Aperiodic Monoids Research Articles

Learning expressions and programs over monoids

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, our results indicate a close connection between both kinds of complexity. We focus on monoids which are either groups or aperiodic, and on the learning model of exact learning from queries. For a group G , we prove that expressions over G are efficiently learnable if G is nilpotent, and impossible to learn efficiently (under cryptographic assumptions) if G is nonsolvable. We present some results for restricted classes of solvable groups, and point out a connection between their efficient learnability and the existence of lower bounds on their computational power in the program model. For aperiodic monoids, our results seem to indicate that the monoid class known as DA captures exactly learnability of expressions by polynomially many Evaluation queries. When using programs instead of expressions, we show that our results for groups remain true, while the situation is quite different for aperiodic monoids.

The many faces of a translation

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 that is closed under reversal. More strikingly, two-way V -transducers are introduced, and the following three models are shown exactly equivalent to Eilenberg's classical notion of a bimachine when V is a group variety or when V is the variety of aperiodic monoids: V -translations, V -single-valued-NFTs and two-way V -transducers.

DUALISABILITY OF FINITE SEMIGROUPS

We describe the inherently non-dualisable finite algebras from some semigroup related classes. The classes for which this problem is solved include the variety of bands, the pseudovariety of aperiodic monoids, commutative monoids, and (assuming a reasonable conjecture in the literature) the varieties of all finite monoids and finite inverse semigroups. The first example of an inherently non-dualisable entropic algebra is also presented.

On a Product of Finite Monoids

In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S 1 ,…, S m , a product &#x25CA m (S m ,…, S 1 , S 0 ). We give a representation of the free objects in the pseudovariety &#x25CA m ( W m ,…, W 1 , W 0 ) generated by these ( m + 1)-ary products where S i &#x2208 W i for all 0 &#x2264 i &#x2264 m . We then give, in particular, a criterion to determine when an identity holds in &#x25CA m ( J 1 ,…, J 1 , J 1 ) with the help of a version of the Ehrenfeucht-Fraisse game ( J 1 denotes the pseudovariety of all semilattice monoids). The union &#x222A m >0 &#x25CA m ( J 1 ,…, J 1 , J 1 ) turns out to be the second level of the Straubing’s dot-depth hierarchy of aperiodic monoids.

Finite Monoids: From Word to Circuit Evaluation

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 $NC^1$, and in this paper analogous ties between evaluating circuits over monoids and the structure of the complexity class P are exhibited. It is shown that circuit evaluation in the case of any nonsolvable monoid is P complete, while circuits over solvable monoids can be evaluated in $DET \subseteq NC^2$. Then the case of aperiodic monoids is completely elucidated: their circuit evaluation problems are either in $AC^0$ or L- or $NL$-complete, depending on the precise algebraic properties of the monoids. Finally, it is shown that the evaluation of circuits over the cyclic group ${\Bbb Z}_q$ for fixed $q \geq 2$ is complete for the logspace counting class $co$-$MOD_qL$, that the problem for p-groups (p a prime) is complete for $MOD_pL$, and that the more general case of nilpotent groups of exponent q belongs to the Boolean closure of $MOD_qL$.

Monoids Respectingn-Chains of Intervals

A new strict hierarchy for the pseudovariety of aperiodic monoids is described. Thenth level of the hierarchy is the pseudovariety of all divisors of monoids of full transformations of finite chains respecting somen-chain of intervals. The levels can be obtained by iterated semidirect product of the first level. The associated hierarchy corresponding to taking partial transformations is also considered and appears intermingled with the above hierarchy.

On semidirect and two-sided semidirect products of finite $\mathcal {J}$trivial monoids

In this paper, using results of Almeida and Weil, we give criteria for the semidirect or two-sided semidirect product of two locally finite pseudovarieties V and W to satisfy an identity u = v. We illustrate these criteria with various semidirect and two-sided semidirect products of pseudovarieties of f-trivial monoids. In particular, let J 1 denote the class of all finite semilattice monoids and let W i be the sequence of pseudovarieties of monoids defined by W 1 = J 1 and W i +1 = J 1 * * W 2 (the two-sided semidirect product of J 1 by W 2 ). Each W k turns out to be perfectly related to the k-move standard Ehrenfeucht-Fraisse game. The union U k>1 W k is then the class A of all finite aperiodic monoids.

Variétés et fonctions rationnelles

We say that a rational (resp. a subsequential) function α from a free monoid into another one is in the variety of monoids V if it may realized by some unambiguous (resp. subsequential) transducer whose monoid of transitions is in V. We characterize these functions when V is the variety of aperiodic monoids, and the variety of groups. In the first case, the period of α −1( L) divides that of L, for each rational language L on the outputs. In the second case, α −1( L) is a group-language for each group language L; equivalently, α is continuous for the pro-finite topology. Examples of such functions are: the multiplication by a given number in a given basis, which is aperiodic; the division, which is a group-function.

