Understanding Gentzen and Frege Systems for QBF

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Understanding Gentzen and Frege Systems for QBF

Olaf Beyersdorff

J´an Pich

School of Computing, University of Leeds, United Kingdom

{o.beyersdorff,j.pich}@leeds.ac.uk

Abstract

Recently Beyersdorff, Bonacina, and Chew [10] introduced a nat-ural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new ex-tended Frege system from [10], denotedEF+_∀red, which is a nat-ural extension of classical extended FregeEF.

Our main results are the following: Firstly, we prove that the standard Gentzen-style systemG∗1 p-simulatesEF+∀redand that G∗1 is strictly stronger under standard complexity-theoretic

hard-ness assumptions.

Secondly, we show a correspondence ofEF+∀redto bounded arithmetic:EF+∀redcan be seen as the non-uniform propositional version of intuitionistic S₂1_{. Specifically, intuitionistic}S₂1 _proofs

of arbitrary statements in prenex form translate to polynomial-size

EF+∀redproofs, andEF+∀redis in a sense the weakest system with this property.

Finally, we show that unconditional lower bounds forEF+_∀red

would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the systemEF+_∀rednaturally unites the central problems from circuit and proof complexity.

Technically, our results rest on a formalised strategy extraction theorem forEF+_∀redakin to witnessing in intuitionisticS₂1_{and a}

normal form forEF+_∀redproofs.

Categories and Subject Descriptors F.2.2 [Analysis of algo-rithms and problem complexity]: Nonnumerical Algoalgo-rithms and Problems—Complexity of proof procedures

General Terms Proof complexity, bounded arithmetic, quantified Boolean formulas

Keywords QBF proof systems, sequent calculus, Frege systems, intuitionistic logic, strategy extraction, lower bounds, simulations

1.

Introduction

Proof complexityaddresses the main question of how hard it is to prove theorems in a given calculus, in particular: what is the length of the shortest proof of a given theorem in a fixed formal system, typically comprised of axioms and rules. This research bears tight and fruitful connections to computational complexity (separating complexity classes in an approach known as Cook’s programme

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[20]), to first-order logic (theories of bounded arithmetic [19, 31]), as well as to practical SAT- and QBF-solving [15].

While the bulk of activity in proof complexity concerns proposi-tional proofs, there has been intense research during the last decade employing proof-complexity methods to further logics, most no-tably non-classical logics (cf. [7]) and proof complexity of quanti-fied Boolean formulas (QBF).

Recent research inQBF proof complexityhas been largely trig-gered by exciting advances in QBF solving—powerful algorithms that solve large classes of formulas from industrial applications. Compared to SAT solving, due to thePSPACEcompleteness of QBF the success of QBF solvers even extends to further fields such as planning [24, 36] and formal verification [5]. To model the strengths of modern QBF solvers, a number of resolution-based proof systems have been recently suggested and analysed from a proof complexity perspective (cf. [3, 8, 9, 11]).

While we have a relatively good understanding of these weak resolution-type systems, much less is known for strong proof sys-tems, and this judgement applies to both propositional and QBF proof complexity. There are two main approaches for designing strong calculi: via sequent-style systems (Gentzen’sLK[25]) and axiom-rule based systems known as Frege or Hilbert-type calculi [20]. In propositional logic, both Gentzen and Frege systems are equivalent from a proof complexity point of view [20, 31].

The situation is more intricate for QBF; and indeed the main aim of the present paper is to shed light on this topic.

Gentzen systems for QBFwere already introduced in the late 80’s by Kraj´ıˇcek and Pudl´ak [32], of which we use slightly modi-fied versionsGiandG∗i due to Cook and Morioka [18]. These

sys-tems are known to be strictly more powerful than QBF resolution [23], but lower bounds are out of reach with current techniques.

As for strong propositional systems, the main source of infor-mation on QBF Gentzen systems stems from their correspondence to Buss’ theories of bounded arithmetic [13, 18, 32]. This corre-spondence allows to translate first-order formulas into sequences of QBFs, and indeed first-order proofs inSi

2orT2ito

polynomial-sizeG∗iorGiproofs, respectively [18, 32], thus providing the main

tool to construct short propositional proofs.

On the other hand, QBF Frege systems were only developed very recently [10]. Their definition is very elegant, adding to

classi-calFregejust one single∀redrule for managing quantifiers,

lead-ing to the QBF systemFrege+_∀red. Alternatively, they can be seen as substitution Frege systems with substitutions allowed just for universally quantified variables.

As for classicalFrege, the strength ofFrege+_∀redcan be cal-ibrated by allowing different classes of formulas (or more directly Boolean circuits [28]) as their underlying objects. With a novel technique [8, 10], uncovering a new and direct relation between circuit complexity and proof complexity, very strong lower bounds have been obtained for QBFFrege, the strongest of which yields an exponential lower bound forAC0_[p]-Frege+

Frege+∀red G∗

0 EF+∀red

G0

G∗

1 G1

p-simulation

‘relaxed’ p-simulation (Theorem 5.2)

Figure 1. The simulation order of QBF Gentzen and Frege systems

AC0_{-Frege[1, 33, 35], while lower bounds for the strongerAC}0_[p]

-Fregeconstitute a major problem, open for more than twenty years.

This exciting development prompts us to target at a better un-derstanding of the new QBF Frege systems. What is their relation to the well-studied QBF Gentzen calculi? Does QBF Frege also admit a correspondence to bounded arithmetic? Can we push lower bounds even beyond the current state-of-the-art bound forAC0_[p]

-Frege+∀redfrom [10]?

In this paper we give answers to all of these three questions. 1.1 Our contributions

Below we summarise our main contributions of this paper, sketch-ing the main results and techniques.

A. Gentzen vs. Frege in QBF: simulations and separations. In classical proof complexityFregeand Gentzen’s sequent system

LKare p-equivalent, i.e., proofs can be efficiently translated be-tween the systems [20]. In contrast, our findings show a more com-plex picture for QBF. We concentrate on the most important stan-dard Gentzen-style systemsG∗0andG∗1 as well as the QBF Frege

systemsFrege+_∀redand EF+_∀red, forming QBF analogues of the classicalFregeand extended Frege systemEFfrom [20].

For these four systems the following picture emerges (cf. Fig-ure 1): We prove thatG∗1p-simulatesEF+∀red(Theorem 5.1) and

likewiseG∗0 p-simulatesFrege+∀red(although the latter under a

slightly more relaxed notion of p-simulation, Theorem 5.2). On the other hand, the converse simulations are unlikely to hold. Under a variety of standard complexity-theoretic assumptions we show that

EF+_∀redis incomparable to bothG∗

0andG0(Theorems 3.1, 3.4,

3.3, 3.5). Hence, unlike in the propositional framework, Gentzen appears to be stronger than Frege in QBF.

While all these separations make use of complexity-theoretic assumptions, it will be very hard to improve these results to un-conditional lower bounds (see C. below). However, since we use a number of unrelated and indeed partly incomparable assumptions, our separations seem very plausible.

B. QBF Frege corresponds to intuitionistic logic.The strongest tool for an understanding of classical Frege as well as proposi-tional and QBF Gentzen systems comes from their correspondence to bounded arithmetic [19, 31]. Here we show such a correspon-dence betweenEF+_∀redand first-order intuitionistic logicIS1₂, introduced in [14, 22]. For this first-order arithmetic formulas are translated into sequences of QBFs [32].

Our main result on the correspondence states that translations of arbitrarily complex prenex theorems inIS1₂admit polynomial-size EF+_∀red proofs (Theorem 7.1). Informally, this says that allIS1_{2consequences can be efficiently derived in}EF+_∀red, and moreover,EF+_∀redis the weakest system with this property.

The second facet of the correspondence is thatIS1_{2 can prove}

the correctness ofEF+_∀redin a suitable encoding (Theorem 7.2),

and in a certain senseEF+_∀redis the strongest proof system that is provably sound in the theoryIS1₂.

Technically, the correspondence as well as the simulation re-sults mentioned under A. above rest on a formalisation of the Strategy Extraction Theorem for QBF Frege systems from [10]. This strategy extraction result states that for formulas provable in

EF+∀redone can compute witnesses for all existential quanti-fiers with Boolean circuits that can be efficiently extracted from theEF+∀redproof.

We provide two formalisations for this result: one in first-order logic, where we formalise strategy extraction inS₂1_{(Theorem 4.1),}

and a second more direct one, where we constructFregeproofs for 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 correspondence toIS1_2.

