On the degree of ambiguity of finite automata

Computational Complexity, 1998

We examine the computational power of modu1a.r r~iimfing, where the modulus m is not a prime power, i n th,e setting of polynomials an boolean variables over %,,,. In particular, we say that a polynomial P weakly i*vprc.wnts a. boolean function f [both have n vari-~i h l ~s ) if for any inputs x and y in we have I ' ( T ) f P(y) whenever f(x) # f(4. Barrington, i ~c i g c l , a.nd Rudich [S] investigated the minimal deqivr of a polynomial representing the OR function in ill.i.7 iua.y, proving an upper bound of O(n'/') (where I' i.7 ihe number of distinct primes dividing m) and ( I lower bound of w ( 1 ) . Here we show a lower bound R(1ngn) when m is a product of two primes and ()((log n)l/('-l)) in general. While many lower bounds nrv known for a much stronger form of representation of a, function b y a polynomial 5, 121, using this liberal ( f i n d , we argue, more natural 5 definition very little is k7rown. While the degree is known to be Q(1ogn) for th.r gcnemlited inner product because of its high comtnun.icaiion complexity [g], our bound is the best known /or On y junction of low communication complexity and f1:i.y m.oduhs not a prime power.

Discrete Applied Mathematics, 1989

We solve the following problem proposed by H. Straubing. Given a two letter alphabet A, what is the maximal number of states f (n) of the minimal automaton of a subset of A n , the set of all words of length n. We give an explicit formula to compute f (n) and we show that 1 = lim n→∞ nf (n)/2 n ≤ lim n→∞ nf (n)/2 n = 2.

Theoretical Computer Science, 2003

For any q > 1, let MOD q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q, t > 1, MOD q is equivalent to MOD t (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC (0) , ACC[q], and ACC, denoted QAC (0) wf , QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, proving that QAC

2017 IEEE International Symposium on Information Theory (ISIT)

This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given m × n binary matrix M , the GM-MDS conjecture, due to Dau et al., states that if M satisfies the so-called MDS condition, then for any field F of size q ≥ n + m − 1, there exists an [n, m]q MDS code whose generator matrix G, with entries in F, fits M (i.e., M is the support matrix of G). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by Yan et al. and Dau et al., that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if M satisfies the MDS condition, then the determinant of a transformation matrix T , such that T V fits M , is not identically zero, where V is a Vandermonde matrix with distinct parameters. In this work, we generalize the TM-MDS conjecture, and present an algebraiccombinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows (m) of M is upper bounded by 5. For this class of special cases of M where the only additional constraint is on m, only cases with m ≤ 4 were previously proven theoretically, and the previously used proof techniques are not applicable to cases with m > 4.

Discrete Mathematics, 2006

For positive integers m and r, one can easily show there exist integers N such that for every map ∆ : In this paper we investigate the minimal such integer, which we call g(m, r). We prove that g(m, 2) = 5(m -1) + 1 for m ≥ 2, that g(m, 3) = 7(m -1) + 1 + m 2 for m ≥ 4, and that g(m, 4) = 10(m -1) + 1 for m ≥ 3. Furthermore, we consider g(m, r) for general r. Along with results that bound g(m, r), we compute g(m, r) exactly for the following infinite families of r: {f2n+3} , {2f2n+3} , {18f2n -7f2n-2} , and {23f2n -9f2n-2} , where here fi is the ith Fibonacci number defined by f0 = 0 and f1 = 1.