Article

# Elements of Automata Theory

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

## Abstract

Automata theory lies at the foundation of computer science, and is vital to a theoretical understanding of how computers work and what constitutes formal methods. This treatise gives a rigorous account of the topic and illuminates its real meaning by looking at the subject in a variety of ways. The first part of the book is organised around notions of rationality and recognisability. The second part deals with relations between words realised by finite automata, which not only exemplifies the automata theory but also illustrates the variety of its methods and its fields of application. Many exercises are included, ranging from those that test the reader, to those that are technical results, to those that extend ideas presented in the text. Solutions or answers to many of these are included in the book.

## No full-text available

... Dans ce cas, la théorie des automates finis [63] ainsi que la théorie des monoïdes finis [51] nous fournissent des outils pour caractériser nos langages : il s'agit de la notion de reconnaissance de langages. Un language L ⊆ A * est reconnu par un monoïde Nerode a prouvé qu'un langage est régulier si, et seulement si, son monoïde syntactique est fini [48]. ...
... In that situation, we find tools to characterise our languages in finite automata theory [63] and finite monoid theory [51]: this is the notion of language recognition. A language L ⊆ A * is recognized by a monoid M if there exists P ⊆ M and a monoid morphism h : A * → M such that h −1 (P ) = L. ...
... We then explain how it is possible to link it to duality theory from Section 1.1 and finally provide a few examples of this connection. For a more thorough introduction to automata theory, we recommend [42] and [63], and for the algebraic treatment of language recognition we refer to [71] and [54]. ...
Thesis
This thesis fits in the area of research that investigates the application of topological duality methods to problems that appear in theoretical computer science. One of the eventual goals of this approach is to derive results in computational complexity theory by studying appropriate topological objects which characterise them. The link which relates these two seemingly separated fields is logic, more precisely a subdomain of finite model theory known as logic on words. It allows for a description of complexity classes as certain families of languages, possibly non-regular, on a finite alphabet. Very few is known about the duality theory relative to fragments of first-order logic on words which lie outside of the scope of regular languages. The contribution of our work is a detailed study of such a fragment. Fixing an integer k ≥ 1, we consider the Boolean algebra BΣ1[Nu_k ]. It corresponds to the fragment of logic on words consisting in Boolean combinations of sentences defined by using a block of at most k existential quantifiers, letter predicates and uniform numerical predicates of arity l ∈ {1, ..., k}. We give a detailed study of the dual space of this Boolean algebra, for any k ≥ 1, and provide several characterisations of its points. In the particular case where k = 1, we are able to construct a family of ultrafilter equations which characterise the Boolean algebra BΣ1[Nu_1 ].
... In the theory of formal languages, one of standard techniques is to investigate series in non-commuting variables [78]. In particular, it is known [3] that the so called regular languages, i.e. the languages recognized by finite state automata [77], give rise to non-commutative rational series. In this section, we consider the Padé approximation to the characteristic series of the so called Fibonacci language recovering its rational function representation. ...
... where 1 represents the empty word, can be read-out from the corresponding deterministic finite state automaton [3,77] visualized on Fig. 1. ...
... Their ratio agrees with the series F (t) due to the formula (for more details on topology in the space of formal series, see [77]) ...
Article
Full-text available
We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev–Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.
... The only construction on automata that does (without destroying their structure) is the circulation of labels involved for instance in the synchronisation of transducers (cf. [10,11]) or as a preliminary for the minimisation of sequential transducers (cf. [10,12]). ...
... [10,11]) or as a preliminary for the minimisation of sequential transducers (cf. [10,12]). ...
... We essentially follow the definitions and notation of [10]. The model of weighted automaton used in this paper is more restricted though, for both theoretical and computational efficiency. ...
Article
This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms — which generalises and unifies the notion of bisimulation to the whole class of weighted automata — and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm.
