We show the following hardness results for monotone learning and approximation of monotone circuit size: 1) Under the Randomised Exponential-Time Hypothesis (rETH), it requires time n^{Ω(log n)} to PAC-learn monotone formulas with n input bits and size s(n) = n by monotone circuits of size n^{(log n)^{1-ε}}, for every ε > 0. 2) Under the Randomised Exponential-Time Hypothesis (rETH), for any δ > 0, there is a polynomially bounded function m such that m^{1-δ}-multiplicatively approximating the minimum monotone circuit size of a monotone function consistent with a sequence of m(n) labelled examples {(x_i, b_i)} over n-bit inputs requires time m^{Ω(log(m))}. Our results are shown by a novel application of lifting arguments in proof and communication complexity to hardness of monotone learning, by building on the seminal result of Atserias and Müller [Atserias and Müller, 2020] on hardness of automating Resolution proofs.

We study whether lower bounds against constant-depth algebraic circuits computing the Permanent over finite fields (Limaye-Srinivasan-Tavenas, J. ACM 2025; Forbes, CCC 2024) are hard to prove in certain proof systems. We focus on a DNF formula that expresses that such lower bounds are hard for constant-depth algebraic proofs. Using an adaptation of the diagonalization framework of Santhanam and Tzameret (SIAM J. Comput. 2025), we show unconditionally that this family of DNF formulas does not admit polynomial-size propositional AC0[p]-Frege proofs infinitely often. This rules out the possibility that the DNF family is easy, and establishes that its status is either that of a hard tautology for AC0[p]-Frege or else unprovable (not a tautology). While it remains open whether the DNFs in question are tautologies, we provide evidence in this direction. In particular, under the plausible assumption that certain weak properties of multilinear algebra, specifically those involving tensor rank, do not admit short constant-depth algebraic proofs, the DNFs are tautologies. We also observe that several weaker variants of the DNF formula are provably tautologies, and we show that the question of whether the DNFs are tautologies connects to conjectures of Razborov (ICALP 1996) and Krajicek (J. Symb. Log. 2004). Our result has two additional features. (i) Existential depth amplification: the DNF formula is parameterised by a constant depth d bounding the depth of the algebraic proofs. We show that there exists some fixed depth d such that if there are no small depth-d algebraic proofs of certain circuit lower bounds for the Permanent, then there are no such small algebraic proofs in any constant depth. (ii) Necessity: we show that our result is a necessary step towards establishing lower bounds against constant-depth algebraic proofs, and more generally against any sufficiently strong proof system.

A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser [ 16 ] posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an unconditional polynomial-time randomized algorithm B such that, for infinitely many values of n , \(B(1^n)\) outputs a canonical n -bit prime \(p_n\) with high probability. More generally, we prove that for every dense property Q of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying Q . This improves upon a subexponential-time construction of Oliveira and Santhanam [ 49 ]. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell [ 11 ], using a variant of the Shaltiel–Umans generator [ 51 ].

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.

Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly , such as PSPACE/poly and E/poly . We show the following: (Unconditional Lower Bounds for Learning) Building on Klivans et al. (2013), we prove unconditionally that BPE/poly cannot be weakly learned in polynomial time over the uniform distribution, even with membership and equivalence queries. (Robustness of Learning) For the concept classes EXP/poly and PSPACE/poly , we unconditionally show that worst-case and average-case learning are equivalent, that PAC -learnability and learnability over the uniform distribution are equivalent, and that membership queries do not help in either case. (Reducing Succinct Search to Decision for Learning) For the decision problems R Kt and R KS capturing the complexity of learning EXP/poly and PSPACE/poly , respectively, we show a succinct search to decision reduction: for each of these problems, the problem is in BPP iff there is a probabilistic polynomial-time algorithm computing circuits encoding proofs for positive instances of the problem. This is shown via a more general result giving succinct search to decision results for PSPACE, EXP and NEXP , which might be of independent interest. (Implausibility of Oblivious Strongly Black-Box Reductions showing NP -hardness of learning NP/poly ) We define a natural notion of hardness of learning with respect to oblivious strongly blackbox reductions. We show that learning PSPACE/poly is PSPACE hard with respect to oblivious strongly black-box reductions. On the other hand, if learning NP/poly is NP -hard with respect to oblivious strongly black-box reductions, the polynomial hierarchy collapses.

