79 Search Results for "Saxena, Nitin"


Volume

LIPIcs, Volume 182

40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

FSTTCS 2020, December 14-18, 2020, BITS Pilani, K K Birla Goa Campus, Goa, India (Virtual Conference)

Editors: Nitin Saxena and Sunil Simon

Document
Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials

Authors: Pranav Bisht and Nitin Saxena

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP'10; Volkovich RANDOM'17; Bhargava, Saraf and Volkovich FOCS'18, JACM'20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring. To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest.

Cite as

Pranav Bisht and Nitin Saxena. Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2022.9,
  author =	{Bisht, Pranav and Saxena, Nitin},
  title =	{{Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.9},
  URN =		{urn:nbn:de:0030-drops-174012},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.9},
  annote =	{Keywords: Multivariate polynomial factorization, derandomization, sparse polynomials, symmetric polynomials, factor-sparsity, convex polytopes}
}
Document
Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits

Authors: C. S. Bhargav, Sagnik Dutta, and Nitin Saxena

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)


Abstract
We show that any product-depth Δ algebraic circuit for the Iterated Matrix Multiplication Polynomial IMM_{n,d} (when d = O(log n/log log n)) must be of size at least n^Ω(d^{1/(φ²)^Δ}) where φ = 1.618… is the golden ratio. This improves the recent breakthrough result of Limaye, Srinivasan and Tavenas (FOCS'21) who showed a super polynomial lower bound of the form n^Ω(d^{1/4^Δ}) for constant-depth circuits. One crucial idea of the (LST21) result was to use set-multilinear polynomials where each of the sets in the underlying partition of the variables could be of different sizes. By picking the set sizes more carefully (depending on the depth we are working with), we first show that any product-depth Δ set-multilinear circuit for IMM_{n,d} (when d = O(log n)) needs size at least n^Ω(d^{1/φ^Δ}). This improves the n^Ω(d^{1/2^Δ}) lower bound of (LST21). We then use their Hardness Escalation technique to lift this to general circuits. We also show that our lower bound cannot be improved significantly using these same techniques. For the specific two set sizes used in (LST21), they showed that their lower bound cannot be improved. We show that for any d^o(1) set sizes (out of maximum possible d), the scope for improving our lower bound is minuscule: there exists a set-multilinear circuit that has product-depth Δ and size almost matching our lower bound such that the value of the measure used to prove the lower bound is maximum for this circuit. This results in a barrier to further improvement using the same measure.

Cite as

C. S. Bhargav, Sagnik Dutta, and Nitin Saxena. Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2022.18,
  author =	{Bhargav, C. S. and Dutta, Sagnik and Saxena, Nitin},
  title =	{{Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.18},
  URN =		{urn:nbn:de:0030-drops-168161},
  doi =		{10.4230/LIPIcs.MFCS.2022.18},
  annote =	{Keywords: polynomials, lower bounds, algebraic circuits, proof barrier, fibonacci numbers}
}
Document
Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits

Authors: Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena

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


Abstract
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ^[k] Π Σ ∧) and sum-product-of-constant-degree-polynomials (Σ^[k] Π Σ Π^[δ]), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha,Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work. We give the first polynomial-time PIT for Σ^[k] Π Σ ∧. Further, we give the first quasipolynomial time blackbox PIT for both Σ^[k] Π Σ ∧ and Σ^[k] Π Σ Π^[δ]. No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.

Cite as

Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 11:1-11:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{dutta_et_al:LIPIcs.CCC.2021.11,
  author =	{Dutta, Pranjal and Dwivedi, Prateek and Saxena, Nitin},
  title =	{{Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{11:1--11:27},
  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.2021.11},
  URN =		{urn:nbn:de:0030-drops-142857},
  doi =		{10.4230/LIPIcs.CCC.2021.11},
  annote =	{Keywords: Polynomial identity testing, hitting set, depth-4 circuits}
}
Document
Variety Evasive Subspace Families