... Semilinear sets. We summarize a few results from the theory of semilinear sets, and refer to [Sak09,Chapter II.7.3] and [DIV12a] for more details. ...
... By Dickson's Lemma (see [Sak09,Lemma II.7.1]), the set L has finitely many minimal elements a 1 , . . . , a k with respect to the partial order on N d , and it follows that L = k i=1 a i + L 0 . ...
... Proof. By[Sak09, Proposition II.7.15] a subset of N d is semilinear if and only if it is rational. By[Sak09, Theorem II.7.3], the rational subsets of N d form a boolean algebra.One can show that every semilinear set is a finite union of simple linear sets. ...
Preprint
Full-text available
A multivariate, formal power series over a field $K$ is a B\'ezivin series if all its coefficient can be expressed as a sum of at most $r$ elements from a finitely generated subgroup $G \le K^*$; it is a P\'olya series if one can take $r=1$. We give explicit structural descriptions of $D$-finite B\'ezivin series and $D$-finite P\'olya series over fields of characteristic $0$. A $D$-finite series is finitary if its coefficients are contained in a finitely generated subalgebra of $K$. The class of finitary $D$-finite series contains all algebraic series and their diagonals. We obtain a characterization of multivariate finitary $D$-finite series whose Hadamard sub-inverse is also finitary $D$-finite. The proofs use unit equations and results on semilinear sets.
... We collect them in the following proposition (cf. for instance [KN01,Sak09a,Sip13]). 2) Let A 1 = (Q 1 , A, I 1 , ∆ 1 , F 1 ) and A 2 = (Q 2 , A, I 2 , ∆ 2 , F 2 ) be two NFA's over A. Then, the intersection L(A 1 ) ∩ L(A 2 ) is accepted by the NFA A = (Q 1 × Q 2 , A, I 1 × I 2 , ∆, F 1 × F 2 ) where ∆ = {((q 1 , q 2 ), a, (q 1 , q 2 )) | (q 1 , a, q 1 ) ∈ ∆ 1 , (q 2 , a, q 2 ) ∈ ∆ 2 }. ...
... Trivially we get L(A 1 ) = {a 1 }, L(A 2 ) = {a 2 }, and L(A 3 ) = {a 3 }, hence we can consider the normalized DFA's A 1 , A 2 , and A 3 (cf. for instance [KN01,Sak09a,Sip13]) as the ones corresponding to EPIL formulas #(p r (1) ∧ p d (1)), #(p r (1) ∧ p d (2)), and #(p r (1) ∧ p d (3)), respectively. Then, we construct the DFA A ψ,r , depicted in Figure 8, which corresponds to FOEIL sentence ψ w.r.t. ...
... We collect them in the following proposition (cf. for instance [DKV09,Sak09a]). ...
Article
One of the key aspects in component-based design is specifying the software architecture that characterizes the topology and the permissible interactions of the components of a system. To achieve well-founded design there is need to address both the qualitative and non-functional aspects of architectures. In this paper we study the qualitative and quantitative formal modelling of architectures applied on parametric component-based systems, that consist of an unknown number of instances of each component. Specifically, we introduce an extended propositional interaction logic and investigate its first-order level which serves as a formal language for the interactions of parametric systems. Our logics achieve to encode the execution order of interactions, which is a main feature in several important architectures, as well as to model recursive interactions. Moreover, we prove the decidability of equivalence, satisfiability, and validity of first-order extended interaction logic formulas, and provide several examples of formulas describing well-known architectures. We show the robustness of our theory by effectively extending our results for parametric weighted architectures. For this, we study the weighted counterparts of our logics over a commutative semiring, and we apply them for modelling the quantitative aspects of concrete architectures. Finally, we prove that the equivalence problem of weighted first-order extended interaction logic formulas is decidable in a large class of semirings, namely the class (of subsemirings) of skew fields.
