Denis Therien

• McGill University

In computations realized by finite automata, a rich understanding has come from comparing the algebraic structure of the machines to the combinatorics of the languages being recognized. In this expository paper, we will first survey some basic ideas that have been useful in this model. In the second part, we sketch how this dual approach can be gen...

Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \textACC0\text{ACC}^0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class is not known to be learnable in any reasonable...

Unlike the wreath product, the block product is not associative at the level of varieties. All decomposition theorems involving block products, such as the bilateral version of Krohn–Rhodes' theorem, have always assumed a right-to-left bracketing of the operands. We consider here the left-to-right bracketing, which is generally weaker. More precise...

In order to systematize existing results, we propose to analyze the learnability of boolean functions computed by an algebraically defined model, programs over monoids. The expressiveness of the model, hence its learning complexity, depends on the algebraic structure of the chosen monoid. We identify three classes of monoids that can be identified,...

Abstract. We consider generalized first-order sentences over < using both ordinary and modular quantifiers. It is known that the languages definable by such sentences are exactly the regular languages whose syntactic monoids contain only solvable groups. We show that any sentence in this logic is equivalent to one using three variables only, and we...

Let 0 ≤ r q. We will define a new kind of quantifier $(q,r)\exists^{(q,r)}. Informally, $(q,r) xf\exists^{(q,r)} x\phi means “the number of positions x that satisfy Φ is congruent to r modulo q”. Formally, let w be a V-structure over A, and let x Ï nx \notin \nu. We obtain | w |\left | w \right | different ( nÈ{ x } )\left ( \nu \cup \left \{ x \ri...

We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain...

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this class...

We study languages with bounded communication complexity in the multiparty "input on the forehead model" with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

Let CC(m) be the class of circuits in which all gates are MODm gates. In this paper we prove lower bounds for circuits in CC(m) and related classes. • Circuits in which all gates are MODm gates need ( n) gates to compute the MODq func- tion, when m and q are co-prime. No non-trivial bounds were known before for computing MODq functions. Our argumen...

A letter e ∈Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The Crane Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in first-order with arbitrary numerical predicates (\({\bf FO}[\mathit{Arb}]\)) is in...

We give a new proof of recent results of Grolmusz and Tardos on the computing power of constant-depth circuits consisting of a single layer of MODmMOD_m gates followed by a fixed number of layers of MODpkMOD_{p^k} -gates, where p is prime.

The algebraic theory of finite automata has developed into a well-structured field, with the notion of variety serving as the unifying concept. Of course, the class of regular languages is very restricted and, for that reason, has not so far played any significant role in computational complexity. The connections between automata and complexity tha...

Without Abstract

Algebraic techniques are used to prove that any circuit constructed with MOD q gates that computes the AND function must use (n) gates at the first level. This constitutes the first non-trivial lower bound on the circuit size required to compute AND with MOD q gates that is valid for arbitrary q.

First-order translations have recently been characterized as the maps computed by aperiodic single-valued non-deterministic finite transducers (NFTs). It is shown here that this characterization lifts to "V-translations" and "V-single-valued-NFTs", where V is an arbitrary monoid pseudovariety that is closed under reversal. More strikingly, two-way...

We study the problem of learning an unknown function represented as an expression or a program over a known finite monoid. As in other areas of computational complexity where programs over algebras have been used, the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid. Indeed, o...

We 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 Pippenger's 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 r...

A new model, non-uniform deterministic finite automata (NUDFA's) 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 NC 1 [Ba86], N...

We study languages with bounded communication complexity in the multiparty “input on the forehead model” with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

In this paper we prove that the class of functions expressible by first order formulas with only two variables coincides with the class of functions computable by AC⁰ circuits with a linear number of gates. We then investigate the feasibility of using Ehrenfeucht-Fraisse games to prove lower bounds for that class of circuits, as well as...

The algebraic theory of finite automata has been one of the most successful tools to study and classify regular languages. These very same tools can in fact be used to understand more powerful models of computation and we discuss here the impact that semigroup theory can have in computational complexity.

In this document we describe the original motivation and goals of the seminar as well as the sequence of talks given during the seminar. @InProceedings{schwentick_et_al:DSP:2007:977, author = {Thomas Schwentick and Denis Th{'e}rien and Heribert Vollmer}, title = {06451 Executive Summary -- Circuits, Logic, and Games }, booktitle = {Circuits, Logic,...

From 08.11.06 to 10.11.06, the Dagstuhl Seminar 06451 ``Circuits, Logic, and Games'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the semi...

