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REPORTS > KEYWORD > ALGEBRAIC CIRCUITS:
Reports tagged with Algebraic circuits:
TR13-185 | 24th December 2013
Fu Li, Iddo Tzameret

#### Generating Matrix Identities and Proof Complexity Lower Bounds

Revisions: 3

Motivated by the fundamental lower bounds questions in proof complexity, we investigate the complexity of generating identities of matrix rings, and related problems. Specifically, for a field $\mathbb{F}$ let $A$ be a non-commutative (associative) $\mathbb{F}$-algebra (e.g., the algebra Mat$_d(\mathbb{F})\;$ of $d\times d$ matrices over $\mathbb{F}$). We say that a non-commutative ... more >>>

TR16-101 | 1st July 2016
Toniann Pitassi, Iddo Tzameret

#### Algebraic Proof Complexity: Progress, Frontiers and Challenges

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight ... more >>>

TR17-007 | 19th January 2017
Michael Forbes, Amir Shpilka, Ben Lee Volk

#### Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds

Revisions: 1

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been ... more >>>

TR17-009 | 19th January 2017
Joshua Grochow, Mrinal Kumar, Michael Saks, Shubhangi Saraf

#### Towards an algebraic natural proofs barrier via polynomial identity testing

We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov-Rudich ... more >>>

TR17-163 | 2nd November 2017
Michael Forbes, Amir Shpilka

#### A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

In this paper we study the complexity of constructing a hitting set for $\overline{VP}$, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given ... more >>>

TR18-184 | 5th November 2018
Iddo Tzameret, Stephen Cook

#### Uniform, Integral and Feasible Proofs for the Determinant Identities

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over $GF(2)$ in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the ... more >>>

TR19-019 | 19th February 2019
Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi

#### Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\mathop{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas (MFCS ... more >>>

TR19-065 | 1st May 2019
Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon

#### Derandomization from Algebraic Hardness: Treading the Borders

Revisions: 2

A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new construction of such a HSG assuming that we have ... more >>>

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