... The relation realized by the transducer is the set of pairs (x, y) where x and y are the input and output labels of an accepting run. It is a classical result that if the relation realized by some transducer is a function, it is also realized by a real-time transducer which is easily obtained from the first one [16,Prop. 1.1,p. ...
... We now introduce weighted automata. In this paper we only consider weighted automata whose weights are rational numbers with the usual addition and multiplication (see [16,Chap. III] for a complete introduction). ...
Preprint
Full-text available
We consider finite state non-deterministic but unambiguous transducers with infinite inputs and infinite outputs, and we consider the property of Borel normality of sequences of symbols. When these transducers are strongly connected, and when the input is a Borel normal sequence, the output is a sequence in which every block has a frequency given by a weighted automaton over the rationals. We provide an algorithm that decides in cubic time whether an unambiguous transducer preserves normality.
... In the literature trimming methods are known for finite-state automata [Sak09, Prop. 1 [Sak09,p. 408], and wta over strong bimonoids [DFKV22,Thm. ...
... A (Σ, B)-wta A is called trim if each of its states is useful; in a similar way, we define weak-trim and local-trim by employing weakly useful and local-useful, respectively. We note that the notions of trim in [Sak09] and in [DFKV22] correspond to the notions weak-trim and local-trim, respectively. In the next observation we formally compare the different trim properties (cf. Figure 7.1). ...
Preprint
This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata.
... Semilinear sets. We summarize a few results from the theory of semilinear sets, and refer to [Sak09,Chapter II.7.3] and [DIV12a] for more details. ...
... By Dickson's Lemma (see [Sak09,Lemma II.7.1]), the set L has finitely many minimal elements a 1 , . . . , a k with respect to the partial order on N d , and it follows that L = k i=1 a i + L 0 . ...
Article
Full-text available
A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup $G \le K^*$ ; it is a Pólya series if one can take $r=1$ . We give explicit structural descriptions of D -finite Bézivin series and D -finite Pólya series over fields of characteristic $0$ , thus extending classical results of Pólya and Bézivin to the multivariate setting.
... Leiss inductive automaton construction [49] leads to the same automaton [30]. This automaton is also called standard as it has a unique initial state which is non-returning [70,71]. Below we will see its connection with the partial derivative automaton, A PD . ...
... Derivatives of weighted rational expressions, that represent formal power series with coefficients in a semiring, have also been extensively studied [50,29,70,71]. In this case, derivatives are also connected with the notion of quotient of a series. ...
Chapter
The notions of derivative and partial derivative of regular expressions revealed themselves to be very powerful and have been successfully extended to many other formal language classes and algebraic structures. Although the undisputed elegance of this formalism, its efficient practical use is still a challenging research topic. Here we give a brief historical overview and summarise some of these aspects.
... To describe our objects in the sequel, we use regular languages and regular expressions [Sak09]. Recall that if A is an alphabet and L and L are two languages of words on A, then L.L is the language of the words of the form uu where u ∈ L and u ∈ L . ...
... A longest saturated chain between the minimal element and the maximal element of L is a maximal saturated chain. Let us describe the set of join-irreducible and meet-irreducible elements of Tr(n) by using the regular expression notation [Sak09] recalled in Section 2.1. ...
... Among them, the most common are Petri nets, process algebras, and finite automata [15]. Generally, automata-based specification consists in connecting some states by means of transitions [16]. Formally, states are valuations of internal variables of a system. ...
... First, the server view follows. The notation is intuitive: server types are defined (lines 2,9,16). The formal parameters specify the agents and other servers used. ...
