We show that for any unsatisfiable CNF formula φ that requires resolution refutation width at least w, and for any 1-stifling gadget g (for example, g = MAJ₃), (1) every resolution-over-parities (Res(⊕)) refutation of the lifted formula φ∘g of size at most S has depth at least Ω(w²/log S); (2) every Res(⊕) refutation of the lifted formula φ∘g has size Ω(w²). The first result substantially extends and simplifies all previously known lifting theorems for bounded-depth Res(⊕). The lifting result of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026] requires gadgets of logarithmic size and applies only to refutations of depth at most O(nlog n), whereas our result applies to nearly quadratic depth. The liftings of Bhattacharya and Chattopadhyay [Sreejata Kishor Bhattacharya and Arkadev Chattopadhyay, 2025] and of Byramji and Impagliazzo [Farzan Byramji and Russell Impagliazzo, 2025] apply to nearly quadratic depth as well, but rely on a much stronger assumption of (Ω(n),Ω(n))-DT-hardness, which is far less standard than large resolution width. Our proof combines the random-walk-with-restarts method of Alekseev and Itsykson [Yaroslav Alekseev and Dmitry Itsykson, 2025] with a new idea: the random walk is defined relative to the structure of the refutation graph, rather than by a distribution on inputs induced by the formula. Using this technique, we substantially strengthen the supercritical size-depth tradeoff of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026], both by improving the depth lower bound and by reducing the size of the separating formulas to polynomial in the number of variables, with the latter resolving an open question posed in [Dmitry Itsykson and Alexander Knop, 2026]. In particular, we construct a family of polynomial-size formulas that admit polynomial-size resolution refutations, while any Res(⊕) refutation of depth o(n²/log⁴ n) necessarily has superpolynomial size. Our second result yields a pure quadratic lower bound on the size of Res(⊕) refutations, improving upon the previously known near-quadratic lower bound of [Farzan Byramji and Russell Impagliazzo, 2025].

Alekseev and Itsykson (STOC 2025) proved the existence of an unsatisfiable CNF formula such that any resolution over parities (Res(⊕)) refutation must either have exponential size (in the formula size) or superlinear depth (in the number of variables). In this paper, we extend this result by constructing a formula with the same hardness properties, but which additionally admits a resolution refutation of quasi-polynomial size. This establishes a supercritical tradeoff between size and depth for resolution over parities. The proof builds on the framework of Alekseev and Itsykson and relies on a lifting argument applied to the supercritical tradeoff between width and depth in resolution, proposed by Buss and Thapen (IPL 2026).

Strong lower bounds of the form 2(1−є)n, where n is the number of variables and є>0 is arbitrarily small (i.e., bounds consistent with the Strong ETH), are exceptionally rare in proof complexity. The seminal work of Beck and Impagliazzo (STOC 2013) achieved such a bound for regular resolution, and the strongest extension known prior to our work was proved for O(є)-regular resolution by Bonacina and Talebanfard (Algorithmica, 2017).

Abstract. The proof system resolution over parities ([Formula: see text]) operates with disjunctions of linear equations (linear clauses) over [Formula: see text]; it extends the resolution proof system by incorporating linear algebra over [Formula: see text]. Over the years, several exponential lower bounds on the size of tree-like [Formula: see text] refutations have been established. However, proving a superpolynomial lower bound on the size of dag-like [Formula: see text] refutations remains a highly challenging open question. We prove an exponential lower bound for regular [Formula: see text]. Regular [Formula: see text] is a subsystem of dag-like [Formula: see text] that naturally extends regular resolution. This is the first known superpolynomial lower bound for a fragment of dag-like [Formula: see text] which is exponentially stronger than tree-like [Formula: see text]. In the regular regime, resolving linear clauses [Formula: see text] and [Formula: see text] on a linear form [Formula: see text] is permitted only if, for both [Formula: see text], the linear form [Formula: see text] does not lie within the linear span of all linear forms that were used in resolution rules during the derivation of [Formula: see text]. Namely, we show that the size of any regular [Formula: see text] refutation of the binary pigeonhole principle [Formula: see text] is at least [Formula: see text]. A corollary of our result is an exponential lower bound on the size of a strongly read-once linear branching program solving a search problem. This resolves an open question raised by Gryaznov, Pudlák, and Talebanfard [ Proceedings of the 37 th Computational Complexity Conference, LIPIcs Leibniz Int. Proc. Inform. 234, S. Lovett, ed., Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022, pp. 1–16]. As a byproduct of our technique, we prove that the size of any tree-like [Formula: see text] refutation of the weak binary pigeonhole principle [Formula: see text] is at least [Formula: see text] using Prover-Delayer games. We also give a direct proof of a width lower bound: we show that any dag-like [Formula: see text] refutation of [Formula: see text] contains a linear clause [Formula: see text] with [Formula: see text] linearly independent equations.

