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Understanding Gentzen and Frege Systems for QBF

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Understanding Gentzen and Frege Systems for QBF Understanding Gentzen and Frege Systems for QBF Olaf Beyersdorff Jan´ Pich School of Computing, University of Leeds, United Kingdom fo.beyersdorff,[email protected] Abstract [19]), to first-order logic (theories of bounded arithmetic [18, 30]), Recently Beyersdorff, Bonacina, and Chew [9] introduced a natural as well as to practical SAT- and QBF-solving [14]. class of Frege systems for quantified Boolean formulas (QBF) While the bulk of activity in proof complexity concerns proposi- and showed strong lower bounds for restricted versions of these tional proofs, there has been intense research during the last decade systems. Here we provide a comprehensive analysis of the new employing proof-complexity methods to further logics, most no- tably non-classical logics (cf. [6]) and proof complexity of quanti- extended Frege system from [9], denoted EF + 8red, which is a natural extension of classical extended Frege EF. fied Boolean formulas (QBF). Our main results are the following: Firstly, we prove that the Recent research in QBF proof complexity has been largely trig- ∗ gered by exciting advances in QBF solving—powerful algorithms standard Gentzen-style system G1 p-simulates EF + 8red and that ∗ that solve large classes of formulas from industrial applications. G1 is strictly stronger under standard complexity-theoretic hard- ness assumptions. Compared to SAT solving, due to the PSPACE completeness of QBF the success of QBF solvers even extends to further fields such Secondly, we show a correspondence of EF + 8red to bounded as planning [23, 34] and formal verification. To model the strengths arithmetic: EF + 8red can be seen as the non-uniform propositional 1 1 of modern QBF solvers, a number of resolution-based proof sys- version of intuitionistic S2 . Specifically, intuitionistic S2 proofs of arbitrary statements in prenex form translate to polynomial-size tems have been recently suggested and analysed from a proof com- plexity perspective (cf. [3, 7, 8, 10]). EF + 8red proofs, and EF + 8red is in a sense the weakest system with this property. While we have a relatively good understanding of these weak resolution-type systems, much less is known for strong proof sys- Finally, we show that unconditional lower bounds for EF + 8red would imply either a major breakthrough in circuit complexity or tems, and this judgement applies to both propositional and QBF in classical proof complexity, and in fact the converse implications proof complexity. There are two main approaches for designing strong calculi: via sequent-style systems (Gentzen’s LK [24]) and hold as well. Therefore, the system EF + 8red naturally unites the central problems from circuit and proof complexity. axiom-rule based systems known as Frege or Hilbert-type calculi Technically, our results rest on a formalised strategy extraction [19]. In propositional logic, both Gentzen and Frege systems are 1 equivalent from a proof complexity point of view [19, 30]. theorem for EF + 8red akin to witnessing in intuitionistic S and a 2 The situation is more intricate for QBF; and indeed the main normal form for EF + 8red proofs. aim of the present paper is to shed light on this topic. Categories and Subject Descriptors F.2.2 [Analysis of algo- Gentzen systems for QBF were already introduced in the late rithms and problem complexity]: Nonnumerical Algorithms and 80’s by Kraj´ıcekˇ and Pudlak´ [31], of which we use slightly modi- ∗ Problems—Complexity of proof procedures fied versions Gi and Gi due to Cook and Morioka [17]. These sys- tems are known to be strictly more powerful than QBF resolution General Terms Proof complexity, bounded arithmetic, quantified [22], but lower bounds are out of reach with current techniques. Boolean formulas As for strong propositional systems, the main source of infor- Keywords QBF proof systems, sequent calculus, Frege systems, mation on QBF Gentzen systems stems from their correspondence intuitionistic logic, strategy extraction, lower bounds, simulations to Buss’ theories of bounded arithmetic [12, 17, 31]. This corre- spondence allows to translate first-order formulas into sequences i i of QBFs, and indeed first-order proofs in S2 or T2 to polynomial- 1. Introduction ∗ size Gi or Gi proofs, respectively [17, 31], thus providing the main Proof complexity addresses the main question of how hard it is to tool to construct short propositional proofs. prove theorems in a given calculus, in particular: what is the length On the other hand, QBF Frege systems were only developed of the shortest proof of a given theorem in a fixed formal system, very recently [9]. Their definition is very elegant, adding to classi- typically comprised of axioms and rules. This research bears tight cal