Circuits, Matrices, and Nonassociative Computation

We consider the complexity of various computational problems over nonassociative algebraic structures. Specifically, we look at the problem of evaluating circuits, formulas, and words, over both nonassociative structures themselves and over matrices with elements in these structures. Extending past work, we show that such problems can characterize a wide variety of complexity classes up to and including NP. As an example, the word (i.e., iterated multiplication) problems involving a sequence of O(logkn) matrices over a structure ( S; +, ·) in which (S; +) is a monoid or an aperiodic monoid are complete for NCk+1 and for ACk, respectively, and a word problem variant involving matrices of size O(logkn) is complete for SCk.

A Classification of Aperiodic Power Monoids

A picture is given of the poset of pseudovarieties generated by the power monoids of the members of given pseudovarieties of aperiodic monoids. It provides answers to various questions concerning power pseudovarieties, showing, in particular, that their classifying poset is a countably infinite lattice. Several related pseudovarieties are shown to have decidable membership problems.

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx t=x t+q. The varietyCom 1,1 is the variety of finite semilattices also denoted byJ 1. In this paper, we consider the product varietyJ 1*Com t,q generated by all semidirect products of the formM*N withM∈J 1 andN∈Com t,q . We give a complete sequence of equations forJ 1*Com t,q implying complete sequences of equations forJ 1*(Com∩A),J 1*(Com∩G) andJ 1*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups.

Equations and monoid varieties of dot-depth one and two

Each level of the Straubing's hierarchy of aperiodic monoids can be parametrized in a natural way. This paper studies this parametrization for dot-depth one and two monoids. For level one, it is shown that the mth level is defined by a finite sequence of equations if and only if m = 1, 2 or 3. For level two, and for m ⩾ 1, a sequence of equations is given which is satisfied in the mth level and shown to ultimately define the 1st level.

Some results on the dot-depth hierarchy

In this paper we pursue the study of the decidability of the dot-depth hierarchy. We give an effective lower bound for the dotdepth of an aperiodic monoid. The main tool for this is the study of a certain operation on varieties of finite monoids in terms of Mal'cev product. We also prove the equality of two decidable varieties which were known to contain all dot-depth two monoids. Finally, we restrict our attention to inverse monoids, and we prove that the class of inverse dot-depth two monoids is locally finite.

Some results on the generalized star-height problem

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 star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by ( Z /2 Z ) n . In the same direction, we show that one of the languages that were conjectured to be of star-height 2 during the past ten years is in fact of star-height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M ∘ ( G ∘ N ), where M and N are aperiodic monoids, and G is a commutative group, the L is of star-height ≤ 1. Finally we show that every rational language is the inverse image, under some morphism between free monoids, of a language of (resticted) star-height 1.

Games, equations and dot-depth two monoids

Given any finite alphabet A and positive integers m 1,…, m k , congruences on A ∗, denoted by ~( m 1,…, m k ) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A ∗/~( m 1,…, m k . A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For | A|≥2, a necessary and sufficient condition is given for A ∗/~( m 1,…, m k ) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A ∗/~( m 1,…, m k ) are discussed.

The membership problem in aperiodic transformation monoids

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 monoids are classified. The collection of all varieties V forms an infinite lattice under the inclusion ordering, with the subfamily of varieties that are contained in A forming an infinite sublattice. For each V ⊆ A , the associated problem MEMB( V ) of testing membership in transformation monoids that belong to V , is considered. Remarkably, the computational complexity of each such problem turns out to look familiar. Moreover, only five possibilities occur as V ranges over the whole aperiodic sublattice: With one family of NP-hard exceptions whose exact status is still unresolved, any such MEMB( V ) is either PSPACE-complete, NP-complete, P-complete or in AC 0 . These results thus uncover yet another surprisingly tight link between the theory of monoids and computational complexity theory.

Two-sided wreath product of categories

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 appear as building blocks in decompositions results. Also, a two-sided version of the wreath product may be used as the connecting operation. This paper combines the two ideas: the new operation, called the block product, is defined directly as acting on categories and basic properties are presented. As an application, an open problem in the theory of regular languages is solved.

Programs over aperiodic monoids

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 such monoids over which it is not possible to count modulo p.

Séries formelles et algèbres syntactiques

The notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleene's theorem, Schützenberger's theorem about aperiodic monoid and Eilenberg's theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schützenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schützenberger's theorem on aperiodic monoids.

A generalization of finiteness

We explore an analogy between the family β1, of finite/cofinite languages and the family ψ1 of languages whose syntactic monoids are J-trivial. It is shown that (a) J-trivial monoids, (b)L-trivial monoids, (c) R-trivial monoids, and (d) a recently studied family, that we callK, of aperiodic monoids are natural generalization of the families of syntactic semigroups of (a) finite/cofinite languages, (b) definite languages, (c) reverse definite languages, and (d) generalized definite languages, respectively. In the case of alphabets of one and two letters, the languages corresponding to the familyK of monoids are characterized, illustrating the above-mentioned analogy explicitly.