The proof system resolution over parities (Res(⊕)) operates with disjunctions of linear equations (linear clauses) over GF(2); it extends the resolution proof system by incorporating linear algebra over GF(2). Over the years, several exponential lower bounds on the size of tree-like refutations have been established. However, proving a superpolynomial lower bound on the size of dag-like Res(⊕) refutations remains a highly challenging open question. We prove an exponential lower bound for regular Res(⊕). Regular Res(⊕) is a subsystem of dag-like Res(⊕) that naturally extends regular resolution. This is the first known superpolynomial lower bound for a fragment of dag-like Res(⊕) which is exponentially stronger than tree-like Res(⊕). In the regular regime, resolving linear clauses C1 and C2 on a linear form f is permitted only if, for both i∈ {1,2}, the linear form f does not lie within the linear span of all linear forms that were used in resolution rules during the derivation of Ci. Namely, we show that the size of any regular Res(⊕) refutation of the binary pigeonhole principle BPHPnn+1 is at least 2Ω(∛n/logn). A corollary of our result is an exponential lower bound on the size of a strongly read-once linear branching program solving a search problem. This resolves an open question raised by Gryaznov, Pudlak, and Talebanfard (CCC 2022). As a byproduct of our technique, we prove that the size of any tree-like Res(⊕) refutation of the weak binary pigeonhole principle BPHPnm is at least 2Ω(n) using Prover-Delayer games. We also give a direct proof of a width lower bound: we show that any dag-like Res(⊕) refutation of BPHPnm contains a linear clause C with Ω(n) linearly independent equations.

Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a class of circuits - because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottom-up compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottom-up compilation resides in the bottom-up paradigm itself.

A canonical communication problem Search(φ) is defined for every unsatisfiable CNF φ: an assignment to the variables of φ is partitioned among the communicating parties, they are to find a clause of φ falsified by this assignment. Lower bounds on the randomized k-party communication complexity of Search(φ) in the number-on-forehead (NOF) model imply tree-size lower bounds, rank lower bounds, and size-space tradeoffs for the formula φ in the semantic proof system T^{cc}(k,c) that operates with proof lines that can be computed by k-party randomized communication protocol using at most c bits of communication [Göös and Pitassi, 2014]. All known lower bounds on Search(φ) (e.g. [Beame et al., 2007; Göös and Pitassi, 2014; Russell Impagliazzo et al., 1994]) are realized on ad-hoc formulas φ (i.e. they were introduced specifically for these lower bounds). We introduce a new communication complexity approach that allows establishing proof complexity lower bounds for natural formulas. First, we demonstrate our approach for two-party communication and apply it to the proof system Res(⊕) that operates with disjunctions of linear equalities over 𝔽₂ [Dmitry Itsykson and Dmitry Sokolov, 2014]. Let a formula PM_G encode that a graph G has a perfect matching. If G has an odd number of vertices, then PM_G has a tree-like Res(⊕)-refutation of a polynomial-size [Dmitry Itsykson and Dmitry Sokolov, 2014]. It was unknown whether this is the case for graphs with an even number of vertices. Using our approach we resolve this question and show a lower bound 2^{Ω(n)} on size of tree-like Res(⊕)-refutations of PM_{K_{n+2,n}}. Then we apply our approach for k-party communication complexity in the NOF model and obtain a Ω(1/k 2^{n/2k - 3k/2}) lower bound on the randomized k-party communication complexity of Search(BPHP^{M}_{2ⁿ}) w.r.t. to some natural partition of the variables, where BPHP^{M}_{2ⁿ} is the bit pigeonhole principle and M = 2ⁿ+2^{n(1-1/k)}. In particular, our result implies that the bit pigeonhole requires exponential tree-like Th(k) proofs, where Th(k) is the semantic proof system operating with polynomial inequalities of degree at most k and k = 𝒪(log^{1-ε} n) for some ε > 0. We also show that BPHP^{2ⁿ+1}_{2ⁿ} superpolynomially separates tree-like Th(log^{1-ε} m) from tree-like Th(log m), where m is the number of variables in the refuted formula.

