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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We study close connections between Indistinguishability Obfuscation (IO) and the Minimum Circuit Size Problem (MCSP), and argue that efficient algorithms/construction for MCSP and IO create a synergy. Some of our main results are:
- If there exists a perfect (imperfect) IO that is computationally secure against nonuniform polynomial-size circuits, then for all k ∈ ℕ: NP ∩ ZPP^{MCSP} ⊈ SIZE[n^k] (MA ∩ ZPP^{MCSP} ⊈ SIZE[n^k]).
- In addition, if there exists a perfect IO that is computationally secure against nonuniform polynomial-size circuits, then NEXP ∩ ZPEXP^{MCSP} ⊈ P/poly.
- If MCSP ∈ BPP, then statistical security and computational security for IO are equivalent.
- If computationally-secure perfect IO exists, then MCSP ∈ BPP iff NP = ZPP.
- If computationally-secure perfect IO exists, then ZPEXP ≠ BPP.
To the best of our knowledge, this is the first consequence of strong circuit lower bounds from the existence of an IO. The results are obtained via a construction of an optimal universal distinguisher, computable in randomized polynomial time with access to the MCSP oracle, that will distinguish any two circuit-samplable distributions with the advantage that is the statistical distance between these two distributions minus some negligible error term. This is our main technical contribution. As another immediate application, we get a simple proof of the result by Allender and Das (Inf. Comput., 2017) that SZK ⊆ BPP^{MCSP}.

Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. Synergy Between Circuit Obfuscation and Circuit Minimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{impagliazzo_et_al:LIPIcs.APPROX/RANDOM.2023.31, author = {Impagliazzo, Russell and Kabanets, Valentine and Volkovich, Ilya}, title = {{Synergy Between Circuit Obfuscation and Circuit Minimization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {31:1--31:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.31}, URN = {urn:nbn:de:0030-drops-188569}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.31}, annote = {Keywords: Minimal Circuit Size Problem (MCSP), Circuit Lower Bounds, Complexity Classes, Indistinguishability Obfuscation, Separation of Classes, Statistical Distance} }

Document

**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

For every prime p > 0, every n > 0 and κ = O(log n), we show the existence of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over 𝔽_p with M extension variables, each depending on at most κ original variables requires size exp(Ω(n²)/10^κ(M + n log n))

Russell Impagliazzo, Sasank Mouli, and Toniann Pitassi. Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{impagliazzo_et_al:LIPIcs.CCC.2023.7, author = {Impagliazzo, Russell and Mouli, Sasank and Pitassi, Toniann}, title = {{Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {7:1--7:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.7}, URN = {urn:nbn:de:0030-drops-182774}, doi = {10.4230/LIPIcs.CCC.2023.7}, annote = {Keywords: Proof complexity, Algebraic proof systems, Polynomial Calculus, Extension variables, AC⁰\lbrackp\rbrack-Frege} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Connections between proof complexity and circuit complexity have become major tools for obtaining lower bounds in both areas. These connections - which take the form of interpolation theorems and query-to-communication lifting theorems - translate efficient proofs into small circuits, and vice versa, allowing tools from one area to be applied to the other. Recently, the theory of TFNP has emerged as a unifying framework underlying these connections. For many of the proof systems which admit such a connection there is a TFNP problem which characterizes it: the class of problems which are reducible to this TFNP problem via query-efficient reductions is equivalent to the tautologies that can be efficiently proven in the system. Through this, proof complexity has become a major tool for proving separations in black-box TFNP. Similarly, for certain monotone circuit models, the class of functions that it can compute efficiently is equivalent to what can be reduced to a certain TFNP problem in a communication-efficient manner. When a TFNP problem has both a proof and circuit characterization, one can prove an interpolation theorem. Conversely, many lifting theorems can be viewed as relating the communication and query reductions to TFNP problems. This is exciting, as it suggests that TFNP provides a roadmap for the development of further interpolation theorems and lifting theorems.
In this paper we begin to develop a more systematic understanding of when these connections to TFNP occur. We give exact conditions under which a proof system or circuit model admits a characterization by a TFNP problem. We show:
- Every well-behaved proof system which can prove its own soundness (a reflection principle) is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a well-behaved proof system which proves its own soundness.
- Every well-behaved monotone circuit model which admits a universal family of functions is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a well-behaved monotone circuit model with a universal problem. As an example, we provide a TFNP characterization of the Polynomial Calculus, answering a question from [Mika Göös et al., 2022], and show that it can prove its own soundness.

Sam Buss, Noah Fleming, and Russell Impagliazzo. TFNP Characterizations of Proof Systems and Monotone Circuits. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 30:1-30:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buss_et_al:LIPIcs.ITCS.2023.30, author = {Buss, Sam and Fleming, Noah and Impagliazzo, Russell}, title = {{TFNP Characterizations of Proof Systems and Monotone Circuits}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {30:1--30:40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.30}, URN = {urn:nbn:de:0030-drops-175332}, doi = {10.4230/LIPIcs.ITCS.2023.30}, annote = {Keywords: Proof Complexity, Circuit Complexity, TFNP} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points.
Focusing on constant dimension d, we show that any k-quantifier, d-dimensional such problem is solvable in O(n^{k-1} log^{d-1} n) time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, (3k-3)-dimensional problem in this class that requires time Ω(n^{k-1-o(1)}).
Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time O(nlog^{d-1} n), and k-quantifier problems with k > 3 reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination VCND_d (Given three sets of vectors X,Y and Z of dimension d,d and 2d, respectively, is there an x ∈ X and a y ∈ Y so that their concatenation x∘y is not dominated by any z ∈ Z, where vector u is dominated by vector v if u_i ≤ v_i for each coordinate 1 ≤ i ≤ d), and determine it as the "unique" candidate to be complete for this class (under fine-grained assumptions).

Haozhe An, Mohit Gurumukhani, Russell Impagliazzo, Michael Jaber, Marvin Künnemann, and Maria Paula Parga Nina. The Fine-Grained Complexity of Multi-Dimensional Ordering Properties. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{an_et_al:LIPIcs.IPEC.2021.3, author = {An, Haozhe and Gurumukhani, Mohit and Impagliazzo, Russell and Jaber, Michael and K\"{u}nnemann, Marvin and Nina, Maria Paula Parga}, title = {{The Fine-Grained Complexity of Multi-Dimensional Ordering Properties}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.3}, URN = {urn:nbn:de:0030-drops-153869}, doi = {10.4230/LIPIcs.IPEC.2021.3}, annote = {Keywords: Fine-grained complexity, First-order logic, Orthogonal vectors} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes.
In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, and Avi Wigderson. On the Power and Limitations of Branch and Cut. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 6:1-6:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fleming_et_al:LIPIcs.CCC.2021.6, author = {Fleming, Noah and G\"{o}\"{o}s, Mika and Impagliazzo, Russell and Pitassi, Toniann and Robere, Robert and Tan, Li-Yang and Wigderson, Avi}, title = {{On the Power and Limitations of Branch and Cut}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {6:1--6:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.6}, URN = {urn:nbn:de:0030-drops-142809}, doi = {10.4230/LIPIcs.CCC.2021.6}, annote = {Keywords: Proof Complexity, Integer Programming, Cutting Planes, Branch and Cut, Stabbing Planes} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.

Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the Pseudo-Deterministic Query Complexity of NP Search Problems. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldwasser_et_al:LIPIcs.CCC.2021.36, author = {Goldwasser, Shafi and Impagliazzo, Russell and Pitassi, Toniann and Santhanam, Rahul}, title = {{On the Pseudo-Deterministic Query Complexity of NP Search Problems}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {36:1--36:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.36}, URN = {urn:nbn:de:0030-drops-143104}, doi = {10.4230/LIPIcs.CCC.2021.36}, annote = {Keywords: Pseudo-determinism, Query complexity, Proof complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas.
We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest.
We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.

Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Lifting for Constant-Depth Circuits and Applications to MCSP. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{carmosino_et_al:LIPIcs.ICALP.2021.44, author = {Carmosino, Marco and Hoover, Kenneth and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina}, title = {{Lifting for Constant-Depth Circuits and Applications to MCSP}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {44:1--44:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.44}, URN = {urn:nbn:de:0030-drops-141135}, doi = {10.4230/LIPIcs.ICALP.2021.44}, annote = {Keywords: circuit complexity, constant-depth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Computational pseudorandomness studies the extent to which a random variable Z looks like the uniform distribution according to a class of tests ℱ. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a high entropy distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent.
We consider three notions of computational entropy which are known to be equivalent when the test class ℱ is closed under taking majorities. This equivalence constitutes (essentially) the so-called dense model theorem of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao’s proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where ℱ is not closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is false.

Russell Impagliazzo and Sam McGuire. Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{impagliazzo_et_al:LIPIcs.ITCS.2021.2, author = {Impagliazzo, Russell and McGuire, Sam}, title = {{Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {2:1--2:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.2}, URN = {urn:nbn:de:0030-drops-135417}, doi = {10.4230/LIPIcs.ITCS.2021.2}, annote = {Keywords: Computational entropy, dense model theorem, coin problem} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p >= 2 (the circuit class AC^0[p]). Namely, we show that MCSP requires d-depth AC^0[p] circuits of size at least exp(N^{0.49/d}), where N=2^n is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N-bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2+N^{-0.49}, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC^0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC^0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC^1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC^0 circuit with MCSP-oracle gates.

Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal. AC^0[p] Lower Bounds Against MCSP via the Coin Problem. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{golovnev_et_al:LIPIcs.ICALP.2019.66, author = {Golovnev, Alexander and Ilango, Rahul and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina and Tal, Avishay}, title = {{AC^0\lbrackp\rbrack Lower Bounds Against MCSP via the Coin Problem}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {66:1--66:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.66}, URN = {urn:nbn:de:0030-drops-106422}, doi = {10.4230/LIPIcs.ICALP.2019.66}, annote = {Keywords: Minimum Circuit Size Problem (MCSP), circuit lower bounds, AC0\lbrackp\rbrack, coin problem, hybrid argument, MKTP, biased random boolean functions} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Suppose Alice and Bob are communicating in order to compute some function f, but instead of a classical communication channel they have a pair of walkie-talkie devices. They can use some classical communication protocol for f where in each round one player sends a bit and the other one receives it. The question is whether talking via walkie-talkie gives them more power? Using walkie-talkies instead of a classical communication channel allows players two extra possibilities: to speak simultaneously (but in this case they do not hear each other) and to listen at the same time (but in this case they do not transfer any bits). The motivation for this kind of a communication model comes from the study of the KRW conjecture. We show that for some definitions this non-classical communication model is, in fact, more powerful than the classical one as it allows to compute some functions in a smaller number of rounds. We also prove lower bounds for these models using both combinatorial and information theoretic methods.