We introduce and study the following natural total search problem, which we call the heavy element avoidance (Heavy Avoid) problem: for a distribution on$N$bits specified by a Boolean circuit sampling it, and for some parameter$\delta(N)\geq 1/$poly$(N)$fixed in advance, output an$N$-bit string that has probability less than$\delta(N)$. We show that the complexity of Heavy Avoid is closely tied to frontier open questions in complexity theory about uniform randomized lower bounds and derandomization. Among other results, we show: 1)For a wide range of circuit classes$\mathcal{C}$, including$\text{ACC}^{0}, \text{TC}^{0},\text{NC}^{1}$and general Boolean circuits, EX P does not have uniform randomized C-circuits if and only if Heavy Avoid for uniform implicit C -samplers has efficient deterministic algorithms infinitely often. This gives the first algorithmic characterization of lower bounds for EXP against uniform randomized low-depth circuits. We show similar algorithmic characterizations for lower bounds in PSPACE, NP and$\text{EXP}^{\text{NP}}$. 2)Unconditionally, there are polynomial-time pseudodeterministic algorithms that work infinitely often for several variants of Heavy Avoid, such as for uniform samplers of small randomness complexity. In contrast, the existence of a similar algorithm that solves Heavy Avoid for arbitrary polynomial-time samplers would solve a long-standing problem about hierarchies for probabilistic time. 3)If there is a time and depth efficient deterministic algorithm for Heavy Avoid, then$BPP=P$. Without the depth-efficiency requirement in the assumption, we still obtain a non-trivial form of infinitely-often deterministic simulation of randomized algorithms. These results are shown using non-black-box reductions, and we argue that the use of non-black-box reductions is essential here. The full version is available on ECCC [1].

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 randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser [1] posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an unconditional polynomial-time randomized algorithm B such that, for infinitely many values of $n, B\left(1^{n}\right)$ outputs a canonical n-bit prime $p_{n}$ with high probability. More generally, we prove that for every dense property Q of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying Q. This improves upon a subexponential-time construction of Oliveira and Santhanam [2]. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell [3], using a variant of the Shaltiel-Umans generator [4].

We consider the range avoidance problem (called Avoid): given the description of a circuit with more output gates than input gates, find a string that is not in the range of the circuit. This problem is complete for the class APEPP that corresponds to explicit constructions of objects whose existence follows from the probabilistic method (Korten, FOCS 2021). Motivated by applications in explicit constructions and complexity theory, we initiate the study of the range avoidance problem for weak circuit classes, and obtain the following results: 1)Generalising Williams’s connections between circuitanalysis algorithms and circuit lower bounds (J. ACM 2014), we present a framework for solving $\mathscr{C}$-Avoid in FPNPusing circuit-analysis data structures for $\mathscr{C}$, for “typical” multi-output circuit classes $\mathscr{C}$. As an application, we present a non-trivial FPNPrange avoidance algorithm for De Morgan formulas./inlp>An important technical ingredient is a construction of rectangular PCPs of proximity, building on the rectangular PCPs by Bhangale, Harsha, Paradise, and Tal (FOCS 2020).2)Using the above framework, we show that circuit lower bounds for ENPare equivalent to circuit-analysis algorithms with ENPpreprocessing. This is the first equivalence result regarding circuit lower bounds for ENP. Our equivalences have the additional advantages that they work in both infinitely-often and almost-everywhere settings, and that they also hold for larger (e.g., subexponential) size bounds.3)Complementing the above results, we show that in some settings, solving $\mathscr{C}$-Avoid would imply breakthrough lower bounds, even for very weak circuit classes $\mathscr{C}$. In particular, an algorithm for AC0-Avoid with polynomial stretch implies lower bounds against NC1, and an algorithm for $NC_{4}^{0}$-Avoid with very small stretch implies lower bounds against NC1and branching programs.4)We show that Avoid is in FNP if and only if there is a propositional proof system that breaks every non-uniform proof complexity generator. This result connects the study of range avoidance with fundamental questions in proof complexity.

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: 1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof. 2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function. 3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0. 4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets. 5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20].

We show broad equivalences in the average-case complexity of many different meta-complexity problems, including Kolmogorov complexity, time-bounded Kolmogorov complexity, and the Minimum Circuit Size Problem. These results hold for a wide range of parameters (various thresholds, approximation gaps, weak or strong average-case hardness, etc.) and complexity notions, showing the theory of meta-complexity is very *robust* in the average-case setting.

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.

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, the deterministic query complexity, $D(f)$, is at most quartic in the quantum query complexity, $Q(f)$: $D(f) = O(Q(f)^4)$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We also use the result to resolve the quantum analogue of the Aanderaa-Karp-Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $Q(f) = Ω(n)$, which is also optimal.

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity, K^t, is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures: - We show, perhaps surprisingly, that the KT complexity is bounded-error average-case hard if and only if there exist one-way functions in constant parallel time (i.e. NC⁰). This result crucially relies on the idea of randomized encodings. Previously, a seminal work of Applebaum, Ishai, and Kushilevitz (FOCS'04; SICOMP'06) used the same idea to show that NC⁰-computable one-way functions exist if and only if logspace-computable one-way functions exist. - Inspired by the above result, we present randomized average-case reductions among the NC¹-versions and logspace-versions of K^t complexity, and the KT complexity. Our reductions preserve both bounded-error average-case hardness and zero-error average-case hardness. To the best of our knowledge, this is the first reduction between the KT complexity and a variant of K^t complexity. - We prove tight connections between the hardness of K^t complexity and the hardness of (the hardest) one-way functions. In analogy with the Exponential-Time Hypothesis and its variants, we define and motivate the Perebor Hypotheses for complexity measures such as K^t and KT. We show that a Strong Perebor Hypothesis for K^t implies the existence of (weak) one-way functions of near-optimal hardness 2^{n-o(n)}. To the best of our knowledge, this is the first construction of one-way functions of near-optimal hardness based on a natural complexity assumption about a search problem. - We show that a Weak Perebor Hypothesis for MCSP implies the existence of one-way functions, and establish a partial converse. This is the first unconditional construction of one-way functions from the hardness of MCSP over a natural distribution. - Finally, we study the average-case hardness of MKtP. We show that it characterizes cryptographic pseudorandomness in one natural regime of parameters, and complexity-theoretic pseudorandomness in another natural regime.