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

Abstract In 2004 Atserias, Kolaitis, and Vardi proposed $\text {OBDD}$ -based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false $\text {OBDD}$ from $\text {OBDD}$ s representing clauses of the initial formula. All $\text {OBDD}$ s in such proofs have the same order of variables. We initiate the study of $\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in $\text {OBDD}$ s. At first we consider a proof system $\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from $\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of $\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for $\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from $\text {OBDD}(\land )$ to $\text {OBDD}(\land , \text{reordering})$ . In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on $\text {OBDD}$ s and symbolic quantifier elimination (we denote algorithms based on this approach as $\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an $\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over $\mathbb {F}_2$ ), but we show that there are formulas representing systems of linear equations over $\mathbb {F}_2$ that are hard for $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over $\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.

We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.

We consider satisfiable Tseitin formulas TS_{G,c} based on d-regular expanders G with the absolute value of the second largest eigenvalue less than d/3. We prove that any nondeterministic read-once branching program (1-NBP) representing TS_{G,c} has size 2^{\Omega(n)}, where n is the number of vertices in G. It extends the recent result by Itsykson at el. [STACS 2017] from OBDD to 1-NBP. On the other hand it is easy to see that TS_{G,c} can be represented as a read-2 branching program (2-BP) of size O(n), as the negation of a nondeterministic read-once branching program (1-coNBP) of size O(n) and as a CNF formula of size O(n). Thus TS_{G,c} gives the best possible separations (up to a constant in the exponent) between 1-NBP and 2-BP, 1-NBP and 1-coNBP and between 1-NBP and CNF.

In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based proof systems that additionally contain a rule that allows to change the order in OBDDs. At first we consider a proof system OBDD(and, reordering) that uses the conjunction (join) rule and the rule that allows to change the order. We exponentially separate this proof system from OBDD(and)-proof system that uses only the conjunction rule. We prove two exponential lower bounds on the size of OBDD(and, reordering)-refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for OBDD(and)-proofs and the second one extends the result of Tveretina et al. from OBDD(and) to OBDD(and, reordering). In 2004 Pan and Vardi proposed an approach to the propositional satisfiability problem based on OBDDs and symbolic quantifier elimination (we denote algorithms based on this approach as OBDD(and, exists)-algorithms. We notice that there exists an OBDD(and, exists)-algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time. In contrast, we show that there exist formulas representing systems of linear equations over F_2 that are hard for OBDD(and, exists, reordering)-algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over F_2 that correspond to some checksum matrices of error correcting codes.

