Derandomizing polynomial identity tests means proving circuit lower bounds

Abstract

We show that derandomizing Polynomial Identity Testing is, essentially, equivalent to proving circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or, even, nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) NEXP ⊂ P/poly or (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic formula computes an identically zero polynomial.Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP = P (or, even, coRP ⊆ ∩ >0NTIME(2 n ), infinitely often), then NEXP is not computable by polynomial-size arithmetic circuits. Thus, establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.

Deterministic polynomial identity tests for multilinear bounded-read formulae

“…In fact, an efficient deterministic blackbox identity test for arithmetic formulae of depth three or four already implies such lower bounds (Agrawal & Vinay 2008;Gupta et al 2013). Conversely, the well-known hardness versus randomness trade-offs imply that sufficiently strong Boolean circuit lower bounds yield efficient deterministic polynomial identity tests, and there are a couple of similar results starting from arithmetic circuit lower bounds as well (Dvir et al 2009;Kabanets & Impagliazzo 2004). cc (2015) PIT for multilinear bounded-read formulae 3…”

“…Efficiently derandomizing identity testing implies Boolean or arithmetic formula/circuit lower bounds that have been elusive for half a century (Aaronson & van Melkebeek 2011;Kabanets & Impagliazzo 2004;Kinne et al 2012). In fact, an efficient deterministic blackbox identity test for arithmetic formulae of depth three or four already implies such lower bounds (Agrawal & Vinay 2008;Gupta et al 2013).…”

Reconstructive Dispersers and Hitting Set Generators

“…The output of a PRG can thus be substituted for true randomness to obtain efficient simulations of BPP, by enumerating over all seeds. However, the existence of a uniform family of PRGs useful for derandomizing BPP implies circuit lower bounds that seem well beyond our current abilities to prove (and more recently, it has been shown that BPP = P itself implies circuit lower bounds [15]). Thus, in the absence of circuit lower bounds, the goal is to construct PRGs under a hardness assumption, and then we can hope for a family of constructions that represent a "best-possible" tradeoff between the hardness assumption required and the deterministic simulation implied by the PRG.…”

Building Above Read-Once Polynomials: Identity Testing and Hardness of Representation

“…However, obtaining deterministic polynomial time algorithms for PIT remained open since then. In 2004, Kabanets and Impagliazzo [11] showed that a deterministic polynomial time algorithm for PIT implies lower bounds (either NEXP ⊂ P/poly or permanent does not have polynomial size arithmetic circuits), thus making it one of the central problems in algebraic complexity. Following [11], intense efforts over the last decade have been directed towards de-randomizing PIT [21,25].…”