For a complexity class C and language L, a constructive separation of “L is not in C” gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every C-algorithm attempting to decide L. We study the questions: Which lower bounds can be made constructive? What are the consequences of constructive separations? We build a case that “constructiveness” serves as a dividing line between many weak lower bounds we know how to prove, and strong lower bounds against P, ZPP, and BPP. Put another way, constructiveness is the opposite of a complexity barrier: it is a property we want lower bounds to have. Our results fall into three broad categories. 1. For many separations, making them constructive would imply breakthrough lower bounds. Our first set of results shows that, for many well-known lower bounds against streaming algorithms, one-tape Turing machines, and query complexity, as well as lower bounds for the Minimum Circuit Size Problem, making these lower bounds constructive would imply break-through separations ranging from “EXP not equal to BPP” to even “P not equal to NP”. 2. Most conjectured uniform separations can be made constructive. Our second set of results shows that for most major open problems in lower bounds against P, ZPP, and BPP, including “P not equal to NP”, “P not equal to PSPACE”, “P not equal to PP”, “ZPP not equal to EXP”, and “BPP not equal to NEXP”, any proof of the separation would further imply a constructive separation. Our results generalize earlier results for “P not equal to NP” [Gutfreund, Shaltiel, and Ta-Shma, CCC 2005] and “BPP not equal to NEXP” [Dolev, Fandina and Gutfreund, CIAC 2013]. Thus any proof of these strong lower bounds must also yield a constructive version, compared to many weak lower bounds we currently know. 3. Some separations cannot be made constructive. Our third set of results shows that certain complexity separations cannot be made constructive. We observe that for all super-polynomially growing functions$\mathbf{t}$, there are no constructive separations for detecting high t-time Kolmogorov complexity (a task which is known to be not in P) from any complexity class, unconditionally. We also show that under plausible conjectures, there are languages in NP -$\mathbf{P}$for which there are no constructive separations from any complexity class.

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of BPTIME: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

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.

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.

The fundamental Minimum Circuit Size Problem is a well-known example of a problem that is neither known to be in 𝖯 nor known to be NP-hard. Kabanets and Cai [Kabanets and Cai, 2000] showed that if MCSP is NP-hard under "natural" m-reductions, superpolynomial circuit lower bounds for exponential time would follow. This has triggered a long line of work on understanding the power of reductions to MCSP. Nothing was known so far about consequences of NP-hardness of MCSP under general Turing reductions. In this work, we consider two structured kinds of Turing reductions: parametric honest reductions and natural reductions. The latter generalize the natural reductions of Kabanets and Cai to the case of Turing-reductions. We show that NP-hardness of MCSP under these kinds of Turing-reductions imply superpolynomial circuit lower bounds for exponential time.

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.

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 study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades. Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.

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 introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are: 1) We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings. 2) We show that the security of Goldreich's PRG and OWF is closely related to two other long-standing problems: Specifically, to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits. We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best "hard" candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.

We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [Eran Gat and Shafi Goldwasser, 2011]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed output with high probability. We explore pseudo-determinism in the settings of learning and approximation. Our goal is to simulate known randomized algorithms in these settings by pseudo-deterministic algorithms in a generic fashion - a goal we succinctly term pseudo-derandomization. Learning. In the setting of learning with membership queries, we first show that randomized learning algorithms can be derandomized (resp. pseudo-derandomized) under the standard hardness assumption that E (resp. BPE) requires large Boolean circuits. Thus, despite the fact that learning is an algorithmic task that requires interaction with an oracle, standard hardness assumptions suffice to (pseudo-)derandomize it. We also unconditionally pseudo-derandomize any {quasi-polynomial} time learning algorithm for polynomial size circuits on infinitely many input lengths in sub-exponential time. Next, we establish a generic connection between learning and derandomization in the reverse direction, by showing that deterministic (resp. pseudo-deterministic) learning algorithms for a concept class C imply hitting sets against C that are computable deterministically (resp. pseudo-deterministically). In particular, this suggests a new approach to constructing hitting set generators against AC^0[p] circuits by giving a deterministic learning algorithm for AC^0[p]. Approximation. Turning to approximation, we unconditionally pseudo-derandomize any poly-time randomized approximation scheme for integer-valued functions infinitely often in subexponential time over any samplable distribution on inputs. As a corollary, we get that the (0,1)-Permanent has a fully pseudo-deterministic approximation scheme running in sub-exponential time infinitely often over any samplable distribution on inputs. Finally, we {investigate} the notion of approximate canonization of Boolean circuits. We use a connection between pseudodeterministic learning and approximate canonization to show that if BPE does not have sub-exponential size circuits infinitely often, then there is a pseudo-deterministic approximate canonizer for AC^0[p] computable in quasi-polynomial time.

