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.

Suppose we are given an infinite sequence of input cells, each initialized with a uniform random symbol from [n]. How hard is it to output a sequence in [n]n that is close to a uniform random permutation? Viola (SICOMP 2020) conjectured that if each output cell is computed by making d probes to input cells, then d≥ω(1). Our main result shows that, in fact, d≥ (logn)Ω(1), which is tight up to the constant in the exponent. Our techniques also show that if the probes are nonadaptive, then d≥ nΩ(1), which is an exponential improvement over the previous nonadaptive lower bound due to Yu and Zhan (ITCS 2024). Our results also imply lower bounds against succinct data structures for storing permutations.

We show that the perfect matching function on n-vertex graphs requires monotone circuits of size 2nΩ(1). This improves on the nΩ(logn) lower bound of Razborov (1985). Our proof uses the standard approximation method together with a new sunflower lemma for matchings.

We exhibit an n-bit partial function with randomized communication complexity O(logn) but such that any completion of this function into a total one requires randomized communication complexity nΩ(1). In particular, this shows an exponential separation between randomized and pseudodeterministic communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

Aaronson (STOC 2010) conjectured that almost k-wise independence fools constant-depth circuits; he called this the generalised Linial-Nisan conjecture. Aaronson himself later found a counterexample for depth-3 circuits. We give here an improved counterexample for depth-2 circuits (DNFs). This shows, for instance, that Bazzi’s celebrated result (k-wise independence fools DNFs) cannot be generalised in a natural way. We also propose a way to circumvent our counterexample: We define a new notion of pseudorandomness called local couplings and show that it fools DNFs and even decision lists.

Random Δ-CNF formulas are one of the few candidates that are expected to be hard for proof systems and SAT algotirhms. Assume we sample m clauses over n variables. Here, the main complexity parameter is clause density, χ := m/n. For a fixed Δ, there exists a satisfiability threshold c_Δ such that for χ > c_Δ a formula is unsatisfiable with high probability. and for χ < c_Δ it is satisfiable with high probability. Near satisfiability threshold, there are various lower bounds for algorithms and proof systems [Eli Ben-Sasson, 2001; Eli Ben-Sasson and Russell Impagliazzo, 1999; Michael Alekhnovich and Alexander A. Razborov, 2003; Dima Grigoriev, 2001; Grant Schoenebeck, 2008; Pavel Hrubes and Pavel Pudlák, 2017; Noah Fleming et al., 2017; Dmitry Sokolov, 2024], and for high-density regimes, there exist upper bounds [Uriel Feige et al., 2006; Sebastian Müller and Iddo Tzameret, 2014; Jackson Abascal et al., 2021; Venkatesan Guruswami et al., 2022]. One of the frontiers in the direction of proving lower bounds on these formulas is the k-DNF Resolution proof system (aka Res(k)). There are several known results for k = 𝒪(√{log n}/{log log n}}) [Nathan Segerlind et al., 2004; Michael Alekhnovich, 2011], that are applicable only for density regime near the threshold. In this paper, we show the first Res(k) lower bound that is applicable in higher-density regimes. Our results work for slightly larger k = 𝒪(√{log n}).

We prove that the sign-rank of the k Hamming Distance matrix on n bits is $2^{O(k)}$, independent of the number of bits n. This strongly refutes the conjecture of Hatami, Hatami, Pires, Tao, and Zhao (random 2022), and Hatami, Hosseini, and Meng (STOC 2023), repeated in several other papers, that the sign-rank should depend on n. This conjecture would have qualitatively separated margin from sign-rank (or, equivalently, bounded-error from unbounded-error randomized communication). In fact, our technique gives constant sign-rank upper bounds for all matrices which reduce to k-Hamming Distance, as well as large-margin matrices recently shown to be irreducible to k-Hamming Distance.

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size–depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: We introduce a new family of unsatisfiable 3-CNF formulas (called bracket formulas) that admit resolution refutations of quasipolynomial size while any refutation of polynomial depth requires exponential size.

The random Δ-CNF model is one of the most important distribution over Δ-SAT instances. It is closely connected to various areas of computer science, statistical physics, and is a benchmark for satisfiability algorithms. Fleming, Pankratov, Pitassi, and Robere and independently Hrubeš and Pudlák showed that when Δ = Θ(logn), any Cutting Planes proof for random Δ-CNF on n variables requires size 2n / n in the regime where the number of clauses guarantees that the formula is unsatisfiable with high probability. In this paper we show tight lower bound 2Ω(n) on size -proofs for random (logn)-CNF formulas. Moreover, our proof is much simpler and self-contained in contrast with previous results based on Jukna’s lower bound for monotone circuits.

Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains Ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors — a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.

A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $ε$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $ε$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $\mathbf{U}^n_k$ for _decision forests_, i.e. every output bit is computed as a decision tree of the inputs. For every $k$ there is an $O(\log n)$-depth decision forest sampling $\mathbf{U}^n_k$ with an inverse-polynomial error [Viola 2012, Czumaj 2015]. We show that for every $ε> 0$ there exists $τ$ such that for decision depth $τ\log (n/k) / \log \log (n/k)$, the error for sampling $\mathbf{U}_k^n$ is at least $1-ε$. Our result is based on the recent robust sunflower lemma [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]. Our second result is about matching a set of $n$-bit strings with the image of a $d$-_local_ circuit, i.e. such that each output bit depends on at most $d$ input bits. We study the set of all $n$-bit strings whose Hamming weight is at least $n/2$. We improve the previously known locality lower bound from $Ω(\log^* n)$ [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] to $Ω(\sqrt{\log n})$, leaving only a quartic gap from the best upper bound of $O(\log^2 n)$.

We present a top-down lower-bound method for depth-4 boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-4 circuits of size exponential in $n^{1 / 3}$. Our proof is an application of robust sunflowers and block unpredictability.

Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson [Michael Alekhnovich et al., 2004] we call a pseudorandom generator ℱ:{0, 1}ⁿ → {0, 1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement b ∉ Im(ℱ) for any string b ∈ {0, 1}^m. In [Michael Alekhnovich et al., 2004] the authors suggested the "functional encoding" of the considered statement for Nisan-Wigderson generator that allows the introduction of "local" extension variables. These extension variables may potentially significantly increase the power of the proof system. In [Michael Alekhnovich et al., 2004] authors gave a lower bound of exp[Ω(n²/{m⋅2^{2^Δ}})] on the length of Resolution proofs where Δ is the degree of the dependency graph of the generator. This lower bound meets the barrier for the restriction technique. In this paper, we introduce a "heavy width" measure for Resolution that allows us to show a lower bound of exp[n²/{m 2^𝒪(εΔ)}] on the length of Resolution proofs of the considered statement for the Nisan-Wigderson generator. This gives an exponential lower bound up to Δ := log^{2 - δ} n (the bigger degree the more extension variables we can use). In [Michael Alekhnovich et al., 2004] authors left an open problem to get rid of scaling factor 2^{2^Δ}, it is a solution to this open problem.

Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.

Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. [László Lovász et al., 1995] who showed the equivalence between regular Resolution and read-once branching programs for "unsatisfied clause search problem" (Search_φ). This connection is widely used, in particular, in the recent breakthrough result about the Clique problem in regular Resolution by Atserias et al. [Albert Atserias et al., 2018]. We study the branching programs with bounded repetitions, so-called (1,+k)-BPs (Sieling [Detlef Sieling, 1996]) in application to the Search_φ problem. On the one hand, it is a natural generalization of read-once branching programs. On the other hand, this model gives a powerful proof system that can efficiently certify the unsatisfiability of a wide class of formulas that is hard for Resolution (Knop [Alexander Knop, 2017]). We deal with Search_φ that is "relatively easy" compared to all known hard examples for the (1,+k)-BPs. We introduce the first technique for proving exponential lower bounds for the (1,+k)-BPs on Search_φ. To do it we combine a well-known technique for proving lower bounds on the size of branching programs [Detlef Sieling, 1996; Detlef Sieling and Ingo Wegener, 1994; Stasys Jukna and Alexander A. Razborov, 1998] with the modification of the "closure" technique [Michael Alekhnovich et al., 2004; Michael Alekhnovich and Alexander A. Razborov, 2003]. In contrast with most Resolution lower bounds, our technique uses not only "local" properties of the formula, but also a "global" structure. Our hard examples are based on the Flow formulas introduced in [Michael Alekhnovich and Alexander A. Razborov, 2003].

We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali–Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller (JACM 2020) that established an analogous result for Resolution.

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.

Building on [Clegg et al. '96], [Impagliazzo et al. '99] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(√(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen '16], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary.

One of the major open problems in proof complexity is to prove lower bounds on AC 0[p]-Frege proof systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove lower bounds on the size for Polynomial Calculus over the {± 1} basis. In this paper we show a technique for proving such lower bounds and moreover we also give lower bounds on the size for Sum-of-Squares over the {± 1} basis.

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.

Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity. Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).

For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols—or, equivalently, that a monotone function associated with F has large monotone circuit complexity. Our result extends to monotone real circuits, which yields new lower bounds for the Cutting Planes proof system.

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 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