Howard Straubing

On uniformity within NC 1

In order to study circuit complexity classes within NC1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC1 uniformity, have appeared in recent research: Immermans families of circuits defined by first-order formulas and a uniformity corresponding to Buss deterministic log-time reductions. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC1 still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Therien, and Thomas. A preliminary version of this work appeared in “Structure of Complexity Theory: Third Annual Conference” pp. 47–59, IEEE Comput. Soc., Washington, DC, 1988.

Finite Automata, Formal Logic, and Circuit Complexity

Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.

Non-uniform automata over groups

Abstract A new model, non-uniform deterministic finite automata (NUDFAs) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC1, NUDFA characterizations of several important subclasses of NC1, and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipsers (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFAs over solvable groups (NUDFAs over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFAs over nilpotent groups and prove some optimal lower bounds for NUDFAs over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in ( D. A. Barrington and D. Therien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).

A generalization of the Schützenberger product of finite monoids

For each n⩾1, an n-ary product ♢ on finite monoids is constructed. This product has the following property: Let Σ be a finite alphabet and Σ∗ the free monoid generated by Σ. For i = 1, …,n, let Ai be a recognizable subset of Σ∗, M(Ai) the syntactic monoid of An and M(A1⋯An) the syntactic monoid of the concatenation product A1⋯An. Then M(A1⋯An)< ♢ (M(A1),…,M(An)). The case n = 2 was studied by Schutzenberger. As an application of the generalized product, I prove the theorem of Brzozowski and Knast that the dot-depth hierarchy of star-free sets is infinite.

Regular languages in NC 1

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 .

Regular languages defined with generalized quantifiers

We study an extension of first-order logic obtained by adjoining quantifiers that count with respect to an integer modulus. It is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups. We obtain an analogous result for regular ω-languages and establish some connections with complexity theory for fixed-depth families of circuits.

Semigroups and languages of dot-depth two

This paper is a contribution to the problem of effectively determining the dot-depth of a star-free language, a problem in the theory of automata and formal languages with close connections to algebra and formal logic. We conjecture an effective criterion for determining whether a given language has dot-depth 2. The condition is shown to be necessary in general, and sufficient for languages over a two-letter alphabet. The condition involves a novel use of categories in the study of semigroup-theoretic problems.

Inversion of matrices over a commutative semiring

It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative semiring, then the equation AB = 1 implies BA = 1. We give two proofs: one algebraic in nature, the other more combinatorial. Both proofs use a generalization of the familiar product law for determinants over a commutative semiring.

On Logical Descriptions of Regular Languages

There are many examples in the research literature of families of regular languages defined by purely model-theoretic means (that is, in terms of the kinds of formulas of predicate logic used to define them) that can be characterized algebraically (that is, in terms of the syntactic monoids or syntactic morphisms of the languages). In fact the existence of such algebraic characterizations appears to be the rule. The present paper gives an explanation of the phenomenon: A generalization of Eilenbergs variety theorem is proved, and then applied to logic. We find that a very wide assortment of families of regular languages defined in model-theoretic terms form varieties in this new sense,and that consequently membership in the family depends only on the syntactic morphism of the language.

Locally trivial categories and unambiguous concatenation

Abstract We use the recently developed theory of finite categories and the two-sided kernel to study the effect of the unambiguous concatenation product of recognizable languages on the syntactic monoids of the languages involved. As a result of this study we obtain an algebraic characterization (originally due to Pin) of the closure of a variety of languages under boolean operations and unambiguous concatenation, and a new proof of a theorem of Straubing characterizing the closure of a variety of languages under boolean operations and concatenation. We also note some connections to the study of the dot-depth hierarchy.