Although intuitionistic bounded arithmetic was already devel-oped by Buss in the mid 80’s [14], no propositional counterpart of this theory was found so far—in sharp contrast to most other arithmetic theories [19]. As we show here, the missing piece in the puzzle is the recent QBF Frege systemEF+∀red.

Indeed, the appealing link betweenIS1_{2 and}EF+∀redcomes via their witnessing properties: similarly asEF+∀redhas strategy extraction for arbitrarily complex QBFs [10], the theoryIS1_2admits

a witnessing theorem for arbitrary first-order formulas [22]. C. Characterising lower bounds for QBF Frege.The main question left open by the recent advances in strong QBF lower bounds [10] is whetherunconditionallower bounds can be obtained

forFrege+∀redor evenEF+∀red. We show here that such a result

would imply either a major breakthrough in circuit complexity (a lower bound for non-uniformNC1 _{or even P/poly) or a major} breakthrough in propositional proof complexity (lower bounds for classicalFregeor evenEF); and in fact the opposite implications hold as well (Theorem 8.1).

This means that the problem of lower bounds for QBF Frege very naturally unites the hardest problem in circuit complexity with the hardest problem in proof complexity. Indeed, by our simula-tions shown in A. this also means that a lower bound for any of the QBF Gentzen systemsGi orG∗i fori ≥ 1would imply either a

circuit lower bound or a lower bound for propositionalFrege. This is conceptually very interesting as a direct connection be-tween progress in circuit complexity and proof complexity has been often postulated (cf. [4]). Our results show that this con-nection directly manifests inFrege+_∀red, thus highlighting that

Frege+_∀redis indeed a natural and important system.

Technically, this result uses a normal form that we achieve

forFrege+_∀redproofs: these can be decomposed into a classical

Fregeproof followed by a number of∀redsteps (Theorem 6.1).

We further show that even∀redsteps suffice that only substitute constants (Theorem 6.3).

Conceptually, our work draws on the close interplay of ideas and techniques from proof complexity, computational complexity, and bounded arithmetic; and it is really the interaction of these areas and techniques that form the technical basis of our results (which enforces us also to include rather extensive preliminaries). 1.2 Organization

In Section 2 we provide background on proof complexity, bounded arithmetic, and QBF Gentzen and Frege systems. We prove the con-ditional separations and the simulations in Sections 3 and 5, re-spectively. Section 4 formalizes strategy extraction in QBF Frege

inS₂1andFrege, and Section 6 derives from this a normalisation

Sec-tion 8 we give the characterizaSec-tion ofFrege+_∀redandEF+_∀red

lower bounds in terms of lower bounds for Boolean circuits or propositionalFrege.

2.

Preliminaries

2.1 Notions from computational complexity

We use standard notation and concepts from computational com-plexity (cf. [2]). In particular, we use the circuit classP/polyof functions computed by polynomial-size Boolean circuits and the class NC1 _{of functions computed by polynomial-size circuits of} logarithmic depth (cf. [37]). We say that a function is hard for

P/polyif it is not computable by a sequence of polynomial-size circuits.

ByFPΣpi[O(logn)]_{we denote the set of functions computed}

by a polynomial-time Turing machine making at most O(logn)

queries to aΣp_i-oracle.FPΣp_{i is defined analogously but without the} restriction on the number of queries.

2.2 Notions from proof complexity

Proof systems.According to [20] aproof systemfor a languageL is a polynomial-time onto functionP :{0,1}∗_{→ L. Each string} φ∈ Lis atheoremand ifP(π) =φ,πis aproofofφinP. Given a polynomial-time functionP : {0,1}∗ _{→ {0,1}}∗_{the fact that} P({0,1}∗_{) ⊆ Lis thesoundness propertyforLand the fact that} P({0,1}∗_{)⊇ Lis thecompleteness propertyforL.}

Proof systems for the languageTAUTof propositional tautolo-gies are calledpropositional proof systemsand proof systems for the languageTQBFof true QBF formulas are calledQBF proof systems. Equivalently, propositional proof systems and QBF proof systems can be defined respectively for the languagesUNSATof unsatisfiable propositional formulas andFQBFof false QBF for-mulas, in this second case we call themrefutational.

Given two proof systemsPandQfor the same languageL,P p-simulatesQ(denotedQ≤pP) if there exists a polynomial-time

functiontsuch that for eachπ∈ {0,1}∗_{,P(t(π)) =Q(π). Two} systems are called p-equivalent if they p-simulate each other.

A proof systemPforLis calledpolynomially boundedif there exists a polynomialpsuch that everyx∈ Lhas aP-proof of size ≤p(|x|).

Frege systems.Fregeproof systems are the common ‘textbook’ proof systems for propositional logic based on axioms and rules [20]. The lines in aFrege proof are propositional formulas built from propositional variables xi and Boolean connectives ¬, ∧,

and∨. AFregesystem comprises a finite set of axiom schemes

and rules, e.g., φ∨ ¬φ is a possible axiom scheme. A Frege

proofis a sequence of formulas where each formula is either a substitution instance of an axiom, or can be inferred from previous formulas by a valid inference rule.Fregesystems are required to be sound and implicationally complete. The exact choice of the axiom schemes and rules does not matter as any twoFregesystems are p-equivalent, even when changing the basis of Boolean connectives [20] and [31, Theorem 4.4.13]. Therefore we can assume w.l.o.g. that modus ponens is the only rule of inference.

UsuallyFregesystems are defined as proof systems where the last formula is the proven formula. Equivalently, we can view them as refutationFregesystems where we start with the negation of the formula that we want to prove and derive a contradiction, and we switch between the two different formulations when convenient.

A number of subsystems and extensions ofFrege have been considered in the literature (cf. [4]). An elegant framework for these systems was introduced by Jeˇr´abek [28], whereC-Frege directly operates with circuits from the setCusing a finite set of derivation

Fregerules. For example, if there are no restrictions onCthenC

-Fregeis p-equivalent to the extended Frege system EF, cf. [28].

IfCis restricted to formulas, i.e.,C = NC1_{, then}

C-Fregeis just

Frege. Throughout the paper, whenever we speak ofEFwe indeed

meanP/poly-FregeandFregestands forNC1-Frege.

Sequent calculus.Gentzen’s sequent calculus [25] is another classical proof system, both for first-order and propositional logic (cf. [31]). Propositional sequent calculusLKoperates with sequents

Γ −→ ∆with the semantic meaningV

φ∈Γφ |= W

ψ∈∆ψ. An

important rule inLKis the cut rule

Γ−→∆, A A,Γ−→∆

(cut rule)

Γ−→∆

whereAis called the cut formula.

LKis well known to be p-equivalent toFrege(cf. [31]). 2.3 Quantified Boolean formulas

Quantified Boolean formulas(QBF) extend propositional formulas by propositional quantifiers∀x. φ(x)with the semantic meaning φ(0)∧φ(1), and∃x. φ(x)meaningφ(0)∨φ(1).

The quantifier complexity of QBFs is captured by setsΣqi and

Πqi, which are defined inductively.Σ q

0= Π

q

0is the set of

quantifier-free propositional formulas,Σqi+1is the closure ofΠ

q

i under

exis-tential quantification, andΠqi+1is the closure ofΣ

q

iunder universal

quantifiers.

Often it is useful to think of a QBF Q1X1. . .QkXk. φ as

agame between theuniversal and the existential player. In the i-th step of the game, the player Qi assigns values to all the

variablesXi. The existential player wins the game iff the matrixφ

evaluates to1under the assignment constructed in the game. The universal player wins iff the matrix φ evaluates to 0. Given a universal variableuwith indexi, astrategy foruis a function from all variables of index< ito{0,1}. A QBF is false iff there exists a winning strategyfor the universal player, i.e. if the universal player has a strategy for all universal variables that wins any possible game [27], [2, Sec. 4.2.2].

2.4 Sequent calculi for QBF

Quantified propositional calculus G, as defined by Cook and Morioka [18], extends Gentzen’s classical propositional sequent calculusLK, cf. [31, Chapter 4.3], by allowing quantified propo-sitional formulas in sequents and by adopting the following extra quantification rules for∀-introduction

φ(x/ψ),Γ−→∆

(∀-l) ∀x. φ,Γ−→∆

Γ−→∆, φ(x/p)

(∀-r)

Γ−→∆,∀x. φ

and∃-introduction φ(x/p),Γ−→∆

(∃-l) ∃x. φ,Γ−→∆

Γ−→∆, φ(x/ψ)

(∃-r).

