing such interpolants based on a resolution calculus for SSAT. As a potential application area of this novel concept of Craig interpolation, we address the symbolic analysis of probabilistic systems. We first investigate the use of interpolation in probabilistic state reachability analysis, turning the falsification procedure employing PBMC into a verification technique for probabilistic safety properties. We furthermore propose an interpolation-based approach to probabilistic region stability, being able to verify that the probability of stabilizing within some region is sufficiently large

On Subclasses of Minimal Unsatisfiable Formulas

We consider the minimal unsatisfiablity problem MU (k) for propositional formulas in conjunctive normal form (CNF) over n variables and n + k clauses, where k is fixed. It will be shown that MU (k) is in NP. Based on the non--deterministic algorithm we prove for MU(2) that after a simplification by resolving over variables occuring at most once positively or at most once negatively such minimal unsatisfiable formulas have a simple and unique form. This leads immediately to an algorithm solving the minimal unsatisfiabilty problem for formulas with n+2 clauses in time O(n 3 ). keywords: Propositional formulas, minimal unsatisfiability, 1 Introduction A propositional formula in conjunctive normal form (CNF) is minimal unsatisfiable if and only if the formula is unsatisfiable and deleting an arbitrary clause will result in a satisfiable formula. The problem whether an arbitrary formula in CNF is minimal unsatisfiable is known to be D P --complete [PaWo 88], where D P is the class..