We address the following question in the average-case complexity: does there exists a language L such that for all easy distributions D the distributional problem (L, D) is easy on the average while there exists some more hard distribution D' such that (L, D') is hard on the average? We consider two complexity measures of distributions: the complexity of sampling and the complexity of computing the distribution function. For the complexity of sampling of distribution, we establish a connection between the above question and the hierarchy theorem for sampling distribution recently studied by Thomas Watson. Using this connection we prove that for every 0 < a < b there exist a language L, an ensemble of distributions D samplable in n^{log^b n} steps and a linear-time algorithm A such that for every ensemble of distribution F that samplable in n^{log^a n} steps, A correctly decides L on all inputs from {0, 1}^n except for a set that has infinitely small F-measure, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}^n for which B correctly decides L has infinitely small D-measure. In case of complexity of computing the distribution function we prove the following tight result: for every a > 0 there exist a language L, an ensemble of polynomial-time computable distributions D, and a linear-time algorithm A such that for every computable in n^a steps ensemble of distributions F , A correctly decides L on all inputs from {0, 1}^n except for a set that has F-measure at most 2^{-n/2} , and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}^n for which B correctly decides L has D-measure at most 2^{-n+1}.

The partial string avoidability problem, also known as partial word avoidability, is stated as follows: given a finite set of strings with possible ``holes'' (undefined symbols), determine whether there exists any two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this paper establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form (CNF) satisfiability problem (SAT), with each clause having infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting constraints (such as clauses, inequalities, polynomials, etc). Two results on their proof complexity are established. First, there is a particular formula that has a short refutation in Resolution with shift, but requires classical proofs of exponential size (Resolution, Cutting Plane, Polynomial Calculus, etc.). At the same time, exponential lower bounds for shifted versions of classical proof systems are established.

The resolution complexity of the perfect matching principle was studied by Razborov [1], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph Gn such that the resolution complexity of the perfect matching principle for Gn is 2Ω(n) w here n is the number of vertices in Gn. This lower bound is tight up to some polynomial. Our result implies the 2Ω(n) lower bounds for the complete graph K2n + 1 and the complete bipartite graph Kn,O(n) that improves the lower bounds following from [1]. We show that for every graph G with n vertices that has no perfect matching there exists a resolution refutation of perfect matching principle for G of size O(n22n). Thus our lower bounds match upper bounds up to a multiplicative constant in the exponent. Our results also imply the well-known exponential lower bounds on the resolution complexity of the pigeonhole principle, the functional pigeonhole principle and the pigeonhole principle over a graph. We also prove the following corollary. For every natural number d, for every n large enough, for every function h : {1, 2, . . . , n} → {1, 2, . . . , d}, we construct a graph with n vertices that has the following properties. There exists a constant D such that the degree of the i-th vertex is at least h(i) and at most D, and it is impossible to make all degrees equal to h(i) by removing the graph’s edges. Moreover, any proof of this statement in the resolution proof system has size 2Ω(n). This result implies well-known exponential lower bounds on the Tseitin formulas as well as new results: for example, the same property of a complete graph. Preliminary version of this paper appeared in proceedings of CSR-2015 [2].

In this paper we study heuristic proof systems and heuristic non-deterministic algorithms. We give an example of a language Y and a polynomial-time samplable distribution D such that the distributional problem (Y, D) belongs to the complexity class H

The existence of a ($p$-)optimal propositional proof system is a major open question in (proof) complexity; many people conjecture that such systems do not exist. Kraj\'{\i}\v{c}ek and Pudl\'{a}k \cite{KP} show that this question is equivalent to the existence of an algorithm that is optimal\footnote{Recent papers \cite{Monroe} call such algorithms \emph{$p$-optimal} while traditionally Levin's algorithm was called \emph{optimal}. We follow the older tradition. Also there is some mess in terminology here, thus please see formal definitions in Sect.~\ref{sec:prelim} below.} on all propositional tautologies. Monroe \cite{Monroe} recently gave a conjecture implying that such algorithm does not exist. We show that in the presence of errors such optimal algorithms \emph{do} exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow the algorithm to claim a small number of false ``theorems'' (according to any polynomial-time samplable distribution on non-tautologies) and err with bounded probability on other inputs. Our result can also be viewed as the existence of an optimal proof system in a class of proof systems obtained by generalizing automatizable proof systems.

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, we show how to construct an infinitely-often one-way function based on f . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.