Γ−→∆,∃x. φ

For the rules∀-l and∃-r,φ(x/ψ)is the result of substitutingψ for all free occurrences ofx in φ. The formula ψ may be any quantifier-free formula (i.e., without bounded variables) that is free for substitution forxinφ(i.e., no free occurrence ofxinφis within the scope of a quantifierQysuch thatyoccurs inψ). The variable pin the rules∀-r and∃-l must not occur free in the bottom sequent. Fori≥0,Giis a subsystem ofGwith cuts restricted to prenex

Σqi∪Π q

i-formulas.G

∗

i denotes the subsystem ofGiallowing only

tree-like proofs.

The systemsGandGiwere originally introduced slightly

differ-ently, cf. [30–32], not restricting the formulasψin∀-l and∃-r to be quantifier-free, and definingGias the systemGallowing onlyΣqi

-formulas in sequents. Hence,Gi’s could not prove all true QBFs.

Notably, (for Cook and Morioka’s definition) Jeˇr´abek and Nguyen [29] showed that the system Gi with cuts restricted to

prenex Σqi-formulas is p-equivalent toGi with cuts restricted to

prenexΠqi-formulas and p-equivalent toGiwith cuts restricted to

(not necessarily prenex)Σqi∪Π q

i-formulas. Moreover these

equiv-alences hold as well for the tree-like versions of these systems. Cook and Morioka [18] also proved that their definition ofGi

is p-equivalent toGifrom [32] fori ≥ 0and prenexΣqi ∪Π q i

-formulas (so by [29] also for non-prenex ones).

On propositional formulasG0is p-equivalent toFregeandG1

is p-equivalent to the Extended Frege systemEF, cf. [31]. Finally, the systemsGiandG∗i have quite constructive

witness-ing properties. Whenever there are polynomial-sizeG∗1 proofs of

formulas∃y. An(x, y)forAn(x, y)∈Σq₁, there exist

polynomial-size circuitsCnwitnessing the existential quantifiers, i.e., the

for-mulaAn(x, Cn(x))holds, cf. [18, Theorem 7]. In case ofG0the

circuits witnessingΣq₁-formulas are fromNC1_{, cf. [18, Theorem 9].} The witnessing theorems can be generalized to systemsG∗i andGi

fori≥1w.r.t.Σqi-formulas and witnessing functions

correspond-ing to higher levels of the polynomial hierarchy. 2.5 Frege systems for QBF

An alternative way how to define reasoning with QBFs was given in [10] by using systems denoted asC-Frege+_∀red.C-Frege+_∀red

is a refutational proof system augmenting the classical C-Frege

system by a∀redrule. Formally, aC-Frege+_∀redrefutation of a QBFQ. φis a sequence of circuitsL1, . . . , Ll∈ CwhereL1=φ,

Ll = ∅, and each Li is derived from previous Lj’s using the

inference rules ofC-Fregeor using the following∀redrule Lj(u)

(∀red) Lj(u/B)

where u is a universal variable that is the innermost (wrt. the quantifier prefixQ) among the variables of Lj, and B ∈ C is

a circuit that contains only variables left of u. In particular, C

-Frege+∀reddoes not manipulate the prefix of the given QBF, so it

proves only QBFs in prenex form.

In principle, variables not quantified in the prefix of a QBF might appear in itsC-Frege+_∀redrefutation as consequences of

C-Fregerules. However, all such variables can be substituted by

arbitrary constants without changing the proven QBF. Therefore, we assume that there are no such ‘redundant’ variables.

If there are no restrictions onC, we denoteC-Frege+_∀redas

EF+_∀red. IfCis restricted to formulas, we speak ofFrege+_∀red. Note thatC-Frege+_∀redis essentially a refutational substitu-tion Frege systemSF, cf. [31], with substitutions allowed only for rightmost universally quantified variables.

In Section 6.1 we will show that in fact restricting the substitut-ing circuitBto constants0,1results in a p-equivalent proof system denotedC-Frege+∀red0,1.

A characteristic property of theC-Frege+∀redsystems is the so calledStrategy Extraction Theorem. The theorem obtained in [10] says that whenever there is aC-Frege+∀redrefutationπof a QBF∃x1∀y1, . . . ,∃xk∀yk. φ(x1, . . . , xk, y1, . . . , yk), then there

areO(|π|)-size witnessing circuitsC1, . . . , Ck∈ Csatisfying n

^

i=1

(y′i↔Ci(x1, . . . , xi, y1′, . . . , y′i−1, π))

→ ¬φ(x1, . . . , xn, y1′, . . . , y′n).

2.6 Bounded arithmetic

In first-order logic we will work with the languageL={0, S,+,·,≤

,x

2

,|x|,#} where the function |x| is intended to mean ‘the length of the binary representation ofx’ andx#y= 2|x|·|y|_.

A quantifier is bounded if it has the form∃x. x≤tor∀x. x≤t forxnot occurring in the termt. A bounded quantifier is sharply bounded ift has the form|s|for some terms. ByΣb

0 (=Πb0 = ∆b

0) we denote the set of all formulas in the languageLwith all

quantifiers sharply bounded. Fori ≥ 0, the setsΣb

i+1andΠbi+1

are defined inductively.Σb

i+1is the closure ofΠbi under bounded

existential and sharply bounded quantifiers, andΠb

i+1is the closure

ofΣb

i under bounded universal and sharply bounded quantifiers.

That is, the complexity of bounded formulas in the languageL (formulas with all quantifiers bounded) is defined by counting the number of alternations of bounded quantifiers, ignoring the sharply bounded ones. Fori >0,∆b

idenotesΣbi∩Πbi.

Bounded formulas capture the polynomial hierarchy: for any i >0thei-th levelΣp_i of the polynomial hierarchy coincides with the sets of natural numbers definable byΣb

i-formulas. Dually for

Πp_i andΠb i.

Buss [13] introduced theories of bounded arithmeticS₂i,T₂ifor i ≥ 1in the languageL. The axioms ofSi

2 consist of a set of

basic axioms defining properties of symbols fromL, cf. [31], and length inductionΣb

i-LIND, which is the following scheme forΣ b i

-formulasA(or equivalently, forA∈Πb

i, in which case we speak

ofΠb i-LIND):

A(0)∧ ∀x.(A(x)→A(x+ 1))→ ∀x. A(|x|).

TheoriesT₂i_{are defined similarly, but here the induction scheme is}

A(0)∧ ∀x.(A(x)→A(x+ 1))→ ∀x. A(x)

forA∈Σb i.

T₂i proves the totality of FPΣpi functions, cf. [31, Theorem

6.1.2]. More precisely, for anyf ∈FPΣpi _{there is a}Σb

i+1-formula

f(x) = y such thatTi

2 ⊢ ∀x∃y. f(x) = y. In the same way,

S₂i _{proves the totality of functions in} FPΣpi[O(logn)]_{, cf. [31,}

Theorem 6.2.2]. By Parikh’s theorem,T₂i_{⊢ ∃}y. f(x) =yimplies

Ti

2 ⊢ ∃y.|y| ≤p(|x|)∧f(x) =yfor some polynomialp, and the

same is true forSi

2(cf. [13, 34]).

S₂i_{can be seen as a first-order non-uniform version of}G∗i,i≥1.

Firstly, forj ≥ 1anyΣb

j-formulaφ(x)can be translated into a

sequencekφ(x)kn_ofΣq

j-formulas, wherendenotes the size of the

inputxin binary (cf. [31, Definition 9.2.1]). Then, fori, j ≥ 1

wheneverSi

2 ⊢ AforA ∈Σbj, there is a polynomialpsuch that

formulaskAkn

haveG∗i-proofs of sizep(n). This also holds forT i

2

in place ofS₂iifG∗i is replaced byGi. The ability to use arbitrary

jis due to Cook and Morioka [18, Theorem 3] who generalized a standard result, cf. [31, Theorem 9.2.6], which worked forj=i.

IfA ∈ Πb

1, we abuse notation and also denote bykAkn the

propositional formulas obtained as inkAkn

, but leaving the uni-versally quantified variables free.S₂1 _⊢AforA∈Πb

1implies that

S₂1_{proves the existence of polynomial-size}G∗1-proofs of

proposi-tional formulaskAkn

, cf. [31, Theorems 9.2.6 and 9.2.7].

3.

Separating Gentzen and Frege for QBF

We start with proving a number of conditional separations between Gentzen and Frege systems for QBF. As we will show later in Section 8, improving these separations to unconditional results tightly corresponds to major open problems in circuit complexity and proof complexity.

3.1 Formulas easy in Gentzen, but hard in Frege

We first provide three different properties that are easy for QBF Gentzen systems, but hard forEF+∀red. Our first conditional result shows that there areΣq

2-formulas with polynomial-sizeG∗1

Theorem 3.1. Leti≥1. Assumef ∈ FPΣpi _{is hard forP}/_poly.

Then formulas k∃y.|y| ≤ p(|x|) ∧f(x) = ykn

, where p is a polynomial and f(x) = y is expressed by a Σb

i+1-formula,

have polynomial-sizeGiproofs and require super-polynomial-size

EF+∀redproofs. Iff ∈FPΣip_{[O(logn)]thenGican be replaced}

byG∗i.

Proof. AsT₂iproves the totality ofFPΣp_{i functions [13], it proves} the totality offand the proof can be transformed into a sequence of polynomial-size Gi proofs [18, 32]. If the totality of f can

be shown by polynomial-size proofs in EF+∀red, then, by the Strategy Extraction Theorem [10],fis inP/poly.

Similarly,S₂i _{proves the totality of}FPΣpi[O(logn)] _functions

and such proofs translate into sequences of polynomial-size G∗i

proofs [13, 18, 32].

It seems that the separation above of G∗1 and EF+∀red by Σq

2-formulas cannot be improved toΣ

q

1-formulas as it is tight in

the following sense. If we had Σq

1-formulas ∃y. An(x, y) with

polynomial-sizeG∗1proofs but without polynomial-sizeEF+∀red

proofs, this would imply thatEFis not polynomially bounded: by the witnessing theorem forG∗1, cf. [18, Theorem 7], there would be

polynomial-size circuitsCnsuch that formulasAn(x, Cn(x))are

true, and so¬An(x, Cn(x))would be hard to refute inEF.

G∗1 andEF+∀redcan be conditionally separated also on the

bounded collection scheme.

Definition 3.2. Thebounded collection schemeBB_(φ)is the for-mula

∃i <|a|,∃w < t(a),∀u < a,∀j <|a|.(φ(i, u)→φ(j,[w]j))

whereφ(i, u)is a formula which can have other free variables,

[w]j is thej-th element of the sequence coded byw, and t(a)

is a concreteL-term depending on the choice of the encoding of sequences.

Roughly,BB_{(φ)says that u’s witnessing φ(i, u) can be}

col-lected in a sequencew:

∀i <|a|,∃u < a, φ(i, u)→ ∃w < t(a),∀j <|a|, φ(j,[w]j).

Theorem 3.3. G∗1 has polynomial-size proofs ofkBB(φ)kn for

allφ ∈ Σb

1. In contrast, there existsφ ∈ Σb1such that formulas

kBB_(φ)kn _{are hard forEF+∀redunless each polynomial-time}

permutation withninputs can be inverted by polynomial-size cir-cuits with probability≥1−1/n.

Proof. The upper bound follows from theS₂1_{-provability of}BB(φ)

forφ ∈ Σb

1, cf.[13, Theorem 14], and its transformation toG∗1

proofs [18, 32].

For the lower bound we will use a result by Cook and Thapen [21] showing that Cook’s theoryPV _{does not prove}BB(φ)for all φ∈Σb

0unless factoring is in probabilistic polynomial time.

Leta = 2n

and φ(i, u) be the formulaf(u) = [y]i for a

polynomial-time permutationf (defined by aΣb

1 formula), andy

encoding a sequence ofnstrings of lengthn.

Assume thatEF+_∀redhas polynomial-size proofs ofkBB(φ)kn

. By the Strategy Extraction Theorem [10] there are polynomial-size circuitsB,Csuch that

∃u <2n. f(u) = [y]C(y)→ ∀j < n. f([B(y)]j) = [y]j.

To invertf we proceed as follows. Givenz ∈ {0,1}n

, pick ran-domlyn stringssi ∈ {0,1}n and leti0 be a position such that Pry∈{0,1}d[C(y) =i0]≤1/nwheredis the number of inputs in

C. Defineyz,sto be the sequence of elementsz, f(s1), . . . , f(sn−1) ordered so that [yz,s]i0 = z and let xz,s be the sequence of

z, s1, . . . , sn−1 ordered so thatf([xz,s]i) = [yz,s]i fori 6= i0.

ThenPrz,s1,...,sn∈{0,1}d[C(yz,s) = i0] ≤ 1/n. Therefore, with

probability ≥ 1−1/n, f([xz,s]C(yz,s)) = [yz,s]C(yz,s) and

f([B(yz,s)]i0) =z.

While the previous two results exhibited formulas easy forG∗1

and hard forEF+∀red, we now show that evenG∗0can proveΣq2

-formulas hard forEF+∀red(modulo hardness of factoring). For this we use a result by Bonet, Pitassi, and Raz [12], who showed thatFrege systems do not admit the so called feasible interpolation property unless factoring of Blum integers is solvable by polynomial-size circuits. (A Blum integer is the product of two distinct primes, which are both congruent 3 modulo 4.)

Theorem 3.4. There are Σq₂-formulas with polynomial-sizeG∗0

proofs. However, assuming factoring of Blum integers is not computable by polynomial-size circuits, these formulas require

EF+_∀redproofs of super-polynomial size.

Proof.In [12] it is shown that there are propositional formulas A0(x, y),A1(x, z)with common variablesxsuch thatA0(x, y)∨

A1(x, z) have polynomial-size Frege proofs but, unless factor-ing of Blum integers is computable by polynomial-size circuits, there are no polynomial-size circuitsC(x)recognizing which of A0(x, y)orA1(x, z)holds for a givenx.

Fregeis p-equivalent toG∗0on propositional formulas [31] and

so it is possible to derive inG∗0the sequents in Figure 2.

Therefore, theΣq₂-formulas

∃b∀y, u.((A0(x, y)∧b)∨(A1(x, u)∧ ¬b))

have polynomial-sizeG∗

0proofs.

If these formulas had polynomial-sizeEF+_∀redproofs, then, by the Strategy Extraction Theorem [10], there would be polynomial-size circuits computingbfromxand thus recognizing which of A0(x, y),A1(x, u)holds.

We remark that the assumptions of Theorems 3.3 and 3.4 are stronger than the assumption of Theorem 3.1. However, while fac-toring forms a good candidate for a one-way function, it is not known if the existence of one-way functions implies the existence of one-way permutations.

3.2 Formulas hard in Gentzen, but easy in Frege

We now give the opposite separation, exhibiting formulas (condi-tionally) hard forG0, but easy forEF+∀red. ThusG∗0 andG0

ap-pear to be incomparable toEF+∀red. Theorem 3.5. IfP/poly6=NC1

then there areΣq₁-formulas with polynomial-sizeEF+_∀redproofs but without polynomial-sizeG0

proofs.

Proof.Letfbe a function inP/poly. ThenEF+∀redhas simple polynomial-size proofs ofΣq

1formulas∃y,∃z. f(x) =y

express-ing the totality offwith auxiliary variableszrepresenting nodes of a polynomial-size circuit computingf. TheEF+_∀redproof re-futes the propositional formulaf(x) 6= yby gradually replacing each variable fromz, yby the circuit it represents.

If the totality offhad polynomial-sizeG0proofs, by theΣq1

witnessing property, cf. [18, Theorem 9],fwould be inNC1_.

Notably, in Section 6 we show thatFrege+∀redandEF+∀red

are p-equivalent to their tree-like versions. This is open forG0and G1, thus providing some further evidence for the incomparability

−→A0(x, y), A1(x, z)

−→(A0(x, y)∧ ¬0)∨(A1(x, u)∧0),(A0(x, v)∧ ¬1)∨(A1(x, z)∧1)

−→ ∀y, u.((A0(x, y)∧ ¬0)∨(A1(x, u)∧0)),(A0(x, v)∧ ¬1)∨(A1(x, z)∧1)

−→ ∀y, u.((A0(x, y)∧ ¬0)∨(A1(x, u)∧0)),∀y, u.((A0(x, y)∧ ¬1)∨(A1(x, u)∧1))

−→ ∃b∀y, u.((A0(x, y)∧ ¬b)∨(A1(x, u)∧b)),∃b∀y, u.((A0(x, v)∧ ¬b)∨(A1(x, z)∧b))

−→ ∃b∀y, u.((A0(x, y)∧ ¬b)∨(A1(x, u)∧b))

Figure 2. TheG∗₀derivation in the proof of Theorem 3.4

4.