Article
Full-text available
Automated verification of distributed systems becomes very important in distributed computing. The graphical insight into the system in the early and late stages of the project is essential. In the design phase, the visual input helps to articulate the collaborative distributed components clearly. The formal verification gives evidence of correctness or malfunction, but in the latter case, graphical simulation of counterexample helps for better understanding design errors. For these purposes, we invented Distributed Autonomous and Asynchronous Automata (DA3), which have the same semantics as the formal verification base – Integrated Model of Distributed Systems (IMDS). The IMDS model reflects the natural characteristics of distributed systems: unicasting, locality, autonomy, and asynchrony. Distributed automata have all of these features because they share the same semantics as IMDS. In formalism, the unified system definition has two views: the server view of the cooperating distributed nodes and the agent view of the migrating agents performing distributed computations. The automata have two formally equivalent forms that reflect two views: Server DA3 for observing servers exchanging messages, and Agent DA3 for tracking agents, which visit individual servers in their progress of distributed calculations. We present the DA3 formulation based on the IMDS formalism and their application to design and verify distributed systems in the Dedan environment. DA3 formalism is compared with other concepts of distributed automata known from the literature.
... Essentially each wsa is a nondeterministic finite-state automaton in which each transition carries a weight (quantity). In order to calculate with weights, an algebraic structure is needed, called weight algebra, and wsa have been investigated over several different weight algebras: semirings [Sch61,Eil74,BR88,KS86,Sak09,DKV09], lattices [Wec78,Rah09], strong bimonoids [DSV10,CDIV10,DV12], valuation monoids [DGMM11,DM12], and multi-cost valuation structures [DP16]. The two operations of these weight algebras, usually called addition and multiplication, are used to calculate the weight of a run on a given input word (by means of multiplication) and to sum up the weights of several runs on the given word (by means of addition). ...
... In this way, a wsa A recognizes a weighted language [[A]] (or: formal power series), i.e., a mapping from the set of input words to the carrier set of the weight algebra. For the theory of wsa we refer to [Sch61,Eil74,SS78,Wec78,BR82,KS86,Kui97,Sak09,DKV09]. ...
Preprint
We consider weighted tree automata over strong bimonoids (for short: wta). A wta $\mathcal{A}$ has the finite-image property if its recognized weighted tree language $[\![\mathcal{A}]\!]$ has finite image; moreover, $\mathcal{A}$ has the preimage property if the preimage under $[\![\mathcal{A}]\!]$ of each element of the underlying strong bimonoid is a recognizable tree language. For each wta $\mathcal{A}$ over a past-finite monotonic strong bimonoid we prove the following results. In terms of $\mathcal{A}$'s structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta $\mathcal{A}$ it is decidable whether $\mathcal{A}$ has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that $\mathcal{A}$ has the preimage property. All our results also hold for weighted string automata.
... This is the reason we went to process view of a model. Finite State Machine (Finite State Automaton) is a notable formalism in the automata theory to represent all the states and the transitions between its states [19]. This formal method generally less powerful in complex and concurrent systems. ...
Article
Full-text available
Over the years, the number of applications supporting enterprise business processes has increased. The challenge of integrating diverse systems is one of the many reasons why many organizations fail to achieve greater automation. To overcome this obstacle, they are turning to Enterprise Application Integration (EAI). Enterprise Application Integration is a process that enables the integration of different applications. This allows the users to easily modify the functionality, share the information among the various applications and reuse the methods. The paper presents a formal method that includes the various levels of EAI. It highlights the various formal methods that can be used to achieve EAIs seamless interoperation. It also supports the concurrent and dynamic system. This paper also proposes a new architecture for EAI that will help them achieve their goals. There are many formal methods for programming languages in software engineering, but most of them are not adequate for the development of complex systems. The author proposes a new methodology based on Petri net which is a graphical representation of semantics.
... This is the reason we went to process view of a model. Finite State Machine (Finite State Automaton) is a notable formalism in the automata theory to represent all the states and the transitions between its states [19]. This formal method generally less powerful in complex and concurrent systems. ...
