Denis Thérien

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).

Classification of finite monoids: the language approach

Abstract The concept of a ∗-variety of congruences is introduced and related to ∗-variety of languages and variety of monoids. A systematic construction of increasingly complex ∗-varieties of congruences is presented. This construction is powerful enough to generate all monoids containing solvable groups. Some hierarchies occurring through this process are shown to correspond to well-known hierarchies of monoids, thus indicating that our construction is natural from an algebraic point of view. Some problems that remain open are also discussed.

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 .

Over words, two variables are as powerful as one quantifier alternation

We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is express ible by both a 2 and a 2 sentence. We thereby establish: UTL = FO2 = 2 \ 2 = UL ; where UTL stands for the string properties expressible in th e temporal logic with ‘eventually in the future’ and ‘eventua lly in the past’ as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of 2 \ 2 = FO2 involves a new combinatorial characterization of these two classes and introduce s a new method of playing Ehrenfeucht-Fraı̈ssé games to verif y identities in semigroups.

Graph congruences and wreath products

Abstract Results on graph congruences have recently been used to decide membership in varieties defined by wreath products. The general technique for doing so is explained. In particular an efficient algorithm is obtained for deciding when a semigroup divides a wreath product of a commutative monoid with locally trivial semigroup.

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.

An algebraic approach to data languages and timed languages

Algebra offers an elegant and powerful approach to understand regular languages and finite automata. Such framework has been notoriously lacking for timed languages and timed automata. We introduce the notion of monoid recognizability for data languages, which includes timed languages as special case, in a way that respects the spirit of the classical situation. We study closure properties and hierarchies in this model and prove that emptiness is decidable under natural hypotheses. Our class of recognizable languages properly includes many families of deterministic timed languages that have been proposed until now, and the same holds for non-deterministic versions.

Algebraic Results on Quantum Automata

AbstractWe use tools from the algebraic theory of automata to investigate the class of languages recognized by two models of Quantum Finite Automata (QFA): Brodsky and Pippengers end-decisive model (which we call BPQFA) and a new QFA model (which we call LQFA) whose definition is motivated by implementations of quantum computers using nucleo-magnetic resonance (NMR). In particular, we are interested in LQFA since NMR was used to construct the most powerful physical quantum machine to date. We give a complete characterization of the languages recognized by LQFA and by Boolean combinations of BPQFA. It is a surprising consequence of our results that LQFA and Boolean combinations of BPQFA are equivalent in language recognition power.

Logics For Context-Free Languages

We define matchings, and show that they capture the essence of context-freeness. More precisely, we show that the class of context-free languages coincides with the class of those sets of strings which can be defined by sentences of the form ∃ bϕ, where ϕ is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.

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.