String matching problems over free partially commutative monoids

AbstractThis paper studies two string matching problems over free partially commutative monoids. We analyze these two problems in detail, and present two efficient polynomial time algorithms for solving them

A cut-invariant law of large numbers for random heaps

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong law of large numbers is proved for heap monoids with Bernoulli measures. Finally, we study a sub-additive version of the Law of large numbers in this framework based on Kingman sub-additive Ergodic Theorem.Comment: 29 pages, 3 figures, 21 reference

Complexity Results for Confluence Problems

Abstract. We study the complexity of the confluence problem for re-stricted kinds of semi–Thue systems, vector replacement systems and general trace rewriting systems. We prove that confluence for length– reducing semi–Thue systems is P–complete and that this complexity reduces to NC2 in the monadic case. For length–reducing vector re-placement systems we prove that the confluence problem is PSPACE– complete and that the complexity reduces to NP and P for monadic sys-tems and special systems, respectively. Finally we prove that for special trace rewriting systems, confluence can be decided in polynomial time and that the extended word problem for special trace rewriting systems is undecidable.

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

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

Decidability Equivalence between the Star Problem and the Finite Power Problem in Trace Monoids

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g. we show the decidability of these problems for recognizable languages which contain at most one non-connected trace

26. Theorietag Automaten und Formale Sprachen 23. Jahrestagung Logik in der Informatik: Tagungsband

Der Theorietag ist die Jahrestagung der Fachgruppe Automaten und Formale Sprachen der Gesellschaft für Informatik und fand erstmals 1991 in Magdeburg statt. Seit dem Jahr 1996 wird der Theorietag von einem eintägigen Workshop mit eingeladenen Vorträgen begleitet. Die Jahrestagung der Fachgruppe Logik in der Informatik der Gesellschaft für Informatik fand erstmals 1993 in Leipzig statt. Im Laufe beider Jahrestagungen finden auch die jährliche Fachgruppensitzungen statt. In diesem Jahr wird der Theorietag der Fachgruppe Automaten und Formale Sprachen erstmalig zusammen mit der Jahrestagung der Fachgruppe Logik in der Informatik abgehalten. Organisiert wurde die gemeinsame Veranstaltung von der Arbeitsgruppe Zuverlässige Systeme des Instituts für Informatik an der Christian-Albrechts-Universität Kiel vom 4. bis 7. Oktober im Tagungshotel Tannenfelde bei Neumünster. Während des Tre↵ens wird ein Workshop für alle Interessierten statt finden. In Tannenfelde werden • Christoph Löding (Aachen) • Tomás Masopust (Dresden) • Henning Schnoor (Kiel) • Nicole Schweikardt (Berlin) • Georg Zetzsche (Paris) eingeladene Vorträge zu ihrer aktuellen Arbeit halten. Darüber hinaus werden 26 Vorträge von Teilnehmern und Teilnehmerinnen gehalten, 17 auf dem Theorietag Automaten und formale Sprachen und neun auf der Jahrestagung Logik in der Informatik. Der vorliegende Band enthält Kurzfassungen aller Beiträge. Wir danken der Gesellschaft für Informatik, der Christian-Albrechts-Universität zu Kiel und dem Tagungshotel Tannenfelde für die Unterstützung dieses Theorietags. Ein besonderer Dank geht an das Organisationsteam: Maike Bradler, Philipp Sieweck, Joel Day. Kiel, Oktober 2016 Florin Manea, Dirk Nowotka und Thomas Wilk