Formalized strategy extraction

In order to prove thatG∗1p-simulatesEF+∀redwe first formalize

the Strategy Extraction Theorem from [10]. We provide two differ-ent formalizations, one inS₂1_{and another one directly in}EF. Both are sufficient for the simulation result. These formalizations guar-antee that the extracted strategy is not just correct, butEF(resp.

C-Frege) provably correct.

Theorem 4.1(Formalized Strategy Extraction). There is a linear-time algorithmAsuch thatS₂1proves the following. Assume thatπ is anEF+∀redrefutation of a QBFψof the form

∃x1∀y2. . .∃xn∀yn. φ(x1, . . . , xn, y1, . . . , yn)

where φ ∈ Σq

0. Then A(π) outputs circuits C1(x1, π), . . . ,

Cn(x1, . . . , xn, y1, . . . , yn−1, π) defining a winning strategy for

the universal player on formulaψ; that is,

∀x1, . . . , xn, y1, . . . , yn. n

^

i=1 (yi↔

Ci(x1, . . . , xi, y1, . . . , yi−1, π))→ ¬φ(x1, . . . , xn, y1, . . . , yn).

Proof. We will inspect the original proof of the Strategy Extraction Theorem from [10], and point out that it essentially uses a Πb

1

-induction on the number of steps in the proofπ, i.e.,Πb

1-LIND

available inS₂1_.

Letπ = (L1, . . . , Ls)be anEF+∀redrefutation of the QBF

Q. φgiven as in Theorem 4.1 and put

πs:=∅, πi:= (Li+1, . . . , Ls)fori < s

φ0 :=φ, φi:=φ∧L1∧ · · · ∧Lifori >0.

We will show by downward induction oni, that fromπiit is

possible to construct in linear time a winning strategy σi₌

{Ci

1(x1, πi), . . . , Cni(x1, . . . , xn, y1, . . . , yn−1, πi)}

for the universal player for the QBFQ. φi. The statement of the

Formalized Strategy Extraction Theorem corresponds to the case i= 0.

In the base case,φs contains a contradiction and the winning

strategy can be defined as the set of trivial circuits{0, . . . ,0}. Assume now thatσi_{is a winning strategy forQ. φ}

i.

IfLiis derived by anEFrule, then we setσi−1:=σi.

Assume now thatLi=Lj[u/B]is the result of an application

of a∀redrule onLjwhereuis the rightmost variable inLj. We

defineCi−1

l :=C i

lifu6=yl, otherwise we set

Cli−1(z) :=

(

B(z) ifLj[u/B](z) = 0

Ci

l(z) ifLj[u/B](z) = 1.

This constructs circuitsCi

lfromπiby a standardO(|πi|)-time

algorithm.

To show that the strategiesσi

are winning for any0≤i≤ |π|, we need to analyze the inductive step.

Assume thatσi_{is the winning strategy for the universal player}

onQ. φi. IfLiis derived by anEFrule, the winning strategy for

Q. φi works also forQ. φi−1 because a falsification ofLi by a

given assignment implies a falsification of one of its predecessors. IfLiis the result of an application of∀red,Cli−1(z)is redefined

only ifLj[u/B](z) = 0. Forz such thatLj[u/B](z) = 1, the

strategy σi _{has to work also for Q. φ}

i−1. Therefore,σi−1 is a

winning strategy for the universal player onQ. φi−1.

The statement that a strategy σ is winning for the universal player onQ. ψis acoNPpredicate (givenπ) expressible as a well-behavedΠb

1-formula. The induction we used is on the number of

steps inπ. Hence, the presented proof is anS₂1-proof.

The statement provable inS₂1_{in Theorem 4.1 is a}coNP predi-cate expressible by aΠb

1-formula. Consequently, translating theS21

proof toEF, the extracted strategy is evenEF-provably correct: Corollary 4.2. Given anEF+∀redrefutationπof a QBF

∃x1∀y2. . .∃xn∀yn. φ(x1, . . . , xn, y1, . . . , yn)

whereφ∈Σq0, we can construct in time|π|O(1)anEFproof of

n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

¬φ(x1, . . . , xn, y1, . . . , yn)

for some circuitsCi.

We will now show the same result as in the last corollary for

Frege+_∀red(and in fact provide an alternative direct proof without

making use of bounded arithmetic forEF+_∀redas well). Theorem 4.3. LetCbe the circuit classNC1

orP/poly.1Given a

C-Frege+∀redrefutationπof a QBF

∃x1∀y2. . .∃xn∀yn. φ(x1, . . . , xn, y1, . . . , yn)

whereφ∈Σq₀, we can construct in time|π|O(1)_{aC-Fregeproof of}

n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

¬φ(x1, . . . , xn, y1, . . . , yn)

for some circuitsCi∈ C.

Proof.Again, we will inspect the original proof of the Strategy Extraction Theorem.

Letπ= (L1, . . . , Ls)be aC-Frege+∀redrefutation of aQBF

Q. φgiven as in Theorem 4.3 and put

πs:=∅, πi:= (Li+1, . . . , Ls)fori < s

φ0:=φ, φi:=φ∧L1∧ · · · ∧Lifori >0.

1_{Indeed, the result can be generalised to further ‘natural’ circuit classesC}

We will show by downward induction oni, that fromπiit is

possible to construct in linear time a winning strategy

σi={C1(i x1, πi), . . . , Cni(x1, . . . , xn, y1, . . . , yn−1, πi)}

for the universal player for theQBFQ. φi. Moreover, formula n

^

l=1

(yl↔Cli(x1, . . . , xl, y1, . . . , yl−1, πi))→

¬φi(x1, . . . , xn, y1, . . . , yn)

denotedσi_(φ

i) will have aC-Fregeproof of sizeK|πi|K for a

constantKdepending only on the choice of theC-Fregesystem. The statement of the theorem corresponds to the casei= 0.

In the base case, φs contains a contradiction so the winning

strategy can be defined as the set of trivial circuits{0, . . . ,0}and it is trivially provably correct.

Assume now thatσi_(φ

i)has aC-Fregeproof of sizeK(s+ 1−

i)|πi|K.

IfLiis derived by aC-Fregerule, thenσi−1:=σi.

Let nowLi=Lj[u/B]be the result of an application of a∀red

rule onLjwhereuis the rightmost variable inLj. Then define

Ci−1

l :=C i

lifu6=yl, otherwise set

Cli−1(z) :=

(

B(z) ifLj[u/B](z) = 0

Ci

l(z) ifLj[u/B](z) = 1.

This constructs strategiesσi_{fromπby aD|π}

i|-time algorithm

for a constantD. W.l.o.g.D < K. In fact, circuitsCi lare inC.

We want to show thatσi−1_(φ

i−1)has aC-Fregeproof of size K(s+ 1−(i−1))|πi−1|K.

IfLiis derived by aC-Fregerule, thenσialso witnesses¬φi−1

because

¬Li→ ¬(L′1∧ · · · ∧L′t)

for some conjunctsL′1, . . . , L′tinφi−1. Note thatCli−1’s are then

Ci

l’s. The implications

¬φi→ ¬φi−1

σi_(φ

i)∧(¬φi→ ¬φi−1)→σi−1(φi−1)

(1)

can be derived by a fixed sequence of C-Frege rules depending only on the choice ofC-Frege. Thus, the common size ofC-Frege

proofs of both these implications is≤K0|πi−1|K0 where w.l.o.g.

K0 < K. Therefore σi−1(φi−1) has a C-Frege proof of size

≤K(s+ 1−i)|πi|K+K1|πi−1|K1≤K(s+ 1−(i−1))|πi−1|K

whereK1 > K0 depends again on a fixed sequence ofC-Frege

rules needed to deriveσi−1_(φ

i−1)from (1) andσi(φi), so w.l.o.g.

K1< K.