We show that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models. We term this phenomenon "hardness magnification". Our examples of hardness magnification include: 1. Let MCSP be the decision problem whose YES instances are truth tables of functions with circuit complexity at most s(n). We show that if MCSP[2^√n] cannot be solved on average with zero error by formulas of linear (or even sub-linear) size, then NP does not have polynomial-size formulas. In contrast, Hirahara and Santhanam (2017) recently showed that MCSP[2^√n] cannot be solved in the worst case by formulas of nearly quadratic size. 2. If there is a c > 0 such that for each positive integer d there is an ε > 0 such that the problem of checking if an n-vertex graph in the adjacency matrix representation has a vertex cover of size (log n)^c cannot be solved by depth-d AC^0 circuits of size m^1+ε, where m = Θ(n^2), then NP does not have polynomial-size formulas. 3. Let (α, β)-MCSP[s] be the promise problem whose YES instances are truth tables of functions that are α-approximable by a circuit of size s(n), and whose NO instances are truth tables of functions that are not β-approximable by a circuit of size s(n). We show that for arbitrary 1/2c, let MKtP[c, s] be the promise problem whose YES instances are strings of Kt complexity at most c(N) and NO instances are strings of Kt complexity greater than s(N). We show that if there is a δ > 0 such that for each ε > 0, MKtP[N^ε, N^ε + 5 log(N)] requires Boolean circuits of size N^1+δ, then EXP is not contained in SIZE (poly). For each of the cases of magnification above, we observe that standard hardness assumptions imply much stronger lower bounds for these problems than we require for magnification. We further explore magnification as an avenue to proving strong lower bounds, and argue that magnification circumvents the "natural proofs" barrier of Razborov and Rudich (1997). Examining some standard proof techniques, we find that they fall just short of proving lower bounds via magnification. As one of our main open problems, we ask whether there are other meta-mathematical barriers to proving lower bounds that rule out approaches combining magnification with known techniques.

We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have demonstrated the central role of this problem and its variations in diverse areas such as cryptography, derandomization, proof complexity, learning theory, and circuit lower bounds. The NP-hardness of computing the minimum numbers of terms in a DNF formula consistent with a given truth table was proved by W. Masek [William J. Masek, 1979] in 1979. In this work, we make the first progress in showing NP-hardness for more expressive classes of circuits, and establish an analogous result for the MCSP problem for depth-3 circuits of the form OR-AND-MOD_2. Our techniques extend to an NP-hardness result for MOD_m gates at the bottom layer under inputs from (Z / m Z)^n.

We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence {pn} of primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often.

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is epsilon_d > 0 such that Parity has correlation at most 1/n^{Omega(1)} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires, and the Generalized Andreev Function has correlation at most 1/2^{n^{Omega(1)}} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires. Previously, only worst-case lower bounds in this setting were known [Impagliazzo/Paturi/Saks, SIAM J. Comp., 1997]. We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth-$d$ threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC^0 circuits with n^{o(1)} general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC^0 with n^{o(1)} threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas ofany depth. Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.

We strengthen the nondeterministic hierarchy theorem for non-deterministic polynomial time to show that the lower bound holds against sub-linear advice. More formally, we show that for any constants d and d' such that 1 <= d < d', and for any time-constructible bound t=o(n^d), there is a language in NTIME(n^d) which is not in NTIME(t)/n^{1/d'}. The best known earlier separation of Fortnow, Santhanam and Trevisan could only handle o(log(n)) bits of advice in the lower bound, and was not tight with respect to the time bounds. We generalize our hierarchy theorem to work for other syntactic complexity measures between polynomial time and polynomial space, including alternating polynomial time with any fixed number of alternations. We also use our technique to derive an almost-everywhere hierarchy theorem for non-deterministic classes which use a sub-linear amount of non-determinism, i.e., the lower bound holds on all but finitely many input lengths rather than just on infinitely many. As one application of our main result, we derive a new lower bound for NP against NP-uniform non-deterministic circuits of size O(n^k) for any fixed k. This result is a significant strengthening of a result of Kannan, which states that not all of NP can be solved with P-uniform circuits of size O(n^k) for any fixed k. As another application, we show strong non-uniform lower bounds for the complexity class RE of languages decidable in randomized linear exponential time with one sided error.

We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems. As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms', and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs. As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model.

The study of the worst-case complexity of the Boolean Satisfiability (SAT) problem has seen considerable progress in recent years, for various types of instances including CNFs, Boolean formulas and constant-depth circuits. We systematically investigate the complexity of solving mixed instances, where different parts of the instance come from different types. Our investigation is motivated partly by practical contexts such as SMT (Satisfiability Modulo Theories) solving, and partly by theoretical issues such as the exact complexity of graph problems and the desire to find a unifying framework for known satisfiability algorithms.

