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.

Atserias and Müller (JACM, 2020) proved that for every unsatisfiable CNF formula $\varphi$, the formula $\operatorname{Ref}(\varphi)$ stating that “$\varphi$ has small Resolution refutations"-does not have subexponential-size Resolution refutations. Conversely, when $\varphi$ is satisfiable, Pudlák (TCS, 2003) showed how to construct a polynomial-size Resolution refutation of $\operatorname{REF}(\varphi)$ given a satisfying assignment of $\varphi$. A question that had remained open is: do all short Resolution refutations of $\operatorname{Ref}(\varphi)$ explicitly leak a satisfying assignment of $\varphi$?We answer this question affirmatively by providing a polynomial-time algorithm that extracts a satisfying assignment for $\varphi$ given any short Resolution refutation of $\operatorname{REF}(\varphi)$. The algorithm follows from a new feasibly constructive proof of the Atserias-Müller lower bound, formalizable in Cook’s theory PV1of bounded arithmetic. This implies that Extended Frege can efficiently prove (a suitable formalization of the statement) that automating Resolution is NP-hard.Motivated by this algorithm, we introduce a new metacomputational problem concerning Resolution lower bounds: the Proof Analysis Problem (PAP). For a fixed proof system Q, the Proof Analysis Problem for Q asks, given a CNF formula $\varphi$ and a Q-proof of a Resolution lower bound for $\varphi$, encoded as $\neg \boldsymbol{REF}(\varphi)$, whether $\varphi$ is satisfiable. In contrast to the Proof Analysis Problem for Resolution, which is in P, we prove that PAP for Extended Frege (EF) is NP-complete. In particular, EF can prove Resolution lower bounds on satisfiable formulas without necessarily revealing a satisfying assignment.Our results yield new insights into proof search and the meta-mathematics of Resolution lower bounds: (i) for every proof system that simulates EF as well as for Resolution, the system is (weakly) automatable if and only if it can be (weakly) automated exclusively on formulas stating Resolution lower bounds; (ii) we provide explicit Ref formulas that are exponentially hard for bounded-depth Frege systems; and (iii) for every strong enough theory of arithmetic T we construct explicit unsatisfiable CNF formulas that are exponentially hard for Resolution but for which T cannot prove even a quadratic Resolution lower bound. This latter result applies to arbitrarily strong theories like PA or ZFC, and does not require any complexity-theoretic assumptions.

A jump operator$J$in proof complexity is a function such that for any proof system$P, J(P)$is a proof system that$P$cannot simulate. Some candidate jump operators were proposed by Krajicek and Pudlak [63] and Krajicek [57], but it is an open problem whether computable jump operators exist or not. In this regard, we introduce a new candidate jump operator based on the power of interactive proofs which given a proof system$P$, [I P,$P]$(I P-randomized implicit proof system based on$P)$is an M A proof system. In the first step, we investigate the relationship between IP - randomized implicit proof systems and Cook-Reckhow proof systems. In particular, we show that if$i$EF (Krajicek's implicit Extended Frege) proves exponential hard on average circuit lower bounds efficiently, then$i$E F simulates [IP, EF]. Moreover, we show that IP-randomized implicit proof systems can be used to prove new connections between different well-studied concepts in complexity theory. Namely, we prove new results about the hardness magnification in proof complexity, the hardness of proving proof complexity lower bounds, the automatability and the feasible disjunction property for Extended Frege using IP - randomized implicit proof systems. One ingredient of our proofs is a formalization of the sum-check protocol [65] in$\mathrm{s}_{2}^{1}$which might be of independent interest. We also look at the general theory of jump operators and consider an old conjecture by Pudlak [78] about finite consistency sentences for first-order theories of arithmetic. In this direction, we prove that certain statements are equivalent, in particular, we prove that the widely believed assumption about the existence of computable jump operators in proof complexity is equivalent to a weaker form of Pudlak's conjecture.

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.

A map g:{0,1}ⁿ → {0,1}^m (m > n) is a hard proof complexity generator for a proof system P iff for every string b ∈ {0,1}^m ⧵ Rng(g), formula τ_b(g) naturally expressing b ∉ Rng(g) requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is Nisan-Wigderson generator. Razborov [A. A. {Razborov}, 2015] conjectured that if A is a suitable matrix and f is a NP∩CoNP function hard-on-average for 𝖯/poly, then NW_{f, A} is a hard proof complexity generator for Extended Frege. In this paper, we prove a form of Razborov’s conjecture for AC⁰-Frege. We show that for any symmetric NP∩CoNP function f that is exponentially hard for depth two AC⁰ circuits, NW_{f,A} is a hard proof complexity generator for AC⁰-Frege in a natural setting. As direct applications of this theorem, we show that: 1) For any f with the specified properties, τ_b(NW_{f,A}) (for a natural formalization) based on a random b and a random matrix A with probability 1-o(1) is a tautology and requires superpolynomial (or even exponential) AC⁰-Frege proofs. 2) Certain formalizations of the principle f_n ∉ (NP∩CoNP)/poly requires superpolynomial AC⁰-Frege proofs. These applications relate to two questions that were asked by Krajíček [J. {Krajíček}, 2019].

Abstract In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

The proof system Res (PC d,R ) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res ( PC 1, R )-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res ( PC 1,𝔽 ) when 𝔽 is a finite field, such as 𝔽 2 . In this article, we investigate Res ( PC d,R ) and tree-like Res ( PC d,R ) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows: (1) We prove almost quadratic lower bounds for Res ( PC d ,𝔽)-refutations for every fixed d . The new lower bounds are for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs with linearly many clauses. (2) We also prove super-polynomial (more than n k for any fixed k ) and also exponential (2 nϵ for an ϵ > 0) lower bounds for tree-like Res ( PC d ,𝔽 )-refutations based on how big d is with respect to n for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs of suitable densities, (c) Pigeonhole principle and Counting mod q principle. The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d =1. The lower bounds for the tree-like systems were known for the case d =1 (except for the Counting mod q principle, in which lower bounds for the case d > 1 were known too). Our lower bounds extend those results to the case where d > 1 and also give new proofs for the case d =1.