TY - JOUR

T1 - On the power of finite automata with both nondeterministic and probabilistic states

AU - Condon, Anne

AU - Hellerstein, Lisa

AU - Pottle, Samuel

AU - Wigderson, Avi

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1998

Y1 - 1998

N2 - We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where Arthur is limited to polynomial time and constant space. Dwork and Stockmeyer [SIAM J. Comput, 19 (1990), pp. 1011-1023] asked whether these npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely, the binary matrix with a row and column for each string of length ≤ n1 where entry [x, y] = 1 if and only if the string xy ε L. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof extends techniques of Frobenius and lohvidov developed for Hankel matrices [Sitzungsber, der Königl, Preuss. Akad. der Wiss., 1894, pp. 407-431], [Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser, Boston, 1982].

AB - We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where Arthur is limited to polynomial time and constant space. Dwork and Stockmeyer [SIAM J. Comput, 19 (1990), pp. 1011-1023] asked whether these npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely, the binary matrix with a row and column for each string of length ≤ n1 where entry [x, y] = 1 if and only if the string xy ε L. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof extends techniques of Frobenius and lohvidov developed for Hankel matrices [Sitzungsber, der Königl, Preuss. Akad. der Wiss., 1894, pp. 407-431], [Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser, Boston, 1982].

KW - Arthur-merlin games

KW - Hankel matrices

KW - Interactive proof systems

KW - Matrix tiling

KW - Nondeterministic probabilistic finite automata

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U2 - 10.1137/S0097539794265578

DO - 10.1137/S0097539794265578

M3 - Article

AN - SCOPUS:0038246725

VL - 27

SP - 739

EP - 762

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 3

ER -