This issue of SICOMP contains seven specially selected papers from the Forty-Fourth Annual ACM Symposium on Theory of Computing, otherwise known as STOC 2012, held May 19 to 22 in New York, New York. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors and subjected to the standard thorough reviewing process of SICOMP. The program committee consisted of Richard Cleve, Parikshit Gopalan, Jason Hartline, Tom Hayes, Anna Karlin, Sanjeev Khanna, Andrew McGregor, Rina Panigrahy, Toniann Pitassi, Ran Raz, Charles Rackoff, Satish Rao, Oded Regev, Dana Ron, Guy Rothblum, Amin Saberi, Rahul Santhanam, Shubhangi Saraf, Daniel Spielman, Madhur Tulsiani, Suresh Venkatasubramanian, Avi Wigderson, and David Williamson. They selected 90 papers out of 303 submissions. We briefly describe the papers that appear here. In “The Multiparty Communication Complexity of Set Disjointness,” Alexander Sherstov presents an $\Omega(n/4^k)^{1/4}$ lower bound on the communication complexity of the $k$-party set disjointness problem. Previously, no polynomial lower bounds were known for $k=\omega(1)$ players. In “Routing in Undirected Graphs with Constant Congestion,” Julia Chuzhoy presents an efficient randomized algorithm that, given a set of demand pairs, routes a polylogarithmic fraction of the maximum number of demand pairs that could be routed on edge-disjoint paths. The routing returned uses each edge at most a constant number of times, whereas the best previous best algorithm guaranteed only polylogarithmic reuse of a single edge. In “Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-$D$ Occur-$k$ Formulas and Depth-$3$ Transcendence Degree-$k$ Circuits,” Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, and Nitin Saxena study the black box identity testing problem in arithmetic complexity. They present an approach using the Jacobian that unifies and generalizes several previous results on polynomial time black box identity testing and is also useful for proving circuit lower bounds. In “The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme,” Yair Bartal, Lee-Ad Gottlieb, and Robert Krauthgamer present an efficient randomized $(1+\epsilon)$-approximation algorithm for the traveling salesman problem when the distances correspond to an arbitrary metric space with bounded intrinsic dimension. In “Computing a Nonnegative Matrix Factorization---Provably,” Sanjeev Arora, Rong Ge, Ravi Kannan, and Ankur Moitra investigate the problem of factorizing a matrix into two nonnegative matrices, an important problem in machine learning, among other areas. They present efficient algorithms for certain natural families of matrices, along with a complementary hardness result. In “Time-Space Trade-offs in Resolution: Superpolynomial Lower Bounds for Superlinear Space,” Paul Beame, Chris Beck, and Russell Impagliazzo give the first size-space tradeoffs for resolution proofs that apply to superlinear space. In “Robustly Solvable Constraint Satisfaction Problems,” Libor Barto and Marcin Kozik characterize constraint satisfaction problems that are robustly satisfiable. Guruswami and Zhou conjectured that a constraint satisfaction problem is robustly satisfiable if and only if it has bounded width. Barto and Kozik confirm this conjecture. We thank the authors and the program committee for their hard work, and we especially thank the reviewers for their work in evaluating and improving the submitted papers.

We consider C-compression games, a hybrid model between computational and communication complexity. A C-compression game for a function f:{0,1}^n -> {0,1} is a two-party communication game, where the first party Alice knows the entire input x but is restricted to use strategies computed by C-circuits, while the second party Bob initially has no information about the input, but is computationally unbounded. The parties implement an interactive communication protocol to decide the value of f(x), and the communication cost of the protocol is the maximum number of bits sent by Alice as a function of n = |x|. We show that any AC_d[p]-compression protocol to compute Majority_n requires communication n / (log(n))^(2d + O(1)), where p is prime, and AC_d[p] denotes polynomial size unbounded fan-in depth-d Boolean circuits extended with modulo p gates. This bound is essentially optimal, and settles a question of Chattopadhyay and Santhanam (2012). This result has a number of consequences, and yields a tight lower bound on the total fan-in of oracle gates in constant-depth oracle circuits computing Majority_n. We define multiparty compression games, where Alice interacts in parallel with a polynomial number of players that are not allowed to communicate with each other, and communication cost is defined as the sum of the lengths of the longest messages sent by Alice during each round. In this setting, we prove that the randomized r-round AC^0[p]-compression cost of Majority_n is n^(Theta(1/r)). This result implies almost tight lower bounds on the maximum individual fan-in of oracle gates in certain restricted bounded-depth oracle circuits computing Majority_n. Stronger lower bounds for functions in NP would separate NP from NC^1. Finally, we consider the round separation question for two-party AC-compression games, and significantly improve known separations between r-round and (r+1)-round protocols, for any constant r.

