Concatenation Hierarchies Research Articles

The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies

We deal with the question of the [Formula: see text]-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids [Formula: see text] is called [Formula: see text]-reducible if, given a finite ordered monoid M, for every inequality of pseudowords that is valid in [Formula: see text], there exists an inequality of [Formula: see text]-words that is also valid in [Formula: see text] and has the same “imprint” in M. Place and Zeitoun have recently proven the decidability of the membership problem for levels [Formula: see text], 1, [Formula: see text] and [Formula: see text] of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels [Formula: see text] and [Formula: see text] are [Formula: see text]-reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels [Formula: see text] and [Formula: see text] are definable by [Formula: see text]-inequalities. Furthermore, in the special case of the Straubing–Thérien hierarchy, using a characterization theorem for level 2 by Place and Zeitoun, we obtain that the level 2 is definable by [Formula: see text]-identities.

Electronic properties and carrier mobilities of nanocarbons formed by non-benzoidal building blocks.

Exotic 1D and 2D carbon nanostructures have been grown in the laboratory in the last few years by means of surface-assisted chemical routes. In these processes, the strategical choice of a molecular precursor plays a dominant role in the determination of the synthesized nanocarbon. Further variations of these techniques are able to produce non-benzoidal carbon quantum-dots (QDs). Considering this experimental scenario as motivation, we propose a series of nanoribbon systems based on concatenating recently synthesized carbon QDs containing pentagonal, hexagonal, and heptagonal rings. We use density functional theory (DFT) simulations to reveal their properties can range from metallic to semiconducting depending on the concatenation hierarchy used to form the nanoribbons. This DFT implementation is based on a LCAO approach to describe valence wavefunctions and most of the simulations employ the PBE-GGA functional. Since this functional is known to underestimate band gaps, we also use the B3LYP functional in a plane-wave DFT approach for a selected case for comparison purposes. These systems show a different gap versus width relationship compared to conventional graphene nanoribbons setups and a particular set of carrier mobility values. We further discuss the interplay between the QD's frontier states and the electronic properties of the nanoribbons in light of their structural details.

Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.

Generic Results for Concatenation Hierarchies

In the theory of formal languages, the understanding of concatenation hierarchies of regular languages is one of the most fundamental and challenging topics. In this paper, we survey progress made on this problem since 1971. We also establish new generic statements regarding this problem.

The ω-inequality problem for concatenation hierarchies of star-free languages

Abstract The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing–Thérien hierarchy.

Bidirectional string assembling systems

We introduce and investigate several variants of a bidirectional string assembling system, which is a computational model that generates strings from copies of assembly units. The underlying mechanism is based on two-sided piecewise assembly of a double-stranded sequence of symbols, where the upper and lower strand have to match. The generative capacities and the relative power of the variants are our main interest. In particular, we prove that bidirectional string assembling system generate languages not represented as any finite concatenation of one-sided string assembling systems. The latter build an infinite, strict and tight concatenation hierarchy. Moreover, it is shown that even the strongest system in question can only generate NL languages, while there are unary regular languages that cannot be derived. Furthermore, a finite strict hierarchy with respect to the different variants considered is shown and closure properties of the languages generated are presented.

New decidable upper bound of the second level in the Straubing-Therien concatenation hierarchy of star-free languages

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.

A counterexample to a conjecture concerning concatenation hierarchies

We give a counterexample to the conjecture which was originally formulated by Straubing in 1986 concerning a certain algebraic characterization of regular languages of level 2 in the Straubing–Thérien concatenation hierarchy of star-free languages.

THE WREATH PRODUCT PRINCIPLE FOR ORDERED SEMIGROUPS1

ABSTRACT Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions of the operations and . Next we turn to concatenation hierarchies. It was shown by Straubing that the n-th level of the dot-depth hierarchy is the variety , where is the variety of locally trivial semigroups and is the n-th level of the Straubing-Thérien hierarchy. We prove that a similar result holds for the half levels. It follows in particular that a level or a half level of the dot-depth hierarchy is decidable if and only if the corresponding level of the Straubing-Thérien hierarchy is decidable.

Algebraic tools for the concatenation product

This paper is a contribution to the algebraic study of the concatenation product. In the first part of the paper, we extend to the ordered case standard algebraic tools related to the concatenation product, like the Schützenberger product and the relational morphisms. We show in a precise way how the ordered Schützenberger product corresponds to polynomial operations on languages. In the second part of the paper, we apply these results to establish a bridge between the three standard concatenation hierarchies, namely the Straubing–Thérien hierarchy, the Brzozowski's (or dot-depth) hierarchy and the group hierarchy.

A conjecture on the concatenation product

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.

Polynomial closure and unambiguous product

This article is a contribution to the algebraic theory of automata, but it also contains an application to Buchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL 0 a 1 L 1 ···a nLn, where thea i’s are letters and theL i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.

Polynomial Closure and Unambiguous Product

This article is a contribution to the algebraic theory of automata, but it also contains an application to Büchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL 0 a 1 L 1 ···a nLn, where thea i’s are letters and theL i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.

Opérations polynomiales et hiérarchies de concaténation

Given a class C of languages, let Pol( C ) be the polynomial closure of C , that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol( C ) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol( C ) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 1 2 and 3 2 of this hierarcy are decidable.