arXiv subjects

Pascal Weil

Publications and source records attributed to Pascal Weil.

At least 37 records · Page 2Linked to original sources

arXiv2014

On the genericity of Whitehead minimality

We show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups.

math.GR↗

arXiv2012

On logical hierarchies within FO^2-definable languages

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier alternations yields an infinite hierarchy whose levels are varieties of languages (and hence admit an algebraic characterization). Using this algebraic approach, we show that the quantifier alternation hierarchy inside FO^{2}[<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Thérien and Vollmer.

cs.LO↗

arXiv2012

The FO^2 alternation hierarchy is decidable

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Thérien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Thérien, and Vollmer.

cs.LO↗

arXiv2011

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

math.GR↗

arXiv2011

Star-Free Languages are Church-Rosser Congruential

The class of Church-Rosser congruential languages has been introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential (belongs to CRCL), if there is a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. To date, it is still open whether every regular language is in CRCL. In this paper, we show that every star-free language is in CRCL. In fact, we prove a stronger statement: For every star-free language L there exists a finite, confluent, and subword-reducing semi-Thue system S such that the total number of congruence classes modulo S is finite and such that L is a union of congruence classes modulo S. The construction turns out to be effective.

cs.FL↗

arXiv2011

An introduction to finite automata and their connection to logic

This is a tutorial on finite automata. We present the standard material on determinization and minimization, as well as an account of the equivalence of finite automata and monadic second-order logic. We conclude with an introduction to the syntactic monoid, and as an application give a proof of the equivalence of first-order definability and aperiodicity.

cs.FL↗

arXiv2010

On the lattice of sub-pseudovarieties of DA

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal'cev products with the pseudovarieties of definite and reverse definite semigroups.

math.GR↗

arXiv2010

Automorphic orbits in free groups: words versus subgroups

We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

math.GR↗

arXiv2009

On finite-index extensions of subgroups of free groups

We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of $H$, and we give a language-theoretic characterization of the lattice of finite-index subgroups of $H$. Finally, we give a polynomial time algorithm to compute the malnormal closure of $H$.

math.GR↗

arXiv2009

On FO2 quantifier alternation over words

We show that each level of the quantifier alternation hierarchy within FO^2[<] -- the 2-variable fragment of the first order logic of order on words -- is a variety of languages. We then use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy -- and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one can effectively compute an integer m such that alpha is equivalent to a formula with at most m+1 alternating blocks of quantifiers, but not to a formula with only m-1 blocks. This is a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels.

cs.LO↗

arXiv2009

Algebraic characterization of logically defined tree languages

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindström quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.

cs.LO↗

arXiv2007

Random generation of finitely generated subgroups of a free group

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $Ø(n)$ in the RAM model and $Ø(n^2\log^2n)$ in the bitcost model.

math.GR↗

arXiv2006

On an algorithm to decide whether a free group is a free factor of another

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.

math.GR↗

arXiv2006

On the complexity of the Whitehead minimization problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem -- to decide whether a word is an element of some basis of the free group -- and the free factor problem can also be solved in polynomial time.

math.GR↗

arXiv2006

Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

math.GR↗

arXiv2006

Algebraic recognizability of regular tree languages

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem à la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and Ségoufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization.

cs.DM↗

arXiv2006

The recognizability of sets of graphs is a robust property

Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of context-free sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) context-free graph grammars, to Vertex Replacement (VR) context-free graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifier-free formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HR-recognizability and VR-recognizability coincide. The same combinatorial condition equates HR-context-free and VR-context-free sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures.

cs.LO↗

arXiv2006

Algebraic recognizability of languages

Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a cornerstone for applications and for theory. We would like to briefly explore why that is, and how this word-related notion extends to more complex models, such as those developed for modeling distributed or timed behaviors.

cs.LO↗