335 research outputs found
Feasible Interpolation for QBF Resolution Calculi
In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF proof systems as well as largely extends the scope of classical feasible interpolation. We apply our technique to obtain new exponential lower bounds to all resolution-based QBF systems for a new class of QBF formulas based on the clique problem. Finally, we show how feasible interpolation relates to the recently established lower bound method based on strategy extraction
Are Short Proofs Narrow? QBF Resolution is not so Simple
The ground-breaking paper âShort Proofs Are Narrow -- Resolution Made Simpleâ by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in their fundamental work, Atserias and Dalmau (J. Comput. Syst. Sci. 2008) show that lower bounds for space again can be obtained via lower bounds for width. In this article, we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBFs). There are a number of different QBF resolution calculi like Q-resolution (the classical extension of propositional resolution to QBF) and the more recent calculi âExp+Res and IR-calc. For these systems, a mixed picture emerges. Our main results show that the relations both between size and width and between space and width drastically fail in Q-resolution, even in its weaker tree-like version. On the other hand, we obtain positive results for the expansion-based resolution systems âExp+Res and IR-calc, however, only in the weak tree-like models. Technically, our negative results rely on showing width lower bounds together with simultaneous upper bounds for size and space. For our positive results, we exhibit space and width-preserving simulations between QBF resolution calculi
Proof complexity of resolution-based QBF calculi
Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths of proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems as long-distance Q-resolution and extensions. With a completely different lower bound argument we show the hardness of the prominent formulas of Kleine BĂŒning et al. [34] for the strong expansion-based calculus IR-calc. Our lower bounds imply new exponential separations between two different types of resolution-based QBF calculi: proof systems for CDCLbased solvers (Q-resolution, long-distance Q-resolution) and proof systems for expansion-based solvers (âExp+Res and its generalizations IR-calc and IRM-calc). The relations between proof systems from the two different classes were not known before
Conformant Planning as a Case Study of Incremental QBF Solving
We consider planning with uncertainty in the initial state as a case study of
incremental quantified Boolean formula (QBF) solving. We report on experiments
with a workflow to incrementally encode a planning instance into a sequence of
QBFs. To solve this sequence of incrementally constructed QBFs, we use our
general-purpose incremental QBF solver DepQBF. Since the generated QBFs have
many clauses and variables in common, our approach avoids redundancy both in
the encoding phase and in the solving phase. Experimental results show that
incremental QBF solving outperforms non-incremental QBF solving. Our results
are the first empirical study of incremental QBF solving in the context of
planning and motivate its use in other application domains.Comment: added reference to extended journal article; revision (camera-ready,
to appear in the proceedings of AISC 2014, volume 8884 of LNAI, Springer
Propagators and Solvers for the Algebra of Modular Systems
To appear in the proceedings of LPAR 21.
Solving complex problems can involve non-trivial combinations of distinct
knowledge bases and problem solvers. The Algebra of Modular Systems is a
knowledge representation framework that provides a method for formally
specifying such systems in purely semantic terms. Formally, an expression of
the algebra defines a class of structures. Many expressive formalism used in
practice solve the model expansion task, where a structure is given on the
input and an expansion of this structure in the defined class of structures is
searched (this practice overcomes the common undecidability problem for
expressive logics). In this paper, we construct a solver for the model
expansion task for a complex modular systems from an expression in the algebra
and black-box propagators or solvers for the primitive modules. To this end, we
define a general notion of propagators equipped with an explanation mechanism,
an extension of the alge- bra to propagators, and a lazy conflict-driven
learning algorithm. The result is a framework for seamlessly combining solving
technology from different domains to produce a solver for a combined system.Comment: To appear in the proceedings of LPAR 2
Genuine Lower Bounds for QBF Expansion
We propose the first general technique for proving genuine lower bounds in expansion-based QBF proof systems. We present the technique in a framework centred on natural properties of winning strategies in the 'evaluation game' interpretation of QBF semantics. As applications, we prove an exponential proof-size lower bound for a whole class of formula families, and demonstrate the power of our approach over existing methods by providing alternative short proofs of two known hardness results. We also use our technique to deduce a result with manifest practical import: in the absence of propositional hardness, formulas separating the two major QBF expansion systems must have unbounded quantifier alternations
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