Kenneth Hoover, Russell Impagliazzo, Ivan Mihajlin, and Alexander V. Smal. Half-Duplex Communication Complexity. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hoover_et_al:LIPIcs.ISAAC.2018.10, author = {Hoover, Kenneth and Impagliazzo, Russell and Mihajlin, Ivan and Smal, Alexander V.}, title = {{Half-Duplex Communication Complexity}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {10:1--10:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.10}, URN = {urn:nbn:de:0030-drops-99583}, doi = {10.4230/LIPIcs.ISAAC.2018.10}, annote = {Keywords: communication complexity, half-duplex channel, information theory} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires n^{epsilon k} time, for some constant epsilon>1/2, to count (note that these conjectures are significantly weaker than the usual ones made on these problems) on randomized machines for all but finitely many input lengths, then we have the following derandomizations:
- BPP can be decided in polynomial time using only n^alpha random bits on average over any efficient input distribution, for any constant alpha>0
- BPP can be decided in polynomial time with no randomness on average over the uniform distribution
This answers an open question of Ball et al. (STOC '17) in the positive of whether derandomization can be achieved from conjectures from fine-grained complexity theory. More strongly, these derandomizations improve over all previous ones achieved from worst-case uniform assumptions by succeeding on all but finitely many input lengths. Previously, derandomizations from worst-case uniform assumptions were only know to succeed on infinitely many input lengths. It is specifically the structure and moderate hardness of the k-Orthogonal Vectors and k-CLIQUE problems that makes removing this restriction possible.
Via this uniform derandomization, we connect the problem-centric and resource-centric views of complexity theory by showing that exact hardness assumptions about specific problems like k-CLIQUE imply quantitative and qualitative relationships between randomized and deterministic time. This can be either viewed as a barrier to proving some of the main conjectures of fine-grained complexity theory lest we achieve a major breakthrough in unconditional derandomization or, optimistically, as route to attain such derandomizations by working on very concrete and weak conjectures about specific problems.

Marco L. Carmosino, Russell Impagliazzo, and Manuel Sabin. Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carmosino_et_al:LIPIcs.ICALP.2018.27, author = {Carmosino, Marco L. and Impagliazzo, Russell and Sabin, Manuel}, title = {{Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.27}, URN = {urn:nbn:de:0030-drops-90316}, doi = {10.4230/LIPIcs.ICALP.2018.27}, annote = {Keywords: Derandomization, Hardness vs Randomness, Fine-Grained Complexity, Average-Case Complexity, k-Orthogonal Vectors, k-CLIQUE} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions.

Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. The Power of Natural Properties as Oracles. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{impagliazzo_et_al:LIPIcs.CCC.2018.7, author = {Impagliazzo, Russell and Kabanets, Valentine and Volkovich, Ilya}, title = {{The Power of Natural Properties as Oracles}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.7}, URN = {urn:nbn:de:0030-drops-88824}, doi = {10.4230/LIPIcs.CCC.2018.7}, annote = {Keywords: natural properties, Minimal Circuit Size Problem (MCSP), circuit lower bounds, hardness of MCSP, learning algorithms, obfuscation, Indistinguishability Obfuscators (IO)} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.
This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.

Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin. Hardness Amplification for Non-Commutative Arithmetic Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carmosino_et_al:LIPIcs.CCC.2018.12, author = {Carmosino, Marco L. and Impagliazzo, Russell and Lovett, Shachar and Mihajlin, Ivan}, title = {{Hardness Amplification for Non-Commutative Arithmetic Circuits}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.12}, URN = {urn:nbn:de:0030-drops-88772}, doi = {10.4230/LIPIcs.CCC.2018.12}, annote = {Keywords: arithmetic circuits, hardness amplification, circuit lower bounds, non-commutative computation} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity.

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, and Robert Robere. Stabbing Planes. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2018.10, author = {Beame, Paul and Fleming, Noah and Impagliazzo, Russell and Kolokolova, Antonina and Pankratov, Denis and Pitassi, Toniann and Robere, Robert}, title = {{Stabbing Planes}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {10:1--10:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.10}, URN = {urn:nbn:de:0030-drops-83418}, doi = {10.4230/LIPIcs.ITCS.2018.10}, annote = {Keywords: Complexity Theory, Proof Complexity, Communication Complexity, Cutting Planes, Semi-Algebraic Proof Systems, Pseudo Boolean Solvers, SAT solvers, Inte} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

The Black-Box Hypothesisstates that any property of Boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an upper bound on its circuit size. If this hypothesis is true, then P neq NP. We focus on the consequences of the hypothesis being false, showing that (under general conditions on the structure of a counterexample) it implies a non-trivial algorithm for CSAT. More specifically, we show that if there is a property F of boolean functions such that F has high sensitivity on some input function f of subexponential circuit complexity (which is a sufficient condition for F being a counterexample to the Black-Box Hypothesis), then CSAT is solvable by a subexponential-size circuit family. Moreover, if such a counterexample F is symmetric, then CSAT is in Ppoly. These results provide some evidence towards the conjecture (made in this paper) that the Black-Box Hypothesis is false if and only if CSAT is easy.

Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, and Shadab Romani. Does Looking Inside a Circuit Help?. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{impagliazzo_et_al:LIPIcs.MFCS.2017.1, author = {Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina and McKenzie, Pierre and Romani, Shadab}, title = {{Does Looking Inside a Circuit Help?}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {1:1--1:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.1}, URN = {urn:nbn:de:0030-drops-80975}, doi = {10.4230/LIPIcs.MFCS.2017.1}, annote = {Keywords: Black-Box Hypothesis, Rice's theorem, circuit complexity, SAT, sensitivity of boolean functions, decision tree complexity} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We generalize the "learning algorithms from natural properties" framework of [CIKK16] to get agnostic learning algorithms from natural properties with extra features. We show that if a natural property (in the sense of Razborov and Rudich [RR97]) is useful also against functions that are close to the class of "easy" functions, rather than just against "easy" functions, then it can be used to get an agnostic learning algorithm over the uniform distribution with membership queries.
* For AC0[q], any prime q (constant-depth circuits of polynomial size, with AND, OR, NOT, and MODq gates of unbounded fanin), which happens to have a natural property with the requisite extra feature by [Raz87, Smo87, RR97], we obtain the first agnostic learning algorithm for AC0[q], for every prime q. Our algorithm runs in randomized quasi-polynomial time, uses membership queries, and outputs a circuit for a given Boolean function f that agrees with f on all but at most polylog(n)*opt fraction of inputs, where opt is the relative distance between f and the closest function h in the class AC0[q].
* For the ideal case, a natural proof of strongly exponential correlation circuit lower bounds against a circuit class C containing AC0[2] (i.e., circuits of size exp(Omega(n)) cannot compute some n-variate function even with exp(-Omega(n)) advantage over random guessing) would yield a polynomial-time query agnostic learning algorithm for C with the approximation error O(opt).

Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Agnostic Learning from Tolerant Natural Proofs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carmosino_et_al:LIPIcs.APPROX-RANDOM.2017.35, author = {Carmosino, Marco L. and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina}, title = {{Agnostic Learning from Tolerant Natural Proofs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {35:1--35:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.35}, URN = {urn:nbn:de:0030-drops-75842}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.35}, annote = {Keywords: agnostic learning, natural proofs, circuit lower bounds, meta-algorithms, AC0\lbrackq\rbrack, Nisan-Wigderson generator} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Impagliazzo and Wigderson (STOC 1997) showed that if E=DTIME(2^O(n)) requires size 2^Omega(n) circuits, then every time T constant-error randomized algorithm can be simulated deterministically in time poly(T). However, such polynomial slowdown is a deal breaker when T=2^(alpha*n), for a constant alpha>0, as is the case for some randomized algorithms for NP-complete problems. Paturi and Pudlak (STOC 2010) observed that many such algorithms are obtained from randomized time T algorithms, for T < 2^o(n), with large one-sided error 1-epsilon, for epsilon=2^(-alpha*n), that are repeated 1/epsilon times to yield a constant-error randomized algorithm running in time T/epsilon=2^((alpha+o(1))*n).
We show that if E requires size 2^Omega(n) nondeterministic circuits, then there is a poly(n)-time epsilon-HSG (Hitting-Set Generator) H:{0,1}^(O(log(n)) + log(1/epsilon) -> {0,1}^n, implying that time T randomized algorithms with one-sided error 1-epsilon can be simulated in deterministic time poly(T)/epsilon. In particular, under this hardness assumption, the fastest known constant-error randomized algorithm for k-SAT (for k > 3) by Paturi et al. (J. ACM 2005) can be made deterministic with essentially the same time bound. This is the first hardness versus randomness tradeoff for algorithms for NP-complete problems. We address the necessity of our assumption by showing that HSGs with very low error imply hardness for nondeterministic circuits with "few" nondeterministic bits.
Applebaum et al. (CCC 2015) showed that "black-box techniques" cannot achieve poly(n)-time computable epsilon-PRGs (Pseudo-Random Generators) for epsilon=n^-omega(1), even if we assume hardness against circuits with oracle access to an arbitrary language in the polynomial time hierarchy. We introduce weaker variants of PRGs with relative error, that do follow under the latter hardness assumption. Specifically, we say that a function G:{0,1}^r -> {0,1}^n is an (epsilon,delta)-re-PRG for a circuit C if (1-epsilon)*Pr[C(U_n)=1] - delta < Pr[C(G(U_r)=1] < (1+epsilon)*Pr[C(U_n)=1] + delta. We construct poly(n)-time computable (epsilon,delta)-re-PRGs with arbitrary polynomial stretch, epsilon=n^-O(1) and delta=2^(-n^Omega(1)). We also construct PRGs with relative error that fool non-boolean distinguishers (in the sense introduced by Dubrov and Ishai (STOC 2006)).
Our techniques use ideas from Paturi and Pudlak (STOC 2010), Trevisan and Vadhan (FOCS 2000), Applebaum et al. (CCC 2015). Common themes in our proofs are "composing" a PRG/HSG with a combinatorial object such as dispersers and extractors, and the use of nondeterministic reductions in the spirit of Feige and Lund (Comp. Complexity 1997).

Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, and Ronen Shaltiel. Pseudorandomness When the Odds are Against You. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 9:1-9:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{artemenko_et_al:LIPIcs.CCC.2016.9, author = {Artemenko, Sergei and Impagliazzo, Russell and Kabanets, Valentine and Shaltiel, Ronen}, title = {{Pseudorandomness When the Odds are Against You}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {9:1--9:35}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.9}, URN = {urn:nbn:de:0030-drops-58375}, doi = {10.4230/LIPIcs.CCC.2016.9}, annote = {Keywords: Derandomization, pseudorandom generator, hitting-set generator, relative error} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Based on Hastad's (1986) circuit lower bounds, Linial, Mansour, and Nisan (1993) gave a quasipolytime learning algorithm for AC^0 (constant-depth circuits with AND, OR, and NOT gates), in the PAC model over the uniform distribution. It was an open question to get a learning algorithm (of any kind) for the class of AC^0[p] circuits (constant-depth, with AND, OR, NOT, and MOD_p gates for a prime p).
Our main result is a quasipolytime learning algorithm for AC^0[p] in the PAC model over the uniform distribution with membership queries. This algorithm is an application of a general connection we show to hold between natural proofs (in the sense of Razborov and Rudich (1997)) and learning algorithms. We argue that a natural proof of a circuit lower bound against any (sufficiently powerful) circuit class yields a learning algorithm for the same circuit class. As the lower bounds against AC^0[p] by Razborov (1987) and Smolensky (1987) are natural, we obtain our learning algorithm for AC^0[p].

Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning Algorithms from Natural Proofs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{carmosino_et_al:LIPIcs.CCC.2016.10, author = {Carmosino, Marco L. and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina}, title = {{Learning Algorithms from Natural Proofs}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {10:1--10:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.10}, URN = {urn:nbn:de:0030-drops-58557}, doi = {10.4230/LIPIcs.CCC.2016.10}, annote = {Keywords: natural proofs, circuit complexity, lower bounds, learning, compression} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization:
- general derandomization of promiseBPP (connected to Boolean circuits),
- derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and
- derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers).
We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion.
Our main results are as follows:
1. We give the first proof that a non-trivial (nondeterministic subexponential-time) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds.
2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam [JS12] (by removing the need for advice).
3. We derive a Boolean circuit lower bound for NEXP intersect coNEXP from the assumption of sufficiently strong non-deterministic derandomization of promiseBPP (without advice), as well as from the assumed existence of an NP-computable non-empty property of Boolean functions useful for proving superpolynomial circuit lower bounds (in the sense of natural proofs of [RR97]); this strengthens the related results of [IKW02].
4. Finally, we turn all of these implications into equivalences for appropriately defined promise classes and for a notion of robust inclusion/separation (inspired by [FS11]) that lies between the classical "almost everywhere" and "infinitely often" notions.

Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Tighter Connections between Derandomization and Circuit Lower Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 645-658, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{carmosino_et_al:LIPIcs.APPROX-RANDOM.2015.645, author = {Carmosino, Marco L. and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina}, title = {{Tighter Connections between Derandomization and Circuit Lower Bounds}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {645--658}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.645}, URN = {urn:nbn:de:0030-drops-53285}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.645}, annote = {Keywords: derandomization, circuit lower bounds, polynomial identity testing, promise BPP, hardness vs. randomness} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Non-relativization of complexity issues can be interpreted
as giving some evidence that these issues cannot be resolved
by "black-box" techniques. In the early 1990's, a sequence of
important non-relativizing results was proved, mainly using
algebraic techniques. Two approaches have been proposed
to understand the power and limitations of these algebraic
techniques: (1) Fortnow gives a construction of a class
of oracles which have a similar algebraic and logical structure,
although they are arbitrarily powerful. He shows that
many of the non-relativizing results proved using algebraic
techniques hold for all such oracles, but he does not show,
e.g., that the outcome of the "P vs. NP" question differs
between different oracles in that class. (2) Aaronson and
Wigderson give definitions of algebrizing separations and
collapses of complexity classes, by comparing classes relative
to one oracle to classes relative to an algebraic extension of
that oracle. Using these definitions, they show both that
the standard collapses and separations "algebrize" and that
many of the open questions in complexity fail to "algebrize",
suggesting that the arithmetization technique is close to its
limits. However, it is unclear how to formalize algebrization
of more complicated complexity statements than collapses
or separations, and whether the algebrizing statements are,
e.g., closed under modus ponens; so it is conceivable that
several algebrizing premises could imply (in a relativizing
way) a non-algebrizing conclusion.
Here, building on the work of Arora, Impagliazzo,
and Vazirani [4], we propose an axiomatic approach to "algebrization",
which complements and clarifies the approaches
of Fortnow and Aaronso&Wigderson. We present logical theories formalizing the notion of algebrizing techniques so that most algebrizing results
are provable within our theories and separations requiring
non-algebrizing techniques are independent of them.
Our theories extend the [AIV] theory formalizing relativization
by adding an Arithmetic Checkability axiom.
We show the following: (i) Arithmetic checkability holds
relative to arbitrarily powerful oracles (since Fortnow's algebraic oracles all satisfy Arithmetic Checkability
axiom); by contrast, Local Checkability of [AIV] restricts the
oracle power to NP cap co-NP. (ii) Most of the algebrizing
collapses and separations from [AW], such as IP = PSPACE,
NP subset ZKIP if one-way functions exist, MA-EXP not in P/poly,
etc., are provable from Arithmetic Checkability. (iii) Many
of the open complexity questions (shown to require nonalgebrizing
techniques in [AW]), such as "P vs. NP", "NP vs.
BPP", etc., cannot be proved from Arithmetic Checkability.
(iv) Arithmetic Checkability is also insufficient to prove one
known result, NEXP = MIP.

Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. An Axiomatic Approach to Algebrization. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{impagliazzo_et_al:DagSemProc.09421.3, author = {Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina}, title = {{An Axiomatic Approach to Algebrization}}, booktitle = {Algebraic Methods in Computational Complexity}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9421}, editor = {Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.3}, URN = {urn:nbn:de:0030-drops-24150}, doi = {10.4230/DagSemProc.09421.3}, annote = {Keywords: Oracles, arithmetization, algebrization} }

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