AssumeLi =Lj[u/B]is the result of an application of∀red

whereu = yl. Then there is a fixed sequence ofC-Frege rules

deriving implications

σi(φi)∧ ¬Lj[u/B]→Cli−1=B∧σ i−1_(φ

i−1)

σi(φi)∧Lj[u/B]→Cli−1=C i l∧σ

i−1_(φ

i−1). The total size of bothC-Frege derivations isK0|πi−1|K0 where

K0depends on the choice ofC-Fregeand the size ofCli−1’s. The

size of allCi−1

l ’s is bounded byK|πi−1| K

. Hence we can assume K0 < K. It follows thatσi−1(φi−1)has aC-Fregeproof of size

≤K(s+ 1−i)|πi|K+K1|πi−1|K1≤K(s+ 1−(i−1))|πi−1|K

where as beforeK1depends on a fixed sequence ofC-Fregerules

needed to simulate a fixed set of ‘cut’ rules, i.e., w.l.o.g.K1 <

K.

5.

Gentzen simulates Frege for QBF

We now apply the formalised Strategy Extraction Theorem from the last section to show that Gentzen systems simulate Frege sys-tems in the QBF context. Frege and Gentzen are well known to be equivalent in the classical propositional case [31]. However, in QBF the opposite simulations (Gentzen by Frege) are very likely false as shown by the conditional separations in Section 3. Theorem 5.1. G∗1p-simulatesEF+∀red.

Proof.By Corollary 4.2, anyEF+_∀redrefutationπof aQBFψ (given as in Corollary 4.2) can be transformed in time|π|O(1)_into

anEFproof of

n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

¬φ(x1, . . . , xn, y1, . . . , yn)

for certain circuitsCi.

Claim 1. There is a|π|O(1)_-sizeG∗

1proof of the following sequent

{yi=Ci(x1, . . . , xi, y1, . . . , yi−1)}ni=1−→

¬φ(x1, . . . , xn, y1, . . . , yn)

where the encoding of circuitsCimight use some auxiliary

vari-ables.

Proof of claim.To see that the claim holds note first that by p-equivalence ofEFandG∗1 (cf. [31]), theEFproof obtained above

can be turned into a|π|O(1)_-sizeG∗

1-proof of the formula

¬

n

^

i=1

yi=Ci(x1, . . . , xi, y1, . . . , yi−1)

!

∨ ¬φ.

This proof can be easily modified so that the∨connective is not introduced, leading to a|π|O(1)_-sizeG∗

1-proof of the sequent

−→ ¬

n

^

i=1

yi=Ci(x1, . . . , xi, y1, . . . , yi−1)

!

,¬φ.

Moving ¬ Vn

i=1 yi=Ci(x1, . . . , xi, y1, . . . , yi−1) from the succedent to the antecedent we obtain

n

^

i=1

(yi=Ci(x1, . . . , xi, y1, . . . , yi−1))−→ ¬φ.

Finally,G∗1 derives the sequent we want by ‘not introducing’∧in

the antecedent. This proves the claim.

Applying∃-r and∃-l introductions,G∗1then derives Γ,∃yn. yn=Cn(x1, . . . , xn, y1, . . . , yn−1)−→

∃yn.¬φ(x1, . . . , xn, y1, . . . , yn)

whereΓ ={yi=Ci(x1, . . . , xi, y1, . . . , yi−1)}ni=1−1.

AsG∗1proves efficiently−→ ∃y. y =C(x)for any circuitC,

we can cut∃yn. yn = Cn(x1, . . . , xn, y1, . . . , yn−1) out of the antecedent and derive

{yi=Ci(x1, . . . , xi, y1, . . . , yi−1)}ni=1−1−→ ∃yn.¬φ.

Now, we use∀-r introduction to obtain

{yi=Ci(x1, . . . , xi, y1, . . . , yi−1)}ni=1−1−→ ∀xn∃yn.¬φ.

In this way we can gradually cut out all formulas from the an-tecedent, quantify all variables and derive¬ψin G∗1 by a proof

To introduce the quantifyer prefix ofψin the previous proof we needed to cutΣq₁-formulas. We would like to use a similar proof to simulateFrege+∀redbyG∗0. However,G∗0is allowed to cut only Σq0-formulas. Therefore we obtain just a simulation ofFrege+∀red

byG∗0where the proven sequent inG∗0contains a nonempty (easily

derivable) antecedent.

Theorem 5.2. There is a polynomial-time functiont such that given anyFrege+_∀redrefutation of a QBFψof the form

∃x1∀y2. . .∃xn∀yn. φ(x1, . . . , xn, y1, . . . , yn)

whereφ∈Σq0,t(π)is aG∗0proof of the sequent

∀x1∃y2. . .∀xn∃yn. n

^

i=1

yi=Ci(x1, . . . , xi, y1, . . . , yi−1)−→ ¬ψ

for some formulasCi. Note that the antecedent has aG∗0 proof of

size|π|O(1)_.

Proof. By Theorem 4.3, anyFrege+_∀redrefutationπof aQBFψ can be transformed in time|π|O(1)_{into aFregeproof of}

n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

¬φ(x1, . . . , xn, y1, . . . , yn)

for certain formulasCi.

Analogously as in the proof of Theorem 5.1, we efficiently obtain a|π|O(1)_-sizeG∗

0proof of

n

^

i=1

yi=Ci(x1, . . . , xi, y1, . . . , yi−1)−→ ¬φ.

Applying rules∃-r,∃-l,∀-l,∀-r (in this order) we derive

∀xn∃yn. n

^

i=1

yi=Ci(x1, . . . , xi, y1, . . . , yi−1)−→ ∀xn∃yn.¬φ.

In this way we efficiently introduce all quantifiers and derive the required sequent inG∗₀.

6.

Normal forms for QBF Frege proofs

In this section we apply results from Section 4 to obtain two normal forms for Frege+_∀red and EF+_∀red proofs. Firstly, we show that anyEF+_∀redrefutation can be efficiently rewritten as anEF

derivation followed essentially just by∀redrules, and the same normalisation applies toFrege+∀red. Secondly, we show that in the∀redrule it is sufficient to only substitute constants.

Theorem 6.1. LetCbe the circuit classNC1orP/poly. For any

C-Frege+∀redrefutationπof a QBFψof the form

∃x1∀y2. . .∃xn∀yn. φ(x1, . . . , xn, y1, . . . , yn)

whereφ∈Σq₀, there is a|π|O(1)

-sizeC-Frege+∀redrefutation of ψstarting with aC-Fregederivation of

n

_

i=1

(yi6=Ci(x1, . . . , xi, y1, . . . , yi−1)) (2)

for some circuitsCi, followed bynapplications of the∀redrule,

gradually replacing the rightmostyibyCi(x1, . . . , xi, y1, . . . , yi−1)

and cuttingyi 6=Ci(x1, . . . , xi, y1, . . . , yi−1)out of the disjunc-tion(2).

Proof.Given aC-Frege+_∀redrefutationπofψ, by Theorem 4.3, there is a|π|O(1)_{-sizeC-Fregeproof of}

n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

¬φ(x1, . . . , xn, y1, . . . , yn).

Havingψfreely available in the refutation,C-Fregecan derive (2) by applying the cut rule (derivable inC-Frege).

The refutation then continues by n applications of the ∀red

rule, which one by one replaces the rightmost variable yi by

Ci(x1, . . . , xi, y1, . . . , yi−1)and eliminates yi6=Ci(x1, . . . , xi, y1, . . . , yi−1)

from the disjunctionW

iyi6=Ci(x1, . . . , xi, y1, . . . , yi−1).

As theFrege(resp.EF) derivation can be efficiently replaced by a tree-likeFrege(resp.EF) proof, cf. [31], and the rest of the

C-Frege+∀red refutation given above is tree-like we obtain the

following.

Corollary 6.2.Frege+_∀redis p-equivalent to tree-likeFrege+_∀red. Likewise,EF+∀redis p-equivalent to tree-likeEF+∀red.

6.1 Substituting constants in∀redis sufficient

Frege+∀redandEF+∀redproofs can be further simplified so that

every∀redrule substitutes only constants instead of general cir-cuits. This shows that the systems are indeed very robustly defined. Theorem 6.3. Frege+∀red is p-equivalent to Frege+∀red0,1.

Likewise,EF+∀redis p-equivalent toEF+∀red0,1.

Proof.LetC be eitherNC1_{orP/poly. It is enough to show that}

anyC-Frege+_∀redrefutation can be transformed efficiently into

a refutation where the∀red rule substitutes only constants. By Theorem 6.1, for anyC-Frege+_∀redrefutationπofQ. φthere is a|π|O(1)_{-sizeC-Fregederivation of}

n

_

i=1

(yi6=Ci(x1, . . . , xi, y1, . . . , yi−1))

fromφ(x1, . . . , xn, y1, . . . , yn). Applying∀red0,1onynwe can

then derive

Cn(x1, . . . , xn, y1, . . . , yn−1)6=c∨

n−1 _

i=1

(yi6=Ci(x1, . . . , xi, y1, . . . , yi−1))

for both constantsc = 0,1. However, there is a polynomial-size

C-Fregeproof of

Cn(x1, . . . , xn, y1, . . . , yn−1) = 1∨

Cn(x1, . . . , xn, y1, . . . , yn−1) = 0,

so we can deriveW

i<n (yi 6= Ci(x1, . . . , xi, y1, . . . , yi−1)). In

this way we can efficiently cut all disjuncts and derive a contradic-tion inC-Frege+∀red0,1.

7.

Intuitionistic logic corresponds to

EF

∀

red

The main information on strong propositional and QBF systems stems from their correspondence to first-order theories of bounded arithmetic (cf. [6, 19, 31]). In this sense,G∗

1corresponds toS21and G1toT21(cf. Section 2.6). Here we will establish such a

correspon-dence between first-order intuitionistic logic andEF+_∀red. In [14] Buss developed an intuitionistic version ofS₂1, denoted

implies the existence of a polynomial-time functionf such that A(x, f(x))holds. This witnessing property resembles the Strategy Extraction Theorem forEF+∀red. Using the formalized Strategy Extraction Theorem we can make the correspondence between these systems formal.2

First, we recall the definition of IS1_{2 by Cook and Urquhart}

[22]. It is equivalent to Buss’ original definition, cf. [14].IS1_{2 is}

a theory in the languageL(likeS₂1_{), with underlying intuitionistic}

predicate logic, cf. [22], a set of basic axioms defining properties of symbols fromL, cf. [22], and a polynomial induction scheme for

Σb+

1 -formulasA:

A(0)∧ ∀x.A(jx 2 k

)→A(x)→ ∀x. A(x).

Here,Σb+

1 -formulas areΣ

b

1-formulas without negation and

impli-cation signs.

S₂1_isΣb

0-conservative overIS12, cf. [22, Corollary 1.7].

We will also use Cook and Urquhart’s conservative extension ofIS1_2denotedIPV_{, cf. [22, Chapter 4 and Theorem 4.12].}IPV

is defined by adding intuitionistic predicate logic to Cook’s the-oryPV_{, cf. [17]. The language of} IPV _{consist of symbols for}

all polynomial-time functions. The hierarchy of formulasΠb i(PV)

is defined analogously asΠb

i but in the language ofIPV. Also,

propositional translationskAkn _forΠb

1(PV)-formulasAare

de-fined analogously as in the case ofA∈Πb

1. Consequently,IPV ⊢

AforA∈Πb

1(PV)implies that propositional formulaskAknhave

polynomial-sizeEFproofs, cf. [31, Theorem 9.2.7].

Cook and Urquhart [22, Corollary 8.18] generalized Buss’ witnessing theorem: wheneverIPV _{⊢ ∀}x∃y. A(x, y) for an ar-bitrarily complex formula A, then there is a polynomial-time functionf (with an IPV function symbolf) such thatIPV _⊢ ∀x. A(x, f(x)).

We are now ready to derive the correspondence betweenIS1₂

andEF+∀red. The correspondence consists of two parts (cf. [6]). For the first part we translate first-order formulasφinto sequences of QBFs [32] and show that translations of provableIS1_2formulas

have shortEF+∀redproofs.

Theorem 7.1. IfIS1₂ proves a statementT in prenex form, then there exist polynomial-sizeEF+_∀redrefutations ofk¬Tkn

.

Proof. By Cook and Urquhart’s improvements of Buss’ witnessing theorem, ifIS1_2provesT of the form

∀x1∃y1. . .∀xn∃yn. T′(x1, . . . , xn, y1, . . . , yn)

forT′ ∈ Σb

0, there is anIPV-functionf1(x1)such thatIPV ⊢

∀x1, x2,∃y2, . . . ,∀xn∃yn. T′(x1, . . . , xn, f1(x1), y2, . . . , yn).

Iterating this argument all existential quantifiers ofT can be wit-nessed provably inIPV _{by polynomial-time functions}f1, . . . , fn.

Therefore,IPV proves theΠb

1(PV)formula

n

^

i=1

(yi↔fi(x1, . . . , xi, y1, . . . , yi−1))→

T′(x1, . . . , xn, y1, . . . , yn) (3)

and the formulask(3)kn_{have polynomial-sizeEFproofs.EF+}

∀red

can now refutek¬Tkn_{in polynomial size by deriving}W

i(yi 6=

2_{It could be tempting to expect that an adequate counterpart toIS}1 2would

be intuitionistic propositional logic. However, intuitionistic propositional logic admits the feasible interpolation property, cf. [16], whileIS1

2 can

(constructively) prove∀x, z.[A(x, y)∨B(x, z)], in principle, without the existence of an efficient interpolant. It is also known, cf. [26], thatIS1₂ ⊢ ∀y. A(x, y)∨∀z. B(x, z)implies the existence of an efficient interpolating circuit, but moving the universal quantifiers inside the disjunction is a priori not allowed in intuitionistic logic.

fi(x1, . . . , xi, y1, . . . , yi−1))and cutting all the disjuncts as in the proof of Theorem 6.1.

The second part of the correspondence consists in proving the soundness of the proof systems in the first-order theory. For this we need to express the correctness ofEF+∀redby QBFs. This is typically done by thereflection principleof a proof systemP, stating that wheneverφhas aP-proof (resp. aP-refutation), then φis true (resp. false).

Here, the Formalized Strategy Extraction Theorem allows us to express the reflection principle ofEF+_∀redby a Πb

1-formula Ref(EF+_∀red). More precisely, we defineRef(EF+_∀red)as the

Πb

1-formula expressing that ifπis a proof of aQBF, then circuits

Ci(x1, . . . , xi, y1, . . . , yi−1, π)obtained as in the Strategy

Extrac-tion Theorem witness the existential quantifiers in theQBFas in the statement of Theorem 4.1.

Theorem 7.2. IS1

2provesRef(EF+∀red).

Proof.The claim follows from Theorem 4.1 together with theΣb

0

-conservativity ofS₂1overIS1₂[22].

Theorem 7.2 implies thatEF+_∀redis the weakest proof system that allows short proofs of allIS1₂ theorems, i.e., whenever Theo-rem 7.1 holds for a ‘decent’ proof systemP in place ofEF+_∀red, thenPp-simulatesEF+_∀redonQBFs: If Theorem 7.1 holds for a proof systemP, then by Theorem 7.2, there are polynomial-size P-proofs ofkRef(EF+_∀red)kn_{. Hence, ifπis anEF+}

∀redproof of aQBFψ, thenPhas|π|O(1)_{-size proofs ofψwith the}

existen-tial quantifiers witnessed by some circuits. ByP being decent we mean thatP can introduce efficiently the existential quantifiers in place of the witnessing circuits and this way proveψefficiently in the size ofπ.

On the other hand,EF+_∀redis intuitively the strongest proof system for whichIS1_{2 proves the reflection principle. Technically,}

this only holds for proof systems that admit the Strategy Extraction Theorem as for other systems we would need to define the reflec-tion principle as a more complex statement.

8.

Characterising QBF Frege lower bounds

We finally address the question of lower bounds forFrege+_∀red

or evenEF+_∀red. Our next result states that achieving such lower bounds unconditionally will either imply a major breakthrough in circuit complexity or a major breakthrough in classical proof complexity.

Theorem 8.1.

1.EF+_∀redis not polynomially bounded if and only ifEFis not polynomially bounded orPSPACE6⊆P/poly.

2.Frege+∀redis not polynomially bounded if and only ifFrege

is not polynomially bounded orPSPACE6⊆NC1 .3

Proof.IfPSPACE6⊆P/polythenEF+∀redis not polynomially bounded by [10, Theorem 5.13]. Clearly, also ifEFis not polyno-mially bounded thenEF+∀redis not polynomially bounded.

In the opposite direction, assume thatEF+∀redis not polyno-mially bounded. Then there is a sequence of true QBFsQ. ψnsuch

that¬Q. ψndo not have polynomial-size refutations inEF+∀red.