Authors: Zeyu Guo

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


Abstract
We introduce the problem of constructing explicit variety evasive subspace families. Given a family ℱ of subvarieties of a projective or affine space, a collection ℋ of projective or affine k-subspaces is (ℱ,ε)-evasive if for every 𝒱 ∈ ℱ, all but at most ε-fraction of W ∈ ℋ intersect every irreducible component of 𝒱 with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers. Using Chow forms, we construct explicit k-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n-space. As one application, we obtain a complete derandomization of Noether’s normalization lemma for varieties of bounded degree in a projective or affine n-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130). As a complement of our explicit construction, we prove a lower bound for the size of k-subspace families that are evasive for degree-d varieties in a projective n-space. When n-k = n^Ω(1), the lower bound is superpolynomial unless d is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

Cite as

Zeyu Guo. Variety Evasive Subspace Families. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 20:1-20:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{guo:LIPIcs.CCC.2021.20,
  author =	{Guo, Zeyu},
  title =	{{Variety Evasive Subspace Families}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{20:1--20:33},
  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.2021.20},
  URN =		{urn:nbn:de:0030-drops-142949},
  doi =		{10.4230/LIPIcs.CCC.2021.20},
  annote =	{Keywords: algebraic complexity, dimension reduction, Noether normalization, polynomial identity testing, pseudorandomness, varieties}
}
Document
Arithmetic Circuit Complexity of Division and Truncation

Authors: Pranjal Dutta, Gorav Jindal, Anurag Pandey, and Amit Sinhababu

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


Abstract
Given polynomials f,g,h ∈ 𝔽[x₁,…,x_n] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s,deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen’s classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)). Further, we show that any univariate polynomial family (f_d)_d, defined by the initial segment of the power series expansion of rational function g_d(x)/h_d(x) up to degree d (i.e. f_d = g_d/h_d od x^{d+1}), where circuit size of g is s_d and degree of g_d is at most d, can be computed by a circuit of size poly(s_d,deg(h_d),log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω (d)), assuming hardness of the integer factorization problem. Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.

Cite as

Pranjal Dutta, Gorav Jindal, Anurag Pandey, and Amit Sinhababu. Arithmetic Circuit Complexity of Division and Truncation. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 25:1-25:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{dutta_et_al:LIPIcs.CCC.2021.25,
  author =	{Dutta, Pranjal and Jindal, Gorav and Pandey, Anurag and Sinhababu, Amit},
  title =	{{Arithmetic Circuit Complexity of Division and Truncation}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{25:1--25:36},
  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.2021.25},
  URN =		{urn:nbn:de:0030-drops-142990},
  doi =		{10.4230/LIPIcs.CCC.2021.25},
  annote =	{Keywords: Arithmetic Circuits, Division, Truncation, Division elimination, Rational function, Algebraic power series, Transcendental power series, Integer factorization}
}
Document
A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization

Authors: Pranjal Dutta, Nitin Saxena, and Thomas Thierauf

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


Abstract
For a polynomial f, we study the sum of squares representation (SOS), i.e. f = ∑_{i ∈ [s]} c_i f_i² , where c_i are field elements and the f_i’s are polynomials. The size of the representation is the number of monomials that appear across the f_i’s. Its minimum is the support-sum S(f) of f. For simplicity of exposition, we consider univariate f. A trivial lower bound for the support-sum of, a full-support univariate polynomial, f of degree d is S(f) ≥ d^{0.5}. We show that the existence of an explicit polynomial f with support-sum just slightly larger than the trivial bound, that is, S(f) ≥ d^{0.5+ε(d)}, for a sub-constant function ε(d) > ω(√{log log d/log d}), implies that VP ≠ VNP. The latter is a major open problem in algebraic complexity. A further consequence is that blackbox-PIT is in SUBEXP. Note that a random polynomial fulfills the condition, as there we have S(f) = Θ(d). We also consider the sum-of-cubes representation (SOC) of polynomials. In a similar way, we show that here, an explicit hard polynomial even implies that blackbox-PIT is in P.

