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

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

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

Verifying and comparing finite state machines for systems that have distributed interfaces

This paper concerns state-based systems that interact with their environment at physically distributed interfaces, called ports. When such a system is used a projection of the global trace, a local trace, is observed at each port. As a result the environment has reduced observational power: the set of local traces observed need not define the global trace that occurred. We consider the previously defined implementation relation ⊆s and prove that it is undecidable whether N ⊆s M and so it is also undecidable whether testing can distinguishing two states or FSMs. We also prove that a form of model-checking is undecidable when we have distributed observations and give conditions under which N ⊆s M is decidable. We then consider implementation relation ⊆sk that concerns input sequences of length κ or less. If we place bounds on κ and the number of ports then we can decide N ⊆sk M in polynomial time but otherwise this problem is NP-hard

On Tools for Completeness of Kleene Algebra with Hypotheses

In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice

Reordering Derivatives of Trace Closures of Regular Languages

We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotiented, are finite. We also define a refined version of the Antimirov reordering derivative operation where parts-of-derivatives (states of the automaton) are nonempty lists of regular expressions rather than single regular expressions. We define the uniform scattering rank of a language and show that, for a regexp whose language has finite uniform scattering rank, the truncation of the (generally infinite) refined Antimirov automaton, obtained by removing long states, is finite without any quotienting, but still accepts the trace closure. We also show that star-connected languages have finite uniform scattering rank

Theory of traces

AbstractThe theory of traces, originated by A. Mazurkiewicz in 1977, is an attempt to provide a mathematical description of the behavior of concurrent systems. Its aim is to reconcile the sequential nature of observations of the system behavior on the one hand and the nonsequential nature of causality between the actions of the system on the other hand.One can see the theory of traces to be rooted in formal string language theory with the notion of partial commutativity playing the central role. Alternatively one can see the theory of traces to be rooted in the theory of labeled acyclic directed graphs (or even in the theory of labeled partial orders).This paper attempts to present a major portion of the theory of traces in a unified way. However, it is not a survey in the sense that a number of new notions are introduced and a number of new results are proved. Although traditionally most of the development in the theory of traces follows the string-language-theoretic line, we try to demonstrate to the reader that the graph-theoretic point of view may be more appropriate.The paper essentially consists of two parts. The first one (Sections 1 through 4) is concerned with the basic theory of traces. The second one (Section 5) presents applications of the theory of traces to the theory of the behavior of concurrent systems, where the basic system model we have chosen is the condition/event system introduced by C.A. Petri

String Diagrammatic Trace Theory

We extend the theory of formal languages in monoidal categories to the multi-sorted, symmetric case, and show how this theory permits a graphical treatment of topics in concurrency. In particular, we show that Mazurkiewicz trace languages are precisely symmetric monoidal languages over monoidal distributed alphabets. We introduce symmetric monoidal automata, which define the class of regular symmetric monoidal languages. Furthermore, we prove that Zielonka's asynchronous automata coincide with symmetric monoidal automata over monoidal distributed alphabets. Finally, we apply the string diagrams for symmetric premonoidal categories to derive serializations of traces.Comment: Paper accepted for MFCS 202