We contribute to the algebraic study of the complexity of constraint satisfaction problems. We give a new sufficient condition on a set of relations Γ over a domain S for the tractability of CSP(Γ): if S is a block-group (a particular class of semigroups) of exponent ω and Γ is a set of relations over S preserved by the operation defined by the pol...

It is well-known that the Σk- and Πk-levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the Δk-levels of these hierarchies: ${\rm Leaf}^{\rm P} (\Delta_k^L) = \Delta_k^p$. The same methods are used to give evidence against an earlier conjecture of Straubing and Thérien abou...

We obtain a logical characterization of an important class of regular languages, denoted \({\mathcal DO}\), and of its most important subclasses in terms of two-variable sentences with ordinary and modular quantifiers but in which all modular quantifiers lie outside the scope of ordinary quantifiers. The result stems from a new decomposition of the...

Finite semigroups, i.e. finites sets equipped with a binary associative operation, have played a role in theoretical computer science for fifty years. They were first observed to be closely related to finite automata, hence, by the famous theorem of Kleene, to regular languages. It was later understood that this association is very deep and the the...

This short note reviews the main contributions of the Ph.D. thesis of Imre Simon. His graduate work had major impact on algebraic theory of automata and thirty years later we are in a good position to appreciate how sensitive he was in selecting good problems, and how clever in solving them!

A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e's from any word in A* does not change its membership or non-membership in L.The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predica...

We develop a new method to analyze the flow of communication in constant-depth circuits. This point of view allows usto prove new lower bounds on the number of wires required to recognize certain languages. We are able to provide explicit languages that can be recognized by AC0 circuits with O(n) gates but not with O(n) wires, and similarly for ACC...

Regular languages are central objects of study in computer science. Although they are quite easy in the traditional space-time framework of sequential computations, the situation is different when other models are considered.In this paper we consider the subclass of regular languages that can be defined via unambiguous concatenation. We show remark...

This contribution wishes to argue in favor of increased interaction between experts on finite monoids and specialists of theory of computation. Developing the algebraic approach to formal computations as well as the computational point of view on monoids will prove to be beneficial to both communities. We give examples of this two-way relationship...

We 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 Pippenger’s end-decisive model, and a new QFA model whose definition is motivated by implementations of quantum computers using nucleo-magnetic resonance (NMR). In particular, we are interes...

The leaf-language mechanism associates a complexity class to a class of regular languages. It is well-known that the Σ k - and Π k -levels of the dot-depth hierarchy and the polynomial hierarchy correspond in this formalism. We extend this correspondence to the Δ k -levels of these hierarchies: LeafP(ΔkL_{k}^{L}) = Δkp_{k}^{p}. These results are ob...

We provide an effective characterization of the until-since hierarchy of linear temporal logic over finite models (strings), that is, we show how to compute for a given temporal property of strings the minimal nesting depth in until and since required to express it. This settles the most prominent classification problem for linear temporal logic. O...

It is known that a finite category can have all its base monoids in a variety V (i.e. be locally V, denoted ℓ V) without itself dividing a monoid in V (i.e. be globally V, denoted gV). This is in particular the case when V=Com, the variety of commutative monoids. Our main result provides a combinatorial characterization of locally commutative categ...

Algebra offers an elegant and powerful approach to understand regular languages and finite automata. Such framework has been notori- ously lacking for timed languages and timed automata. We introduce the notion of monoid recognizability for data languages, which include timed languages as special case, in a way that respects the spirit of the class...

Programs over semigroups are a well-studied model of computation for boolean functions. It has been used successfully to characterize, in algebraic terms, classes of problems that can, or cannot, be solved within certain resources. The interest of the approach is that the algebraic complexity of the semigroups required to capture a class should be...

We show that every regular language L has either constant, logarithmic or linear two-party communication complexity (in a worstcase partition sense). We prove a similar trichotomy for simultaneous communication complexity and a “quadrichotomy” for probabilistic communication complexity.

We show that every regular language L has either constant, logarithmic or linear two-party communication complexity (in a worst-case partition sense). We prove similar classifications for the communication complexity of regular languages for the simultaneous, probabilistic, simultaneous probabilistic and Modp-counting models of communication.

We provide an effective characterization of the “until-since hierarchy” of linear temporal logic, that is, we show how to compute for a given temporal property the minimal nesting depth in “until” and “since” required to express it. This settles the most prominent classi- fication problem for linear temporal logic. Our characterization of the indiv...

The block product of monoids is a bilateral version of the better known wreath product. Unlike the wreath product, block product is not associative. All decomposition theorems based on iterated block products that have appeared until now have assumed right-to-left bracketing of the operands. We here study what happens when the bracketing is made le...

We study the computational complexity of determining whether a systems of equations over a fixed finite monoid has a solution. In [6], it was shown that in the restricted case of groups the problem is tractable if the group is Abelian and NP-complete otherwise. We prove that in the case of an arbitrary finite monoid, the problem is in P if the mono...

Algebra offers an elegant and powerful approach to understand regu- lar languages and finite automata. Such framework has been no toriously lacking for timed languages and timed automata. We introduce the notion of monoid rec- ognizability for data languages, which include timed languages as special case, in a way that respects the spirit of the cl...

In this paper, we consider finite automata with the restriction that whenever the automaton leaves a state it never returns to it. Equivalently we may assume that the states set is partially ordered and the automaton may never move “backwards” to a smaller state. We show that different types of partially ordered automata characterize different lang...

It is known that recognition of regular languages by finite monoids can be generalized to context-free languages and finite groupoids, which are finite sets closed under a binary operation. A loop is a groupoid with a neutral element and in which each element has a left and a right inverse. It has been shown that finite loops recognize exactly thos...

We study the problem of learning an unknown function rep- resented as an expression over a known finite monoid. As in other areas of computational complexity where programs over algebras have been used, the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid. Indeed, our results...

We give an algebraic characterization of the regular languages defined by sentences with both modular and first-order quantifiers that use only two vari- ables.

The formalism of programs over monoids has been studied for its close connection to parallel complexity classes defined by small-depth boolean circuits. We investigate two basic questions about this model. When is a monoid rich enough that it can recognize arbitrary languages (provided no restriction on length is imposed)? When is a monoid weak eno...

A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e's from any word in A* does not change its membership (or non-membership) in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical pred...

We study a model of computation where executing a program on an input corresponds to calculating a product in a finite monoid. We show that in this model, the subsets of 0 ¡ 1 ¢ n that can be recognized by nilpotent groups have exponential cardinality. Translator’s Note: This is a translation of the article “Sur les langages reconnus par des groupe...

The notion of a p-variety arises in the algebraic approach to Boolean circuit complexity. It has great signi cance, since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p-varieties. In this paper, we prove that semigroups of dot-depth one form a pvariety. This example has...

We consider circuits and expressions whose gates carry out multiplication in a nonassociative groupoid such as a quasigroup or loop. We define a class we call the polyabelian groupoids, formed by iterated quasidirect products of Abelian groups. We show that a quasigroup can express arbitrary Boolean functions if and only if it is not polyabelian, i...

This study focuses on read-once automata over groups, where each input is used in a single instruction. In particular, a proof for a rich enough read-once NUDFA N is given. The proof proceeds by bounding the Fourier transform of the distributions involved.

First-order translations have recently been characterized as the maps computed by aperiodic single-valued nondeterministic finite transducers (NFTs). It is shown here that this characterization lifts to “V-translations” and “V-single-valued-NFTs”, where V is an arbitrary monoid pseudovariety. More strikingly, 2-way V-machines are introduced, and th...

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a subclass of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture. In the...

We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property Boolean completeness. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, cir-cuits, an...

We show that functions with convergent real power series can be well approximated by two classes of polynomial-size small-weight threshold circuits: depth-three circuits with threshold gates on all levels, and depth-four circuits with threshold gates on the first two levels and ANDOR gates on the last two. This is done without restricting the input...

D. Therien and T. Wilke (1996) characterized the Until hierarchy of linear temporal logic in terms of aperiodic monoids. Here, a temporal operator able to count modulo q is introduced. Temporal logic augmented with such operators is found decidable as it is shown to express precisely the solvable regular languages. Natural hierarchies are shown to...

We reveal an intimate connection between semidirect products of finite semigroups and substitution of formulas in linear temporal logic. We use this connection to obtain an algebraic characterization of the `until' hierarchy of linear temporal logic; the k-th level of that hierarchy is comprised of all temporal properties that are expressible by a...

We define the counting classes #NC1, GapNC1, PNC1 and C=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1 subseteq #L and that C=NC1 subs...

We define the counting classes #NC1,GapNC1,PNC1, andC=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1⊆#L, thatPNC1⊆L, and thatC=NC1⊆L....

Let M be a finite monoid: define C(k)(M) to be the maximum number of bits that need to be exchanged in the k-party communication game to decide membership in any language recognized by M. We prove the following: a) If M is a group then, for any k, C(k)(M) = O(1) if M is nilpotent of class k − 1 and C(k)(M) = θ(n) otherwise. b) If M is aperiod...

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, V the corresponding variety of languages and V the smallest variety containing V and th...