LetQ. ψnhave the form

∀x1∃y1, . . . ,∀xn∃yn. ψn(x1, . . . , xn, y1, . . . , yn).

3_ByNC1_{we meannon-uniformNC}1_{. Note that by the space hierarchy}

IfPSPACE6⊆P/poly, we are done. Otherwise, there are polynomial-size circuitsCiwitnessing the existential quantifiers inQ. ψn. That

is, for anyx1, . . . , xn, y1, . . . , yn n

^

i=1

(yi↔Ci(x1, . . . , xi, y1, . . . , yi−1))→

ψn(x1, . . . , xn, y1, . . . , yn). (4)

We claim that (4) is a sequence of tautologies without polynomial-sizeEFproofs. Otherwise, having¬ψn,EF can deriveWiyi 6=

Ci(x1, . . . , xi, y1, . . . , yi−1)by a polynomial-size proof, and so as in Theorem 6.1,EF+_∀redcan efficiently refute¬Q. ψn.

The analogous argument works for item 2 of the theorem. This result also essentially answers the main question left open in [10], whether a lower bound for Frege+_∀red can be shown by a different technique than the strategy extraction technique es-tablished in that paper. By Theorem 8.1, any such technique for

Frege+∀redwould immediately transfer to classicalFrege, thus

solvingthemain problem in propositional proof complexity.

9.

Conclusion

In this paper we have undertaken a comprehensive analysis of QBF Frege systems, clarifying their relationships to bounded arith-metic and to Gentzen systems. While the emerging picture clearly shows that Gentzen systems are strictly stronger than Frege in QBF, one question left open by our results is whether the simulation of

Frege+∀redbyG∗0 in Theorem 5.2 can be made to work in the

standard way, i.e., whetherG∗0p-simulatesFrege+∀red.

Acknowledgments

This research was supported by grant no. 48138 from the John Templeton Foundation and EPSRC grant EP/L024233/1.

References

[1] M. Ajtai. The complexity of the pigeonhole-principle.Combinatorica, 14(4):417–433, 1994.

[2] S. Arora and B. Barak. Computational Complexity: A Modern Ap-proach. Cambridge University Press, 2009.

[3] V. Balabanov, M. Widl, and J.-H. R. Jiang. QBF resolution systems and their proof complexities. InSAT’14, pages 154–169, 2014. [4] P. Beame and T. Pitassi. Propositional proof complexity: Past, present,

and future. In G. Paun, G. Rozenberg, and A. Salomaa, editors,

Current Trends in Theoretical Computer Science: Entering the 21st Century, pages 42–70. World Scientific Publishing, 2001.

[5] M. Benedetti and H. Mangassarian. QBF-based formal verification: Experience and perspectives.JSAT, 5(1-4):133–191, 2008.

[6] O. Beyersdorff. On the correspondence between arithmetic theories and propositional proof systems – a survey. Mathematical Logic Quarterly, 55(2):116–137, 2009.

[7] O. Beyersdorff and O. Kutz. Proof complexity of non-classical logics. In N. Bezhanishvili and V. Goranko, editors,Lectures on Logic and Computation - ESSLLI 2010 /ESSLLI 2011, Selected Lecture Notes, pages 1–54. Springer-Verlag, Berlin Heidelberg, 2012.

[8] O. Beyersdorff, L. Chew, and M. Janota. Proof complexity of resolution-based QBF calculi. InProc. Symposium on Theoretical As-pects of Computer Science (STACS’15), pages 76–89. LIPIcs, 2015. [9] O. Beyersdorff, L. Chew, M. Mahajan, and A. Shukla. Feasible

inter-polation for QBF resolution calculi. InProc. International Colloquium on Automata, Languages, and Programming (ICALP’15), pages 180– 192. Springer, 2015.

[10] O. Beyersdorff, I. Bonacina, and L. Chew. Lower bounds: From cir-cuits to QBF proof systems. InProc. ACM Conference on Innovations

in Theoretical Computer Science (ITCS’16), pages 249–260. ACM, 2016.

[11] O. Beyersdorff, L. Chew, M. Mahajan, and A. Shukla. Are short proofs narrow? QBF resolution is not simple. InProc. Symposium on Theoretical Aspects of Computer Science (STACS’16), 2016. [12] M. L. Bonet, T. Pitassi, and R. Raz. On interpolation and

automatiza-tion for Frege systems.SIAM J. Comp., 29(6):1939–1967, 2000. [13] S. R. Buss.Bounded Arithmetic. Bibliopolis, Napoli, 1986. [14] S. R. Buss. The polynomial hierarchy and intuitionistic bounded

arithmetic. InProc. Structure in Complexity Theory Conference, pages 77–103, 1986.

[15] S. R. Buss. Towards NP-P via proof complexity and search.Ann. Pure Appl. Logic, 163(7):906–917, 2012.

[16] S. R. Buss and G. Mints. The complexity of the disjunction and ex-istential properties in intuitionistic logic.Annals of Pure and Applied Logic, 99(1–3):93–104, 1999.

[17] S. A. Cook. Feasibly constructive proofs and the propositional calcu-lus. InProc. 7th Annual ACM Symposium on Theory of Computing, pages 83–97, 1975.

[18] S. A. Cook and T. Morioka. Quantified propositional calculus and a second-order theory for NC1_{.Arch. Math. Log., 44(6):711–749, 2005.}

[19] S. A. Cook and P. Nguyen.Logical Foundations of Proof Complexity. Cambridge University Press, 2010.

[20] S. A. Cook and R. A. Reckhow. The relative efficiency of propositional proof systems.The Journal of Symbolic Logic, 44(1):36–50, 1979. [21] S. A. Cook and N. Thapen. The strength of replacement in weak

arithmetic.ACM Trans. Comput. Log., 7(4):749–764, 2006. [22] S. A. Cook and A. Urquhart. Functional interpretations of feasibly

constructive arithmetic.Ann. Pure Appl. Logic, 63(2):103–200, 1993. [23] U. Egly. On sequent systems and resolution for QBFs. InTheory and Applications of Satisfiability Testing (SAT’12), pages 100–113, 2012. [24] U. Egly, M. Kronegger, F. Lonsing, and A. Pfandler. Conformant

plan-ning as a case study of incremental QBF solving. InArtificial Intelli-gence and Symbolic Computation (AISC’14), pp. 120–131, 2014. [25] G. Gentzen. Untersuchungen ¨uber das logische Schließen.

Mathema-tische Zeitschrift, 39:68–131, 1935.

[26] K. Ghasemloo and J. Pich. A note on natural proofs and intuitionism. available at karlin.mff.cuni.cz/∼pich/natcons.pdf, 2013.

[27] A. Goultiaeva, A. Van Gelder, and F. Bacchus. A uniform approach for generating proofs and strategies for both true and false QBF formulas. InIJCAI, pages 546–553, 2011.

[28] E. Jeˇr´abek. Dual weak pigeonhole principle, Boolean complexity, and derandomization.Annals of Pure and Applied Logic, 129:1–37, 2004. [29] E. Jer´abek and P. Nguyen. Simulating non-prenex cuts in quantified

propositional calculus.Math. Log. Q., 57(5):524–532, 2011. [30] J. Kraj´ıcek and G. Takeuti. On induction-free provability.Ann. Math.

Artif. Intell., 6(1-3):107–125, 1992.

[31] J. Kraj´ıˇcek.Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, 1995.

[32] J. Kraj´ıˇcek and P. Pudl´ak. Quantified propositional calculi and frag-ments of bounded arithmetic.Zeitschrift f¨ur mathematische Logik und Grundlagen der Mathematik, 36:29–46, 1990.

[33] J. Kraj´ıˇcek, P. Pudl´ak, and A. Woods. Exponential lower bounds to the size of bounded depth Frege proofs of the pigeonhole principle.

Random Structures and Algorithms, 7(1):15–39, 1995.

[34] R. Parikh. Existence and feasibility in arithmetic. J. Symb. Log., 36 (3):494–508, 1971.

[35] T. Pitassi, P. Beame, and R. Impagliazzo. Exponential lower bounds for the pigeonhole principle. Computational Complexity, 3:97–140, 1993.

[36] J. Rintanen. Asymptotically optimal encodings of conformant plan-ning in QBF. InAAAI, pages 1045–1050. AAAI Press, 2007. [37] H. Vollmer.Introduction to Circuit Complexity – A Uniform Approach.