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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size <= s(n). We show:
Learning Speedups: If C[s(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2^n/n^{\omega(1)}, then for every k >= 1 and epsilon > 0 the class C[n^k] can be learned to high accuracy in time O(2^{n^epsilon}). There is epsilon > 0 such that C[2^{n^{epsilon}}] can be learned in time 2^n/n^{omega(1)} if and only if C[poly(n)] can be learned in time 2^{(log(n))^{O(1)}}.
Equivalences between Learning Models: We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression.
A Dichotomy between Learnability and Pseudorandomness: In the non-uniform setting, there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning: If for each k >= 1, (depth-d)-C[n^k] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2^n/n^{\omega(1)}, then for each k >= 1, BPE is not contained in (depth-d)-C[n^k]. If for some epsilon > 0 there are P-natural proofs useful against C[2^{n^{epsilon}}], then ZPEXP is not contained in C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes: If there is a k > 0 such that BPE is contained in i.o.Circuit[n^k], then BPEXP is contained in i.o.EXP/O(log(n)). If ZPEXP is contained in i.o.Circuit[2^{n/3}], then ZPEXP is contained in i.o.ESUBEXP.
Hardness Results for MCSP: All functions in non-uniform NC^1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC^0 circuits. In particular, if MCSP is in TC^0 then NC^1 = TC^0.

Igor C. Carboni Oliveira and Rahul Santhanam. Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 18:1-18:49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2017.18, author = {Oliveira, Igor C. Carboni and Santhanam, Rahul}, title = {{Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {18:1--18:49}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.18}, URN = {urn:nbn:de:0030-drops-75327}, doi = {10.4230/LIPIcs.CCC.2017.18}, annote = {Keywords: boolean circuits, learning theory, pseudorandomness} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

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})}.

Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity Helps to Compute Majority. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.23, author = {Oliveira, Igor Carboni and Santhanam, Rahul and Srinivasan, Srikanth}, title = {{Parity Helps to Compute Majority}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.23}, URN = {urn:nbn:de:0030-drops-108453}, doi = {10.4230/LIPIcs.CCC.2019.23}, annote = {Keywords: Computational Complexity, Boolean Circuits, Lower Bounds, Parity, Majority} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

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.

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27, author = {Oliveira, Igor Carboni and Pich, J\'{a}n and Santhanam, Rahul}, title = {{Hardness Magnification near State-Of-The-Art Lower Bounds}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {27:1--27:29}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27}, URN = {urn:nbn:de:0030-drops-108494}, doi = {10.4230/LIPIcs.CCC.2019.27}, annote = {Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity} }

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Track A: Algorithms, Complexity and Games

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

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.
This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time.
Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.

Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira:LIPIcs.ICALP.2019.32, author = {Oliveira, Igor Carboni}, title = {{Randomness and Intractability in Kolmogorov Complexity}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {32:1--32:14}, 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.2019.32}, URN = {urn:nbn:de:0030-drops-106087}, doi = {10.4230/LIPIcs.ICALP.2019.32}, annote = {Keywords: computational complexity, randomness, circuit lower bounds, Kolmogorov complexity} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

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.

Igor Carboni Oliveira and Rahul Santhanam. Majority is Incompressible by AC^0[p] Circuits. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 124-157, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2015.124, author = {Oliveira, Igor Carboni and Santhanam, Rahul}, title = {{Majority is Incompressible by AC^0\lbrackp\rbrack Circuits}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {124--157}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.124}, URN = {urn:nbn:de:0030-drops-50658}, doi = {10.4230/LIPIcs.CCC.2015.124}, annote = {Keywords: interactive communication, compression, circuit lower bound} }