Cite as

Pranjal Dutta, Nitin Saxena, and Thomas Thierauf. A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{dutta_et_al:LIPIcs.ITCS.2021.23,
  author =	{Dutta, Pranjal and Saxena, Nitin and Thierauf, Thomas},
  title =	{{A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{23:1--23:21},
  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.2021.23},
  URN =		{urn:nbn:de:0030-drops-135629},
  doi =		{10.4230/LIPIcs.ITCS.2021.23},
  annote =	{Keywords: VP, VNP, hitting set, circuit, polynomial, sparsity, SOS, SOC, PIT, lower bound}
}
Document
Complete Volume
LIPIcs, Volume 182, FSTTCS 2020, Complete Volume

Authors: Nitin Saxena and Sunil Simon

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
LIPIcs, Volume 182, FSTTCS 2020, Complete Volume

Cite as

40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 1-912, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@Proceedings{saxena_et_al:LIPIcs.FSTTCS.2020,
  title =	{{LIPIcs, Volume 182, FSTTCS 2020, Complete Volume}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{1--912},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020},
  URN =		{urn:nbn:de:0030-drops-132401},
  doi =		{10.4230/LIPIcs.FSTTCS.2020},
  annote =	{Keywords: LIPIcs, Volume 182, FSTTCS 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Nitin Saxena and Sunil Simon

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{saxena_et_al:LIPIcs.FSTTCS.2020.0,
  author =	{Saxena, Nitin and Simon, Sunil},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{0:i--0:xvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.0},
  URN =		{urn:nbn:de:0030-drops-132413},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
The Quest for Mathematical Understanding of Deep Learning (Invited Talk)

Authors: Sanjeev Arora

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Deep learning has transformed Machine Learning and Artificial Intelligence in the past decade. It raises fundamental questions for mathematics and theory of computer science, since it relies upon solving large-scale nonconvex problems via gradient descent and its variants. This talk will be an introduction to mathematical questions raised by deep learning, and some partial understanding obtained in recent years.

Cite as

Sanjeev Arora. The Quest for Mathematical Understanding of Deep Learning (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{arora:LIPIcs.FSTTCS.2020.1,
  author =	{Arora, Sanjeev},
  title =	{{The Quest for Mathematical Understanding of Deep Learning}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.1},
  URN =		{urn:nbn:de:0030-drops-132427},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.1},
  annote =	{Keywords: machine learning, artificial intelligence, deep learning, gradient descent, optimization}
}
Document
Invited Talk
Proofs of Soundness and Proof Search (Invited Talk)

Authors: Albert Atserias

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Let P be a sound proof system for propositional logic. For each CNF formula F, let SAT(F) be a CNF formula whose satisfying assignments are in 1-to-1 correspondence with those of F (e.g., F itself). For each integer s, let REF(F,s) be a CNF formula whose satisfying assignments are in 1-to-1 correspondence with the P-refutations of F of length s. Since P is sound, it is obvious that the conjunction formula SAT(F) & REF(F,s) is unsatisfiable for any choice of F and s. It has been long known that, for many natural proof systems P and for the most natural formalizations of the formulas SAT and REF, the unsatisfiability of SAT(F) & REF(F,s) can be established by a short refutation. In addition, for many P, these short refutations live in the proof system P itself. This is the case for all Frege proof systems. In contrast it was known since the early 2000’s that Resolution proofs of Resolution’s soundness statements must have superpolynomial length. In this talk I will explain how the soundness formulas for a proof system P relate to the computational complexity of the proof search problem for P. In particular, I will explain how such formulas are used in the recent proof that the problem of approximating the minimum proof-length for Resolution is NP-hard (Atserias-Müller 2019). Besides playing a key role in this hardness of approximation result, the renewed interest in the soundness formulas led to a complete answer to the question whether Resolution has subexponential-length proofs of its own soundness statements (Garlík 2019).

Cite as

Albert Atserias. Proofs of Soundness and Proof Search (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{atserias:LIPIcs.FSTTCS.2020.2,
  author =	{Atserias, Albert},
  title =	{{Proofs of Soundness and Proof Search}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.2},
  URN =		{urn:nbn:de:0030-drops-132439},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.2},
  annote =	{Keywords: Proof complexity, automatability, Resolution, proof search, consistency statements, lower bounds, reflection principle, satisfiability}
}
Document
Invited Talk
Convex Optimization and Dynamic Data Structure (Invited Talk)

Authors: Yin Tat Lee

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
In the last three years, there are many breakthroughs in optimization such as nearly quadratic time algorithms for bipartite matching, linear programming algorithms that are as fast as Ax = b. All of these algorithms are based on a careful combination of optimization techniques and dynamic data structures. In this talk, we will explain the framework underlying all the recent breakthroughs. Joint work with Jan van den Brand, Michael B. Cohen, Sally Dong, Haotian Jiang, Tarun Kathuria, Danupon Nanongkai, Swati Padmanabhan, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, Di Wang, Sam Chiu-wai Wong, Guanghao Ye, Qiuyi Zhang.