Frege just one single 8red rule for managing quantifiers, lead- and fruitful connections to computational complexity (separating ing to the QBF system Frege + 8red. Alternatively, they can be seen complexity classes in an approach known as Cook’s programme as substitution Frege systems with substitutions allowed just for universally quantified variables. As for classical Frege, the strength of Frege + 8red can be cal- ibrated by allowing different classes of formulas (or more directly Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without Boolean circuits [27]) as their underlying objects. With a tech- fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice nique [7, 9] uncovering a direct relation between circuit complex- and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to ity and proof complexity very strong lower bounds have been ob- lists, contact the Owner/Author. Request permissions from [email protected] or Publications Dept., ACM, Inc., fax +1 (212) 869-0481. Copyright 20yy held by Owner/Author. Publication Rights Licensed to ACM. tained for QBF Frege, the strongest of which yields an exponen- 0 CONF ’yy Month d–d, 20yy, City, ST, Country tial lower bound for AC [p]-Frege + 8red. In sharp contrast, the Copyright c 20yy ACM 978-1-nnnn-nnnn-n/yy/mm. $15.00 0 DOI: http://dx.doi.org/10.1145/nnnnnnn.nnnnnnn strongest lower bound in the propositional world holds for AC - and in a certain sense EF + 8red is the strongest proof system that 1 is provably sound in the theory IS 2. G1 Technically, the correspondence as well as the simulation results p-simulation mentioned under A. above rest on a formalisation of the Strategy G0 Extraction Theorem for QBF Frege systems from [9]. This strategy ∗ G1 Theorem 5.2 extraction result states that for formulas provable in EF + 8red one (p-simulation is open) can compute witnesses for all existential quantifiers with Boolean ∗ G0 EF + 8red circuits that can be efficiently extracted from the EF + 8red proof. We provide two formalisations for this result: one in first-order 1 Frege + 8red logic, where we formalise strategy extraction in S2 (Theorem 4.1), and a second more direct one, where we construct Frege proofs for Figure 1. The simulation order of QBF Gentzen and Frege systems the witnessing properties (Theorem 4.3). While the second formal- isation applies to more systems and gives the simulation structure detailed in A., the first formalisation is stronger and enables the 1 0 correspondence to IS 2. Frege [1, 32, 33], while lower bounds for the stronger AC [p]- Although intuitionistic bounded arithmetic was already devel- Frege constitute a major problem, open for more than twenty years. oped by Buss in the mid 80’s [13], no QBF counterpart of this This striking development prompts us to target at a better under- theory was found so far—in sharp contrast to most other arithmetic standing of the new QBF Frege systems. What is their relation to theories [18]. As we show here, the missing piece in the puzzle is the well-studied QBF Gentzen calculi? Does QBF Frege also ad- the recent QBF Frege system EF + 8red. 1 mit a correspondence to bounded arithmetic? Can we push lower + 0 Indeed, the appealing link between IS 2 and EF 8red comes bounds even beyond the current state-of-the-art bound for AC [p]- via their witnessing properties: similarly as EF + 8red has strategy Frege + 8red from [9]? 1 extraction for arbitrarily complex QBFs [9], the theory IS 2 admits In this paper we give answers to all of these three questions. a witnessing theorem for arbitrary first-order formulas [21]. 1.1 Our contributions C. Characterising lower bounds for QBF Frege. The main Below we summarise our main contributions of this paper, sketch- question left open by the recent advances in strong QBF lower ing the main results and techniques. bounds [9] is whether unconditional lower bounds can be obtained for Frege + 8red or even EF + 8red. We show here that such a result A. Gentzen vs. Frege in QBF: simulations and separations. would imply either a major breakthrough in circuit complexity (a In classical proof complexity Frege and Gentzen’s sequent system lower bound for non-uniform NC1 or even P=poly) or a major LK are p-equivalent, i.e., proofs can be efficiently translated be- breakthrough in propositional proof complexity (lower bounds for tween the systems [19]. In contrast, our findings show a more com- classical Frege or even EF); and in fact the opposite implications plex picture for QBF. We concentrate on the most important stan- hold as well (Theorem 8.1). ∗ ∗ dard Gentzen-style systems G0 and G1 as well as the QBF Frege This means that the problem of lower bounds for QBF Frege systems Frege + 8red and EF + 8red, forming QBF analogues of very naturally unites the central problem in circuit complexity with the classical Frege and extended Frege system EF from [19].
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