We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is expressible by both a Σ 2 and a Π 2 sentence. We thereby establish: UTL=FO 2 =Σ 2 ∩Π 2 =UL, where UTL stands for the string properties expressible in the temporal logic with ‘eventually in the future’ and ‘eventually in the past’ as t...

We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is expressible by both a Σ2 and a Π2 sentence. We thereby establish: UTL = FO2 = Σ2 at the intersection of all sets Π2, where UTL stands for the string properties expressible in the temporal logic with `eventually in the future' and `eve...

We introduce a simplified cellular automaton (CA) based controller initially developed for a basic stability problem, and successfully adapt it with only slight modifications to a more difficult tracking problem. The results demonstrate that the underlying model is quite robust despite its simplicity, and that CA's are an interesting paradigm in an...

We consider circuits and expressions whose gates carry out multiplication in a non-associative groupoid such as loop. We define a class we call the polyabelian groupoids, formed by iterated quasidirect products of Abelian groups. We show that a loop can express arbitrary Boolean functions if and only if it is not polyabelian, in which case its EXPR...

We study the regular languages recognized by polynomial-length programs over finite semigroups belonging to product varieties V ∗ LI, where V is a variety of finite monoids, and LI is the variety of finite locally trivial semigroups. In the case where the semigroup variety has a particular closure property with respect to programs, we are able to g...

In this paper, we characterize exactly the class of languages that are recognizable by finite loops, i.e. by cancellative binary algebras with an identity. This turns out to be the well-studied class of regular open languages. Our proof technique is interesting in itself: we generalize the operation of block product of monoids, which is so useful i...

Constant-depth polynomial-size threshold circuits are usually classified according to their total depth. For example, the best known threshold circuits for iterated multiplication and division have depths four and three, respectively. In this paper, the complexity of threshold circuits is investigated from a different point of view: explicit AND, O...

The problem of evaluating a circuit whose wires carry values from a finite monoid M and whose gates perform the monoid operation provides a meaningful generalization to the well-studied problem of evaluating a word over M. Evaluating words over monoids is closely tied to the fine structure of the complexity class NC1, and in this paper analogous ti...

We reveal an intimate connection between semidirect products of finite semigroups and substitution of formulas in linear temporal logic. We use this connection to obtain an algebraic characterization of the `until' hierarchy of linear temporal logic; the k-th level of that hierarchy is comprised of all temporal properties that are expressible by a...

The computation tree of a nondeterministic machineMwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular “leaf language”Y, the set of allxfor which the leaf string ofMis contained inY. In this way, in the c...

Leaf languages were used in the context of polynomial time computation to capture complexity classes and to study machine-independent relativizations. In this paper, the expressibility of the leaf language mechanism is investigated in the contexts of logarithmic space and of logarithmic time computation

Algebraic techniques are used to prove that any circuit constructed with MODq gates that computes the AND function must use Ω(n) gates at the first level. The best bound previously known to be valid for arbitraryq was Ω(logn).

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 b, where is first order, b is a binary predicate symbol, and the range of the second order quantifier is res...

We investigate small-depth threshold circuits for iterated multiplication and related problems. One result is that we can solve this problem with an ACo-connection of TC 3 o -languages, i.e. an ACo-connection of languages recognizable by depth-3 threshold circuits. This can be compared to the best known construction, which uses four levels of thres...

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star-height ≤ n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective...

The problem of testing membership in aperiodic or “group-free” transformation monoids is the natural counterpart to the well-studied membership problem in permutation groups. The class A of all finite aperiodic monoids and the class G of all finite groups are two examples of varieties, the fundamental complexity units in terms of which finite monoi...

We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC0, 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 AC0 and to solve in part an...

Concepts from the algebraic theory of finite automata are carried over to the program-over-monoid setting which underlies Barrington's algebraic characterization of the complexity classNC 1. Sets of languages accepted by polynomial-length programs over finite monoids drawn from a given monoid variety V emerge as fundamental language classes: as V r...

The Krohn-Rhodes theorem describes how an arbitrary finite monoid can be decomposed into a wreath product of groups and aperiodic monoids. New tools have recently been introduced to refine and extend this fundamental result. New theorems can be obtained by considering monoids as a special case of categories, thus allowing more general structures to...

A new model, non-uniform deterministic finite automata (NUDFA's) 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 cha...

We consider a model of computation where the execution of a program on an input corresponds to calculating a product in a finite monoid. This model has recently been formalized to give an algebraic point of view on certain types of boolean circuits. We focus our discussion on computations over aperiodic monoids: in particular we exhibit examples of...

Without Abstract

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star-height ? n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective...