Regular Languages in NC¹

Abstract We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC 0 , thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC 0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p , where p is a prime not dividing m , is not AC 0 -reducible to addition mod m , and thus that MAJORITY is not AC 0 -reducible to addition mod m .

[1]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[2]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[3]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[4]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[5]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[6]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[7]  Howard Straubing.  Constant-Depth periodic Circuits , 1991, Int. J. Algebra Comput..

[8]  David A. Mix Barrington,et al.  Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹ , 1989, J. Comput. Syst. Sci..

[9]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[10]  Howard Straubing,et al.  Aperiodic homomorphisms and the concatenation product of recognizable sets , 1979 .

[11]  Howard Straubing,et al.  Non-Uniform Automata Over Groups , 1990, Inf. Comput..

[12]  Richard J. Lipton,et al.  Unbounded fan-in circuits and associative functions , 1983, J. Comput. Syst. Sci..

[13]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[14]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[15]  Janusz A. Brzozowski,et al.  The Dot-Depth Hierarchy of Star-Free Languages is Infinite , 1978, J. Comput. Syst. Sci..

[16]  Richard J. Lipton,et al.  Unbounded Fan-In Circuits and Associative Functions , 1985, J. Comput. Syst. Sci..

[17]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[18]  Bret Tilson,et al.  Categories as algebra: An essential ingredient in the theory of monoids , 1987 .

[19]  Raymond E. Miller,et al.  Varieties of Formal Languages , 1986 .

[20]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[21]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[22]  Howard Straubing,et al.  regular Languages Defined with Generalized Quantifiers , 1988, ICALP.

[23]  G. Lallement.  Semigroups and combinatorial applications , 1979 .

[24]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[25]  Pavel Pudlák,et al.  Threshold circuits of bounded depth , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[26]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[27]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[28]  Howard Straubing,et al.  Families of recognizable sets corresponding to certain varieties of finite monoids , 1979 .