The isomorphism conjecture for constant depth reductions

For any class C closed under TC0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under u-uniform AC0 reductions are isomorphic under u-uniform AC0-computable isomorphisms

Pseudo-random generators and structure of complete degrees

It is shown that, if there exist sets in E (the exponential complexity class) that require 2Ω(n)-sized circuits, then sets that are hard for class P (the polynomial complexity class) and above, under 1-1 reductions, are also hard under 1-1 size-increasing reductions. Under the assumption of the hardness of solving the RSA (Rivest-Shamir-Adleman, 1978) problem or the discrete log problem, it is shown that sets that are hard for class NP (nondeterministic polynomial) and above, under many-1 reductions, are also hard under (non-uniform) 1-1 and size-increasing reductions

DSPACE(n)=<SUP>2</SUP> NSPACE(n): a degree theoretic characterization

It is shown that the following are equivalent. 1. DSPACE(n)=NSPACE(n). 2. There is a nontrivial &#8804; 1-NLm-degree that coincides with &#8804;1-Lm-degree. 3. For every class C closed under log-lin reductions, the &#8804;1-NLm-complete degree of C coincides with the &#8804;1-Lm-complete degree of C

For completeness, sublogarithmic space is no space

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions

Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.Comment: 22 page

On the isomorphism conjecture for 2DFA reductions

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under log-lin reductions: All complete sets for the class C under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic. The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: (1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (2) constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation

Algebraic and Combinatorial Methods in Computational Complexity

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things 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 PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings