1,949 research outputs found

    Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

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    We say that a circuit CC over a field FF functionally computes an nn-variate polynomial PP if for every x{0,1}nx \in \{0,1\}^n we have that C(x)=P(x)C(x) = P(x). This is in contrast to syntactically computing PP, when CPC \equiv P as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-33 and depth-44 arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-33 arithmetic circuits for a polynomial in VNPVNP. 2. Exponential lower bounds for homogeneous depth-44 arithmetic circuits with bounded individual degree for a polynomial in VNPVNP. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-44 arithmetic circuits for the Permanent imply a separation between #P{\#}P and ACCACC. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-44 circuits imply superpolynomial lower bounds for homogeneous depth-55 circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest

    Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

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    Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit dO(1)d^{O(1)}-variate and degree dd polynomial PdVNPP_{d}\in VNP such that if any depth four circuit CC of bounded formal degree dd which computes a polynomial of bounded individual degree O(1)O(1), that is functionally equivalent to PdP_d, then CC must have size 2Ω(dlogd)2^{\Omega(\sqrt{d}\log{d})}. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC0ACC^0 circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC0ACC^0 can also be computed by algebraic ΣΣΠ\Sigma\mathord{\wedge}\Sigma\Pi circuits (i.e., circuits of the form -- sums of powers of polynomials) of 2logO(1)n2^{\log^{O(1)}n} size. Thus they argued that a 2ω(logO(1)n)2^{\omega(\log^{O(1)}{n})} "functional" lower bound for an explicit polynomial QQ against ΣΣΠ\Sigma\mathord{\wedge}\Sigma\Pi circuits would imply a lower bound for the "corresponding Boolean function" of QQ against non-uniform ACC0ACC^0. In their work, they ask if their lower bound be extended to ΣΣΠ\Sigma\mathord{\wedge}\Sigma\Pi circuits. In this paper, for large integers nn and dd such that ω(log2n)dn0.01\omega(\log^2n)\leq d\leq n^{0.01}, we show that any ΣΣΠ\Sigma\mathord{\wedge}\Sigma\Pi circuit of bounded individual degree at most O(dk2)O\left(\frac{d}{k^2}\right) that functionally computes Iterated Matrix Multiplication polynomial IMMn,dIMM_{n,d} (VP\in VP) over {0,1}n2d\{0,1\}^{n^2d} must have size nΩ(k)n^{\Omega(k)}. Since Iterated Matrix Multiplication IMMn,dIMM_{n,d} over {0,1}n2d\{0,1\}^{n^2d} is functionally in GapLGapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC0ACC^0 from GapLGapL

    Three Puzzles on Mathematics, Computation, and Games

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    In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

    Resolution over Linear Equations and Multilinear Proofs

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    We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) interpolation by a communication game technique we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [RazTzameret06]. Specifically, we show the following: 1. Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations. 2. Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the \emph{functional} pigeonhole principle. The latter are different than previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.Comment: 44 page

    A Logical Characterization of Constant-Depth Circuits over the Reals

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    In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202

    Algebraic and Combinatorial Methods in Computational Complexity

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    Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress
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