We study algorithms for the satisfiability problem for quantified Boolean formulas (QBFs), and consequences of faster algorithms for circuit complexity. We show that satisfiability of quantified 3-CNFs with m clauses, n variables, and two quantifier blocks (one existential block and one universal) can be solved deterministically in time . poly(m). For the case of multiple quantifier blocks (alternations), we show that satisfiability of quantified CNFs of size poly(n) on n variables with q quantifier blocks can be solved in 2n−n1/(q + 1)· poly(n) time by a zero-error randomized algorithm. These are the first provable improvements over brute force search in the general case, even for quantified polynomial-sized CNFs with two quantifier blocks. A second zero-error randomized algorithm solves QBF on circuits of size s in 2n–Ω(q) · poly(s) time when the number of quantifier blocks is q. We complement these algorithms by showing that improvements on them would imply new circuit complexity lower bounds. For example, if satisfiability of quantified CNF formulas with n variables, poly(n) size and at most q quantifier blocks can be solved in time 2n–nwq (1/q) then the complexity class NEXP does not have O(log n) depth circuits of polynomial size. Furthermore, solving satisfiability of quantified CNF formulas with n variables, poly(n) size and O(log n) quantifier blocks in time 2n–w(log (n)) time would imply the same circuit complexity lower bound. The proofs of these results proceed by establishing strong relationships between the time complexity of QBF satisfiability over CNF formulas and the time complexity of QBF satisfiability over arbitrary Boolean formulas.

We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is “medium uniform” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: ; For all k, P is not contained in P-uniform SIZE(nk). That is, for all k there is a language Lk∈ P that does not have O(nk)-size circuits constructible in polynomial time. This improves Kannan's lower bound from 1982 that NP is not in P-uniform SIZE(nk) for any fixed k. ; For all k, NP is not in P||NP-uniform SIZE(nk). This also improves Kannan's theorem, but in a different way: the uniformity condition on the circuits is stronger than that on the language itself. ; For all k, LOGSPACE does not have LOGSPACE-uniform branching programs of size nk. 2. Eliminating Non-Uniformity and (Non-Uniform) Circuit Lower Bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniform NC1in ACC0/poly or TC0/poly into a medium-uniform simulation using small advice. This lemma can be used to simplify the proof that faster SAT algorithms imply NEXP circuit lower bounds, and leads to the following new connection: . Consider the following task: given a TC0circuit C of nO(1)size, output yes when C is unsatisfiable, and output no when C has at least 2n-2satisfying assignments. (Behavior on other inputs can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministically in 2n-ω(log n)time, then NEXP ⊄ TC0/poly. The lemma can also be used to derandomize randomized TC0simulations of NC1on almost all inputs: ; Suppose NC1⊆ BPTC0. Then for every ε > 0 and every language L in NC1, there is a (uniform) TC0circuit family of polynomial size recognizing a language L' such that L and L' differ on at most 2nϵinputs of length n, for all n.

We formulate a new connection between instance compressibility [1]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multi-round instance compression. We show that there is no probabilistic multi-round compression protocol for Parity in which the computationally bounded party uses a non-uniform AC0-circuit and transmits at most n/(log(n))ω(1)bits. This result is tight, and strengthens results of Dubrov and Ishai. We also show that a similar lower bound holds for Majority. We also consider the question of round separation, i.e., whether for each r ≥ 1, there are functions which can be compressed better with r rounds of compression than with r - 1 rounds. We answer this question affirmatively for compression using constant-depth polynomial-size circuits. Finally, we prove the first non-trivial lower bounds for 1-round compressibility of Parity by polynomial size ACC0[p] circuits where p is an odd prime.

Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking whether f is identically zero, it suffices to query f on an arbitrary test set of r + 1 points. Could this brute-force method be improved upon by a single point? We develop a framework where such a marginal improvement implies that Permanent does not have polynomial size arithmetic circuits.

We associate to each Boolean language complexity class C the algebraic class a.C consisting of families of polynomials {f_n} for which the evaluation problem over the integers is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a.NEXP does not have poly size constant-free arithmetic circuits. 2. a.NEXP^RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a.MA does not have arithmetic circuits of size n^k. Items 1 and 2 strengthen two results due to (Kabanets and Impagliazzo, 2004). The third item improves a lower bound due to (Santhanam, 2009). We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT is in i.o-NTIME[2^{n^{o(1)}}]/n^{o(1)} if and only if there exists a family of multilinear polynomials in a.NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.

We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search. Our first main result is a simple deterministic algorithm running in time 2n-Ω(n)for satisfiability of formulae of linear size in n, where n is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound. Our second main result is a deterministic algorithm running in time 2n-Ω(n/log(n))for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are "structured", in a certain precise sense, the algorithm can be modified to run in time 2n-Ω(n). To the best of our knowledge, no non-trivial algorithms were known for these problems before. As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size 2n-Ω(n). As a consequence, we get strong superlinear average-case formula size lower bounds for the Parity function.