Cite as

Yin Tat Lee. Convex Optimization and Dynamic Data Structure (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{lee:LIPIcs.FSTTCS.2020.3,
  author =	{Lee, Yin Tat},
  title =	{{Convex Optimization and Dynamic Data Structure}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.3},
  URN =		{urn:nbn:de:0030-drops-132440},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.3},
  annote =	{Keywords: Convex Optimization, Dynamic Data Structure}
}
Document
Invited Talk
Holonomic Techniques, Periods, and Decision Problems (Invited Talk)

Authors: Joël Ouaknine

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades. In this talk, I will give an overview of the area, and in particular will present a select survey of known and original results on decision problems for holonomic sequences and functions. (Holonomic sequences satisfy linear recurrence relations with polynomial coefficients, and holonomic functions satisfy linear differential equations with polynomial coefficients.) I will also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I will relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.

Cite as

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 4:1-4:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{ouaknine:LIPIcs.FSTTCS.2020.4,
  author =	{Ouaknine, Jo\"{e}l},
  title =	{{Holonomic Techniques, Periods, and Decision Problems}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{4:1--4:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.4},
  URN =		{urn:nbn:de:0030-drops-132451},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.4},
  annote =	{Keywords: holonomic techniques, decision problems, recurrence sequences, minimal solutions, Positivity Problem, continued fractions, special functions, periods, exponential periods}
}
Document
Invited Talk
Algorithmic Improvisation for Dependable Intelligent Autonomy (Invited Talk)

Authors: Sanjit A. Seshia

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Algorithmic Improvisation, also called control improvisation or controlled improvisation, is a new framework for automatically synthesizing systems with specified random but controllable behavior. In this talk, I will present the theory of algorithmic improvisation and show how it can be used in a wide variety of applications where randomness can provide variety, robustness, or unpredictability while guaranteeing safety or other properties. Applications demonstrated to date include robotic surveillance, software fuzz testing, music improvisation, human modeling, generating test cases for simulating cyber-physical systems, and generation of synthetic data sets to train and test machine learning algorithms. In this talk, I will particularly focus on applications to the design of intelligent autonomous systems, presenting work on randomized planning for robotics and a domain-specific probabilistic programming language for the design and analysis of learning-based autonomous systems.

Cite as

Sanjit A. Seshia. Algorithmic Improvisation for Dependable Intelligent Autonomy (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 5:1-5:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{seshia:LIPIcs.FSTTCS.2020.5,
  author =	{Seshia, Sanjit A.},
  title =	{{Algorithmic Improvisation for Dependable Intelligent Autonomy}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{5:1--5:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.5},
  URN =		{urn:nbn:de:0030-drops-132465},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.5},
  annote =	{Keywords: Formal methods, synthesis, verification, randomized algorithms, formal specification, testing, machine learning, synthetic data generation, planning}
}
Document
Invited Talk
On Some Recent Advances in Algebraic Complexity (Invited Talk)

Authors: Amir Shpilka

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
Algebraic complexity is the field studying the intrinsic difficulty of algebraic problems in an algebraic model of computation, most notably arithmetic circuits. It is a very natural model of computation that attracted a large amount of research in the last few decades, partially due to its simplicity and elegance, but mostly because of its importance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, deciding whether P = BPP and more, will be easier to solve for arithmetic circuits. In this talk I will give the basic definitions, explain the main questions and how they relate to their Boolean counterparts, and discuss what I view as promising approaches to tackling the most fundamental problems in the field.

Cite as

Amir Shpilka. On Some Recent Advances in Algebraic Complexity (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, p. 6:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{shpilka:LIPIcs.FSTTCS.2020.6,
  author =	{Shpilka, Amir},
  title =	{{On Some Recent Advances in Algebraic Complexity}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{6:1--6:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.6},
  URN =		{urn:nbn:de:0030-drops-132472},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.6},
  annote =	{Keywords: Algebraic Complexity, Arithmetic Circuits, Polynomial Identity Testing}
}
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