Article
Over the years, the number of applications supporting enterprise business processes has increased. The challenge of integrating diverse systems is one of the many reasons why many organizations fail to achieve greater automation. To overcome this obstacle, they are turning to Enterprise Application Integration (EAI). Enterprise Application Integration is a process that enables the integration of different applications. This allows the users to easily modify the functionality, share the information among the various applications and reuse the methods. The paper presents a formal method that includes the various levels of EAI. It highlights the various formal methods that can be used to achieve EAI's seamless interoperation. It also supports concurrent and dynamic systems. This paper also proposes a new architecture for EAI that will help them achieve their goals. There are many formal methods for programming languages in software engineering, but most of them are not adequate for the development of complex systems. The author proposes a new methodology based on Petri net which is a graphical representation of semantics.
... Runtime enforcement (RE) [LBW05,FFM08] is a dynamic verification technique that uses monitors to analyse the runtime behaviour of a system-under-scrutiny (SuS) and transform it in order to conform to some correctness specification. The earliest work in RE [LBW05,LBW09,Sak09,BM11a,KT12] models the behaviour of the SuS as a trace of abstract actions (e.g., α, β, . . . ∈ Act). ...
Article
Full-text available
Runtime enforcement is a dynamic analysis technique that instruments a monitor with a system in order to ensure its correctness as specified by some property. This paper explores bidirectional enforcement strategies for properties describing the input and output behaviour of a system. We develop an operational framework for bidirectional enforcement and use it to study the enforceability of the safety fragment of Hennessy-Milner logic with recursion (sHML). We provide an automated synthesis function that generates correct monitors from sHML formulas, and show that this logic is enforceable via a specific type of bidirectional enforcement monitors called action disabling monitors.
... The classical notion of weighted automaton over a semiring is presented for instance in the reference book [45,Chapter III]. We consider only the case where the weights are in N, and follow the notations of [15, §5.2]. ...
Preprint
Full-text available
Polyregular functions are the class of string-to-string functions definable by pebble transducers (an extension of finite automata) or equivalently by MSO interpretations (a logical formalism). Their output length is bounded by a polynomial in the input length: a function computed by a $k$-pebble transducer or by a $k$-dimensional MSO interpretation has growth rate $O(n^k)$. Boja\'nczyk has recently shown that the converse holds for MSO interpretations, but not for pebble transducers. We give significantly simplified proofs of those two results, extending the former to first-order interpretations by reduction to an elementary property of $\mathbb{N}$-weighted automata. For any $k$, we also prove the stronger statement that there is some quadratic polyregular function whose output language differs from that of any $k$-fold composition of macro tree transducers (and which therefore cannot be computed by any $k$-pebble transducer). In the special case of unary input alphabets, we show that $k$ pebbles suffice to compute polyregular functions of growth $O(n^k)$. This is obtained as a corollary of a basis of simple word sequences whose ultimately periodic combinations generate all polyregular functions with unary input. Finally, we study polyregular and polyblind functions between unary alphabets (i.e. integer sequences), as well as their first-order subclasses.
... This NFA has to be converted to a DFA. However, the resulting DFA from the NFA have been shown to be equivalent [23]. Thompson's construction [25] builds the NFA with a bottom-up approach. ...
Preprint
Due to the large amount of daily scientific publications, it is impossible to manually review each one. Therefore, an automatic extraction of key information is desirable. In this paper, we examine STEREO, a tool for extracting statistics from scientific papers using regular expressions. By adapting an existing regular expression inclusion algorithm for our use case, we decrease the number of regular expressions used in STEREO by about $33.8\%$. We reveal common patterns from the condensed rule set that can be used for the creation of new rules. We also apply STEREO, which was previously trained in the life-sciences and medical domain, to a new scientific domain, namely Human-Computer-Interaction (HCI), and re-evaluate it. According to our research, statistics in the HCI domain are similar to those in the medical domain, although a higher percentage of APA-conform statistics were found in the HCI domain. Additionally, we compare extraction on PDF and LaTeX source files, finding LaTeX to be more reliable for extraction.
