The size of Frege proofs can be characterized in terms of prover-adversary games of Pudlák and Buss. We consider a generalization of prover-adversary games to many standard proof systems and show that some of the major proof complexity lower bounds such as the constant-depth Frege lower bound for the pigeonhole principle based on the method of k-evaluations, the Resolution lower bound for the weak pigeonhole principle based on the method of pseudo-width and Razborov’s Res(k) lower bound for Nisan-Wigderson generators based on expansion and a width lower bound (which is used to derive the Res(k)-hardness of formulas expressing circuit lower bounds) are constructive in the sense that they yield efficient algorithms computing winning strategies of adversaries in the generalized games. This is in contrast with our second result saying that if (a) such a constructive lower bound exists for Extended Frege system EF for formulas expressing succinct circuit lower bounds for SAT and (b) EF is strong enough to prove efficiently the correctness of anticheckers for SAT, then it is easy to separate the canonical pair of EF.

We give a new approach to the fundamental question of whether proof complexity lower bounds for concrete propositional proof systems imply super-polynomial Boolean circuit lower bounds. We observe that any general implication from proof complexity lower bounds for a propositional proof system to super-polynomial Boolean circuit lower bounds implies unconditionally that \({\sf NEXP}\) does not have Boolean circuits of polynomial size. We explore connections that are possible to establish without settling this long-standing and famously hard open question. For any poly-time computable function f , we define the witnessing formulas \(w_n^k(f)\) , which are propositional formulas stating that for any circuit C of size \(n^k\) on n variables and for any formula \(\phi\) of size n , either C computes a satisfying assignment to \(\phi\) or f verifiably refutes that C computes \({\sf SAT}\) on instances of length n . We show that if the witnessing formulas are tautologies, then any super-polynomial lower bound for Extended Frege augmented with \(w_n^k(f)\) axioms implies that \({\sf SAT}\) requires super-polynomial size Boolean circuits. We also give an unconditional equivalence between circuit lower bounds for the Discrete Logarithm problem and proof complexity lower bounds (for propositional formulas efficiently encoding the statement that the Discrete Logarithm problem is computable by small circuits) for a concretely defined strong (non-uniform) propositional proof system. We give consequences of our connections for the meta-mathematics of several major questions in computational complexity, including whether one-way functions can be based on the worst-case hardness of NP, whether there is a dichotomy between one-way functions and worst-case learning with membership queries over the uniform distribution, and whether there are feasibly constructible anti-checkers for Satisfiability. We show that for each of these questions, provability of a positive answer in essentially any standard mathematical theory would imply new connections between propositional proof complexity and circuit complexity. Our results rely on a new notion of “self-provability” of upper bounds, which might be independently interesting, and involve a novel application of random self-reducibility to proof complexity.

Abstract We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small circuits attempting to compute hard functions. As our main result, we prove that if it is possible to find efficiently, in a particular interactive way, errors of many $$p$$ p -size circuits attempting to solve hard problems, then $$p$$ p -size circuits can be PAC learned over the uniform distribution with membership queries by circuits of subexponential size. The opposite implication holds as well. This provides a new characterization of learning algorithms and the natural proofs barrier of Razborov and Rudich. The proof is based on a method of reconstructing Nisan-Wigderson generators introduced by Krajíček (2010) and used to analyze complexity of circuit lower bounds in bounded arithmetic. An interesting consequence of known constructions of learning algorithms from circuit lower bounds is a learning speedup of Oliveira and Santhanam (2016). We present an alternative proof of this phenomenon and discuss its potential to advance the program of hardness magnification.

Folklore in complexity theory suspects that circuit lower bounds against NC1 or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [58]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [45], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [54]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

Abstract We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by Razborov, we show that in many cases it is not possible to combine the approximation method with typical (localizable) hardness magnification theorems to derive strong circuit lower bounds. In particular, one cannot use the approximation method to derive an extremely strong constant-depth circuit lower bound and then magnify it to an $$\textsf{NC}^{1}$$ NC 1 lower bound for an explicit function. To prove this, we show that lower bounds obtained by the approximation method are in many cases localizable in the sense that they imply lower bounds for circuits which are allowed to use arbitrarily powerful oracles with small fan-in.

Hardness magnification reduces major complexity separations (such as EXP ⊈ NC 1 ) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [ 11 , 13 , 14 , 40 , 42 , 43 , 46 ] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q , while Q itself is susceptible to lower bounds, but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: – Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [ 51 ] ? – Can we adapt known lower-bound techniques to establish the desired lower bound for Q ? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit-size problem imply the non-existence of natural proofs. As the non-existence of natural proofs implies the non-existence of efficient learning algorithms, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower-bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier : existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower-bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.

We show unconditionally that Cook’s theory PV formalizing poly-time reasoning cannot prove, for any non-deterministic poly-time machine M defining a language L(M), that L(M) is inapproximable by co-nondeterministic circuits of sub-exponential size. In fact, our unprovability result holds also for a theory which supports a fragment of Jeřábek’s theory of approximate counting APC1. We also show similar unconditional unprovability results for the conjecture of Rudich about the existence of super-bits.

Hardness magnification reduces major complexity separations (such as EXP ⊈ NC^1) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [Igor Carboni Oliveira and Rahul Santhanam, 2018; Dylan M. McKay et al., 2019; Lijie Chen and Roei Tell, 2019; Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019; Igor Carboni Oliveira, 2019; Lijie Chen et al., 2019] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q, while Q itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: - Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [Alexander A. Razborov and Steven Rudich, 1997]? - Can we adapt known lower bound techniques to establish the desired lower bound for Q? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit size problem MCSP imply the non-existence of natural proofs. As a corollary of our result, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier: existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C . Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC 0 [ p ] circuits. In contrast, any non-trivial lower bound for propositional AC 0 [ p ]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.

This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.

We formalize and study the question of whether there are inherent difficulties to showing lower bounds on propositional proof complexity. We establish the following unconditional result: Propositional proof systems cannot efficiently show that truth tables of random Boolean functions lack polynomial size non-uniform proofs of hardness. Assuming a conjecture of Rudich, propositional proof systems also cannot efficiently show that random k-CNFs of linear density lack polynomial size non-uniform proofs of unsatisfiability. Since the statements in question assert the average-case hardness of standard NP problems (MCSP and 3-SAT respectively) against co-nondeterministic circuits for natural distributions, one interpretation of our result is that propositional proof systems are inherently incapable of efficiently proving strong complexity lower bounds in our formalization. Another interpretation is that an analogue of the Razborov-Rudich `natural proofs' barrier holds in proof complexity: under reasonable hardness assumptions, there are natural distributions on hard tautologies for which it is infeasible to show proof complexity lower bounds for strong enough proof systems. For the specific case of the Extended Frege (EF) propositional proof system, we show that at least one of the following cases holds: (1) EF has no efficient proofs of superpolynomial circuit lower bound tautologies for any Boolean function or (2) There is an explicit family of tautologies of each length such that under reasonable hardness assumptions, most tautologies are hard but no propositional proof system can efficiently establish hardness for most tautologies in the family. Thus, under reasonable hardness assumptions, either the Circuit Lower Bounds program toward complexity separations cannot be implemented in EF, or there are inherent obstacles to implementing the Cook-Reckhow program for EF.

We aim to understand inherent reasons for lower bounds for QBF proof systems, and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) `genuine' QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ICALP'16). We prove a strong lower bound for relaxing QU-Res, which also exhibits significant shortcomings of that model. Prompted by this we propose an alternative, improved version, allowing more flexible oracle queries in proofs. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.

Recently Beyersdorff, Bonacina, and Chew [10] introduced a natural 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 extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF.