Unified Characterisations of Resolution Hardness Measures

ACM Trans. Comput. Logic

Mahajan

et al. 2017

Self Cite

The groundbreaking paper 'Short proofs are narrow -resolution made simple' 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 paper we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBF). 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 A Exp+Res and IR-calc. For these systems a mixed picture emerges. Our main results show that both the relations between size and width as well as 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 A Exp+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.

Section: Relations Between Size Width and Space In Classical Resolumentioning

confidence: 99%

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

ACM Trans. Comput. Logic

Mahajan

et al. 2017

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“…for Haken's exponential bound for the pigeonhole principle in dag-like Resolution [36], or the optimal bound in tree-like Resolution [14], and even work for systems stronger than classical Resolution [5] and other measures such as proof space [24] and width [1]. A unified game-theoretic approach was recently established in [17]. Building on the classic game of Pudlák and Impagliazzo [38] for tree-like Resolution, the papers [14,16] devise an asymmetric Prover-Delayer game, which was shown in [15] to even characterise tree-like Resolution size.…”

Section: O Beyersdorff Et Al / Journal Of Computer and System Scienmentioning

confidence: 99%

A Game Characterisation of Tree-like Q-resolution Size

Language and Automata Theory and Applications

Sreenivasaiah

2015

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We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover-Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from [10] for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) [29] for tree-like QU-Resolution.

“…These include the seminal size-width relationship (Ben-Sasson and Wigderson, 2001), the feasible interpolation technique (Krajíček, 1997), or game-theoretic techniques (cf. the recent overview in (Beyersdorff and Kullmann, 2014)). …”

Section: Introductionmentioning

confidence: 99%

Lower Bounds

Bonacina

Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

2016

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A general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from circuit complexity to proof complexity has been effective, a formal connection between the two areas has never been established so far.Here we provide such a formal relation between lower bounds for circuit classes and lower bounds for Frege systems for quantified Boolean formulas (QBF).Starting from a propositional proof system P we exhibit a general method how to obtain a QBF proof system P + ∀red, which is inspired by the transition from resolution to Q-resolution. For us the most important case is a new and natural hierarchy of QBF Frege systems C-Frege + ∀red that parallels the well-studied propositional hierarchy of C-Frege systems, where lines in proofs are restricted to a circuit class C.Building on earlier work for resolution (Beyersdorff, Chew, and Janota, 2015a) we establish a lower bound technique via strategy extraction that transfers arbitrary lower bounds for the circuit class C to lower bounds in C-Frege + ∀red.By using the full spectrum of state-of-the-art circuit lower bounds, our new lower bound method leads to very strong lower bounds for QBF Frege systems: In the propositional case, all these results correspond to major open problems.