In 1982, Kannan showed that SigmaP2does not have nk-sized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan's result, we still cannot prove that PNPdoes not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence - the "algebrization'' barrier - that current techniques have inherent limitations in this respect. We explore questions about fixed-polynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including 1) The following are equivalent: -NP is in SIZE(nk) (has O(nk)-size circuit families) for some k -For each c, PNP[nc]is in SIZE(nk) for some k -ONP/1 is in SIZE(nk) for some k, where ONP is the class of languages accepted obliviously by NP machines, with witnesses for "yes" instances depending only on the input length. 2) For a large number of natural classes C and all k ges C is in SIZE(nk) if and only if C/1 cap P/poly is in SIZE(nk). 3) If there is a d such that MATIME(n) sube NTIME(nd), then PNPdoes not have O(nk) size circuits for any k > 0. 4) One cannot show n2-size circuit lower bounds for oplusP without new nonrelativizing techniques. In particular, the proof that PP nsube SIZE(nk) for all k relies on the (relativizing) result that PPPsube MA rArr PP nsube SIZE(nk), and we give an oracle relative to which PoplusPsube MA and oplusP sube SIZE(n2) both hold.

We study the branching program complexity of the {\em tree evaluation problem}, introduced in \cite{BrCoMcSaWe09} as a candidate for separating \nl\ from\logcfl. The input to the problem is a rooted, balanced $d$-ary tree of height$h$, whose internal nodes are labelled with $d$-ary functions on$[k]=\{1,\ldots,k\}$, and whose leaves are labelled with elements of $[k]$.Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. Deterministic $k$-way branching programs as related to black pebbling algorithms have been studied in \cite{BrCoMcSaWe09}. Here we introduce the notion of {\em fractional pebbling} of graphs to study non-deterministicbranching program size. We prove that this yields non-deterministic branching programs with $\Theta(k^{h/2+1})$ states solving the Boolean problem ``determine whether the root has value 1'' for binary trees - this isasymptotically better than the branching program size corresponding toblack-white pebbling. We prove upper and lower bounds on the fractionalpebbling number of $d$-ary trees, as well as a general result relating thefractional pebbling number of a graph to the black-white pebbling number. We introduce a simple semantic restriction called {\em thrifty} on $k$-way branching programs solving tree evaluation problems and show that the branchingprogram size bound of $\Theta(k^h)$ is tight (up to a constant factor) for all $h\ge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are thrifty as well, and that the bound of $\Theta(k^{h/2+1})$ is tight for non-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty branching programs are optimal among $k$-way branching programs solving the tree evaluation problem - proving this for deterministic programs would separate \lspace\ from \logcfl\, and proving it for non-deterministic programs would separate \nl\ from \logcfl.

We show that for each $k>0$, $\mathsf{MA}/1$ ($\mathsf{MA}$ with 1 bit of advice) does not have circuits of size $n^k$. This implies the first superlinear circuit lower bounds for the promise versions of the classes $\mathsf{MA}$, $\mathsf{AM}$, and $\mathsf{ZPP}_{\parallel}^{\mathsf{NP}}$. We extend our main result in several ways. For each k, we give an explicit language in $(\mathsf{MA}\cap\mathsf{coMA})/1$ which does not have circuits of size $n^k$. We also adapt our lower bound to the average-case setting; i.e., we show that $\mathsf{MA}/1$ cannot be solved on more than $1/2+1/n^k$ fraction of inputs of length n by circuits of size $n^k$. Furthermore, we prove that $\mathsf{MA}$ does not have arithmetic circuits of size $n^k$ for any k. As a corollary to our main result, we obtain that derandomization of $\mathsf{MA}/O(1)$ implies the existence of pseudorandom generators computable using $O(1)$ bits of advice.

The OR-SAT problem asks, given Boolean formulae Φ1,...,Φm each of size at most n, whether at least one of the Φi's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. A number of parametric $\NP$ problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. An approach of Harnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (Buhrman-Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.

Article Circuit lower bounds for Merlin-Arthur classes Author: Rahul Santhanam Simon Fraser University, Vancouver, BC, Canada Simon Fraser University, Vancouver, BC, CanadaView Profile Authors Info & Claims STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computingJune 2007Pages 275–283https://doi.org/10.1145/1250790.1250832Published:11 June 2007Publication History 14citation381DownloadsMetricsTotal Citations14Total Downloads381Last 12 Months6Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function on a smaller number of bits which has greater hardness when measured in terms of input length. A hardness extractor takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function defined on a smaller number of bits which has close to maximum hardness. We prove several positive and negative results about these objects. First, we observe that hardness-based pseudo-random generators can be used to extract deterministic hardness from non-deterministic hardness. We derive several consequences of this observation. Among other results, we show that if E/O(n) has exponential non-deterministic hardness, then E/O{n) has deterministic hardness 2n/n, which is close to the maximum possible. We demonstrate a rare downward closure result: E with sub-exponential advice is contained in non-uniform space 2deltanfor all delta > 0 if and only if there is k > 0 such that P with quadratic advice can be approximated in non-uniform space nk. Next, we consider limitations on natural models of hardness condensing and extraction. We show lower bounds on the advice length required for hardness condensing in a very general model of "relativizing" condensers. We show that non-trivial black-box extraction of deterministic hardness from deterministic hardness is essentially impossible. Finally, we prove positive results on hardness condensing in certain special cases. We show how to condense hardness from a biased function without advice using a hashing technique. We also give a hardness condenser without advice from average-case hardness to worst-case hardness. Our technique uses a connection between hardness condensing and explicit constructions of covering codes

In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment and small-world models, motivated by real-world graphs such as the Internet topology. To address the natural question of which model is best for a particular data set, we propose a model selection criterion for graph models. Since each model is in fact a probability distribution over graphs, we suggest using Maximum Likelihood to compare graph models and select their parameters. Interestingly, for the case of graph models, computing likelihoods is a difficult algorithmic task. However, we design and implement MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We hope that this novel use of ML will objectify comparisons between graph models.

