Contextual partial commutations

We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science

Contextual partial commutations

We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science

Regular languages and partial commutations

[EN] The closure of a regular language under a [partial] commutation I has been extensively studied. We present new advances on two problems of this area: (1) When is the closure of a regular language under [partial] commutation still regular? (2) Are there any robust classes of languages closed under [partial] commutation? We show that the class Pol(G) of polynomials of group languages is closed under commutation, and under partial commutation when the complement of I in A2 is a transitive relation. We also give a su¿cient graph theoretic condition on I to ensure that the closure of a language of Pol(G) under I-commutation is regular. We exhibit a very robust class of languages W which is closed under commutation. This class contains Pol(G), is decidable and can be de¿ned as the largest positive variety of languages not containing (ab)¿. It is also closed under intersection, union, shu¿e, concatenation, quotients, length-decreasing morphisms and inverses of morphisms. If I is transitive, we show that the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, ¿niteness conditions on semigroups and properties of insertion systems. © 2013 Elsevier Inc. All rights reserved[ES] El cierre de un lenguaje regular bajo una conmutación [parcial] $I$ se ha estudiado extensivamente. Presentamos nuevos avances sobre los dos problemas de esta zona: (1) cuando es el cierre de un lenguaje regular bajo ¿conmutación [parcial] todavía regular? (2) Hay alguna clase robusta ¿de idiomas cerraron bajo conmutación [parcial]? Demostramos que la clase \PolG de polinomios de grupo idiomas está cerrada bajo conmutación y bajo conmutación parcial cuando el complemento de $$I en $A ^ 2$ es una relación transitiva. También damos un gráfico suficiente condición teórica en $$I para asegurarse de que el cierre de un lenguaje de \PolG$$ bajo $lo$-conmutación es regular. Exhibimos un muy robusto clase de idiomas \cW que es cerrado bajo conmutación. Esta clase contiene \PolG$$, es decidible y puede definirse como el más grande positiva variedad de idiomas que no contengan $(ab) ^ *$. También es cerrado bajo intersección, Unión, shuffle, concatenación, cocientes, longitud decreciente morfismos e inversas de morfismos. Si $$I es transitivo, demostramos que el cierre de un lenguaje de \cW bajo $Lo$-conmutación es regular. Las pruebas son no triviales y se combinan varias técnicas avanzadas, incluyendo el tipo de Ramsey combinatoria argumentos, propiedades algebraicas de la monoid sintáctica, finito condiciones sobre semigrupos y propiedades de los sistemas de inserción.The first author was supported by the project Automatas en dispositivos moviles: interfaces de usuario y realidad aumentada (PAID 2019-06-11) supported by Universidad Politecnica de Valencia. The third author was supported by the project ANR 2010 BLAN 0202 02 FREC.Cano Gómez, A.; Guaiana, G.; Pin, J. (2013). Regular languages and partial commutations. Information and Computation. 230:76-96. https://doi.org/10.1016/j.ic.2013.07.003S769623

Contextual partial commutations

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Application

Solving word equations modulo partial commutations

AbstractIt is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz

Implementation relations for testing through asynchronous channels

This paper concerns testing from an input output transition system (IOTS) model of a system under test that interacts with its environment through asynchronous first in first out (FIFO) channels. It explores methods for analysing an IOTS without modelling the channels. If IOTS M produces sequence $\sigma$ then, since communications are asynchronous, output can be delayed and so a different sequence might be observed. Thus M defines a language Tr(M) of sequences that can be observed when interacting with M through FIFO channels. We define implementation relations and equivalences in terms of Tr(M): an implementation relation says how IOTS N must relate to IOTS M in order for N to be a correct implementation of M. It is important to use an appropriate implementation relation since otherwise the verdict from a test run might be incorrect and because it influences test generation. It is undecidable whether IOTS N conforms to IOTS M and so also whether there is a test case that can distinguish between two IOTSs. We also investigate the situation in which we have a finite automaton P and either wish to know whether $Tr(M) \cap L(P)$ is empty or whether Tr(M) \cap \tr(P) is empty and prove that these are undecidable. In addition, we give conditions under which conformance and intersection are decidable.This work was partially supported by EPSRC grant EP/G04354X/1:The Birth, Life and Death of Semantic Mutants

Transitive factorizations of free partially commutative monoids and Lie algebras

Let \M(A,\theta) be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet $B\subset A$ such that the right factor of a bisection \M(A,\theta)=\M(B,\theta_B).T be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of \M(A,\theta) and associated bases of $L_K(A,\theta)$

Forward Analysis and Model Checking for Trace Bounded WSTS

We investigate a subclass of well-structured transition systems (WSTS), the bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth. Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all $\omega$-regular properties on the set of infinite traces of the system

The Tutte-Grothendieck group of a convergent alphabetic rewriting system

The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial, and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski's theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between non commutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the later is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was even not mention by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a non convergent rewriting system, outside the scope of Brylawski's work.Comment: 17 page