Randomized polynomial time identity testing for noncommutative circuits

Abstract

In this paper we show that the black-box polynomial identity testing for noncommutative polynomials f ∈ F z 1 , z 2 , · · · , z n of degree D and sparsity t, can be done in randomized poly(n, log t, log D) time. As a consequence, if the black-box contains a circuit C of size s computing f ∈ F z 1 , z 2 , · · · , z n which has at most t non-zero monomials, then the identity testing can be done by a randomized algorithm with running time polynomial in s and n and log t. This makes significant progress on a question that has been open for over ten years.The earlier result by Bogdanov and Wee [BW05], using the classical Amitsur-Levitski theorem, gives a randomized polynomial-time algorithm only for circuits of polynomially bounded syntactic degree. In our result, we place no restriction on the degree of the circuit.Our algorithm is based on automata-theoretic ideas introduced in [AMS08, AM08]. In those papers, the main idea was to construct deterministic finite automata that isolate a single monomial from the set of nonzero monomials of a polynomial f in F z 1 , z 2 , · · · , z n . In the present paper, since we need to deal with exponential degree monomials, we carry out a different kind of monomial isolation using nondeterministic automata.

Identity Testing for +-Regular Noncommutative Arithmetic Circuits

“…Now, we can design small nondeterministic substitution automata that can nondeterministically effect this transformation from P i to P i,I for each i. guess the locations in I. The rest of the proof is similar to the proof of Theorem 2 in [AMR16].…”

“…Specifically, C may compute a nonzero exponential degree noncommutative polynomial, but it is not clear if we can test that by evaluating C on matrices of polynomial dimension. Also, the black-box PIT result of [AMR16] cannot be applied here since C can compute polynomials of double-exponential sparsity.…”

“…By Lemma 2 there is a set I ⊆ [D] of size at most s − 1 such that C = s i=1 R i = 0 if and only if C = s i=1 R i,I = 0. Similar to the result in [AMR16], we will use a small size nondeterministic automaton to guess this subset I of locations, and substitute suitable commuting variables at all locations in [D] \ I. It will turn out that the transition matrices for each variable x i corresponding to this automaton will give us the desired black-box substitution.…”

“…In [AMR16], we made partial progress on this problem: we gave an efficient randomized blackbox polynomial identity test for noncommutative arithmetic circuits that compute a polynomial with exponentially many monomials [AMR16]. However, in general noncommutative circuits of size O(s) can compute polynomials with 2 2 s monomials.…”

Efficient Black-Box Identity Testing over Free Group Algebra

“…The Amitsur-Levitzki theorem states that a nonzero noncommutative polynomial p ∈ F X of degree < 2k cannot be an identity for the matrix ring M k (F). Additionally, it is shown that a nonzero noncommutative polynomial does not vanish on matrices of dimension logarithmic in the sparsity of the polynomial, yielding a randomized polynomial time algorithm for noncommutative circuits computing a nonzero polynomial of exponential degree and exponential sparsity [AJMR17].…”