We show that for any constant a, ZPP/b(n) strictly contains ZPTIME(na)/b(n) for some b(n) = O(log n log log n). Our techniques are very general and give the same hierarchy for all common semantic time classes including RTIME, NTIME ∩ coNTIME, UTIME, MATIME, AMTIME and BQTIME.We show a stronger hierarchy for RTIME: For every constant c, RP/1 is not contained in RTIME(nc)/(log n)1/2c. To prove this result we first prove a similar statement for NP by building on Zák's proof of the nondeterministic time hierarchy.

We derive a stronger consequence of $\mathsf{EXP}$ (deterministic exponential time) having polynomial-size circuits than was known previously, namely that for each language $L \in \mathsf{P}$ (polynomial time), and for each efficiently decidable error-correcting code E having nontrivial relative distance, there is a simulation of L in Merlin-Arthur polylogarithmic time that fools all deterministic polynomial-time adversaries for inputs that are codewords of E. Using the connection between circuit lower bounds and derandomization, we obtain uniform assumptions for derandomizing $\mathsf{BPP}$ (probabilistic polynomial time). Our results strengthen the space-randomness tradeoffs of Sipser [J. Comput. System Sci., 36 (1988), pp. 379--383], Nisan and Wigderson [J. Comput. System Sci.}, 49 (1994), pp. 149--167], and Lu [Comput. Complexity, 10 (2001), pp. 247--259]. We also consider a more quantitative notion of simulation, where the measure of success of the simulation is the fraction of inputs of a given length on which the simulation works. Among other results, we show that if there is no polynomial-time bound t such that $\mathsf{P}$ can be simulated well by Merlin-Arthur machines operating in time t, then for any $\epsilon > 0$ there is a simulation of $\mathsf{BPP}$ in $\mathsf{P}$ that works for all but $2^{n^{\epsilon}}$ inputs of length n. This is a uniform strengthening of a recent result of Goldreich and Wigderson [ Proceedings of the 6th International Workshop on Randomization and Approximation Techniques in Computer Science, 2002, pp. 209--223]. Finally, we give an unconditional simulation of multitape Turing machines operating in probabilistic time t by Turing machines operating in deterministic time o(2 t ). We show similar results for randomized $\mathsf{NC}^{1}$ circuits. Our proofs are based on a combination of techniques in the theory of derandomization with results on holographic proofs.

We show a hierarchy for probabilistic time with one bit of advice, specifically we show that for all real numbers 1 /spl les/ /spl alpha/ /spl les/ /spl beta/, BPTIME(n/sup /spl alpha//)/l /spl sube/ BPTIME(n/sup /spl beta//)/l. This result builds on and improves an earlier hierarchy of Barak using O(log log n) bits of advice. We also show that for any constant d > 0, there is a language L computable on average in BPP but not on average in BPTIME (n/sup d/). We build on Barak's techniques by using a different translation argument and by a careful application of the fact that there is a PSPACE-complete problem L such that worst-case probabilistic algorithms for L take only slightly more time than average-case algorithms.

Gives an extension of the result due to Paul, Pippenger, Szemeredi and Trotter (1983) that deterministic linear time (DTIME) is distinct from nondeterministic linear time (NTIME). We show that NTIME[n/spl radic/log*(n)] /spl ne/ DTIME[n/spl radic/log*(n)]. We show that if the class of multi-pushdown graphs has {o(n), o[n/log(n)]} segregators, then NTIME[n log(n)] /spl ne/ DTIME[n log(n)]. We also show that at least one of the following facts holds: (1) P /spl ne/ L, and (2) for all polynomially bounded constructible time bounds t, NTIME(t) /spl ne/ DTIME(t). We consider the problem of whether NTIME(t) is distinct from NSPACE(t) for constructible time bounds t. A pebble game on graphs is defined such that the existence of a "good" strategy for the pebble game on multi-pushdown graphs implies a "good" simulation of nondeterministic time-bounded machines by nondeterministic space-bounded machines. It is shown that there exists a "good" strategy for the pebble game on multi-pushdown graphs if the graphs have sublinear separators. Finally, we show that nondeterministic time-bounded Turing machines can be simulated by /spl Sigma//sub 4/ machines with an asymptotically smaller time bound, under the assumption that the class of multi-pushdown graphs has sublinear separators.