Verification of Automata with Storage Mechanisms

An important question in computer science is to ask, whether a given system conforms to a specification. Often this question is equivalent to ask whether a finite automaton with certain memory like a stack or queue can reach some given state. In this thesis we focus this reachability problem of automata having one or more lossy or reliable stacks or queues as their memory. Unfortunately, the reachability problem is undecidable or of high complexity in most of these cases. We circumvent this by several approximation methods. So we extend the exploration algorithm by Boigelot and Godefroid under-approximating the reachability problem of queue automata. We also study some automata having multiple stacks with a restricted behavior. These “asynchronous pushdown systems” have an efficiently decidable reachability problem. To show our results we first have to gain knowledge of several algebraic properties of the so-called transformation monoid of the studied storage mechanisms.An important research topic in computer science is the verification, i.e., the analysis of systems towards their correctness. This analysis consists of two parts: first we have to formalize the system and the desired properties. Afterwards we have to find algorithms to check whether the properties hold in the system. In many cases we can model the system as a finite automaton with a suitable storage mechanism, e.g., functional programs with recursive calls can be modeled as automata with a stack (or pushdown). Here, we consider automata with two variations of stacks and queues: 1. Partially lossy queues and stacks, which are allowed to forget some specified parts of their contents at any time. We are able to model unreliable systems with such memories. 2. Distributed queues and stacks, i.e., multiple such memories with a special synchronization in between. Often we can check the properties of our models by solving the reachability and recurrent reachability problems in our automata models. It is well-known that the decidability of these problems highly depends on the concrete data type of our automata’s memory. Both problems can be solved in polynomial time for automata with one stack. In contrast, these problems are undecidable if we attach a queue or at least two stacks to our automata. In some special cases we are still able to verify such systems. So, we will consider only special automata with multiple stacks - so-called asynchronous pushdown automata. These are multiple (local) automata each having one stack. Whenever these automata try to write something into at least one stack, we require a read action on these stacks right before these actions. We will see that the (recurrent) reachability problem is decidable for such asynchronous pushdown automata in polynomial time. We can also semi-decide the reachability problem of our queue automata by exploration of the configration space. To this end, we can join multiple consecutive transitions to so-called meta-transformations and simulate them at once. Here, we study meta-transformations alternating between writing words from a given regular language into the queues and reading words from another regular language from the queues. We will see that such metatransformations can be applied in polynomial time. To show this result we first study some algebraic properties of our stacks and queues.Ein wichtiges Forschungsthema in der Informatik ist die Verifikation, d.h., die Analyse von Systemen bezüglich ihrer Korrektheit. Diese Analyse erfolgt in zwei Schritten: Zuerst müssen wir das System und die gewünschten Eigenschaften formalisieren. Anschließend benötigen wir Algorithmen zum Testen, ob das System die Eigenschaften erfüllt. Oftmals können wir das Systemals endlichen Automaten mit geeignetem Speichermechanismus modellieren, z.B. rekursive Programme sind im Wesentlichen Automaten mit einem Stack. Hier betrachten wir Automaten mit zwei Varianten von Stacks und Queues: 1. Partiell vergessliche Stacks und Queues, welche bestimmte Teile ihrer Inhalte jederzeit vergessen können. Diese können für unzuverlässige Systeme verwendet werden. 2. Verteilte Stacks und Queues, d.h., mehrere Stacks und Queues mit vordefinierter Synchronisierung. Häufig lassen sich die Eigenschaften unserer Modelle mithilfe des (wiederholten) Erreichbarkeitsproblems in unseren Automaten lösen. Dabei ist bekannt, dass die Entscheidbarkeit dieser Probleme oftmals stark vom konkreten Datentyp des Speichers abhängt. Beide Probleme können für Automaten mit einem Stack in Polynomialzeit gelöst werden. Sie sind jedoch unentscheidbar, wenn wir Automaten mit einer Queue oder zwei Stacks betrachten. In bestimmten Spezialfällen sind aber dennoch in der Lage diese Systeme zu verifizieren. So können wir beispielsweise bestimmte Automaten mit mehreren Stacks betrachten - so genannte Asynchrone Kellerautomaten. Diese bestehen aus mehreren (lokalen) Automaten mit jeweils einem Stack. Wann immer diese Automaten etwas in mind. einen Stack schreiben, müssen sie unmittelbar zuvor von diesen Stacks etwas lesen. Das (wiederholte) Erreichbarkeitsproblem ist in asynchronen Kellerautomaten in Polynomialzeit entscheidbar. Wir können zudem das Erreichbarkeitsproblem von Queueautomaten durch Exploration des Konfigurationsraums semi-entscheiden. Hierzu können wir mehrere aufeinanderfolgende Transitionen zu so genannten Meta-Transformationen zusammenfassen und diese in einem Schritt simulieren. Hier betrachten wir Meta-Transformationen, die zwischen dem Lesen und Schreiben von Wörtern aus zwei gegebenen regulären Sprachen alternieren. Diese Meta-Transformationen können in Polynomialzeit ausgeführt werden. Für dieses Ergebnis müssen wir jedoch zunächst verschiedene algebraische Eigenschaften der Queues betrachten