... Transdutores vêm sendo estudados desde os primórdios da Teoria dos Autômatos e têm várias aplicações por exemplo em processamento de linguagem natural e Biologia Computacional [Mihov and Schulz 2019]. Consulte [Sakarovitch 2009] e [Muscholl and Puppis 2019] para definições desse e de outros conceitos discutidos neste texto, bem como um panorama de resultados fundamentais. ...
Conference Paper
A possibilidade de se decompor um transdutor bidirecional k-valorado T em k transdutores bidirecionais funcionais é um problema em aberto. Consideramos dois casos particulares desse problema: aquele em que T é k-ambíguo na entrada, e aquele em que imagens distintas de uma mesma palavra pela relação realizada por T têm comprimentos distintos. Discutimos como essas decomposições podem ser obtidas utilizando construções que apresentamos para resolver outros problemas: decompor um transdutor unidirecional k-valorado, e construir uma uniformização de um transdutor bidirecional.
... In der Welt der I/O-TSe entspricht dieser Systemkopplung die Kopplung über identische I/O-Zeichen: Die Ausgabezeichen des "sendenden" TS sind die Eingabezeichen des "empfangenden" TS. Die Transitionsrelation des Produkt-TS der interagierenden TSe 1 Diese I/O-TS werden in der Literatur auch Transducer und im deterministischen Fall auch Mealy-Automat genannt [26]. ergibt sich als entsprechende systematische Restriktion der unbeschränkten Produkt-Transitionsrelation gemäß der Anforderung, dass eingehende Zeichen zu verarbeiten sind (s. ...
Article
Full-text available
Zusammenfassung Dieser Artikel ist der erste einer Viererreihe, die eine Diskussion über die Grundlagen der Informatik anregen soll. Dabei steht das Begriffsnetzwerk, das den Kern der Informatik ausmacht, im Zentrum. Dieses Begriffsnetzwerk steht in dem Spannungsfeld, zum einen weitgehend in unserem alltagssprachlichen Verständnis verankert zu sein, aber zum anderen mittels des spezifischen informatischen Ansatzes, sich auf das Unterscheidbare zu fokussieren, auch auf genuin informatischen Konzepten aufzubauen. In diesem ersten Teil wird die Informatik als eine Theorie der Interaktion vorgestellt und als ihre zentralen Begriffe werden die Zustandsfunktion, das System, die Unterscheidbarkeit und damit die Information, die Berechenbarkeit sowie eine ganze Reihe mathematischer Konzepte eingeführt, allen voran das der Relation, der Funktion, der mathematischen Struktur, der Äquivalenz und der Komposition. Der grundlegende Bezug zum Informationskonzept als Abstraktion vom Ununterscheidbaren macht die Informatik zu einer Strukturwissenschaft und bindet damit die Klarheit ihrer Begrifflichkeit letztlich an deren Bezug zur Mathematik. Drei wichtige Konsequenzen werden aufgeführt. Erstens gibt es keine zwei verschiedenen Welten der Informatik und der Physik, die auf geheimnisvolle Weise „wechselwirken“. Sondern: Es gibt nur eine Welt, die wir unterschiedlich beschreiben. Zweitens lässt sich mit Klärung der Frage, was aus Sicht der Informatik eigentlich zwischen Systemen, die über Signale miteinander kommunizieren, „transportiert“ wird – nämlich die Information – auch feststellen, was insbesondere nicht transportiert wird – nämlich irgendeine Bedeutung. Und drittens ergibt sich ein gewisser „virtueller“ Charakter rein informatisch beschreibbarer Systeme, der dort endet, wo die Abstraktion, dass die Struktur von Systemen in der Interaktion invariant bleibt und es allein auf die Unterscheidbarkeit ankommt, ihre Gültigkeit verliert.
... For automata with weights in a semiring K, this is no longer true. More generally, a K-automaton is unambiguous if (i) between each two states p and q and for every word w there is at most one path from p to q labeled by w, and (ii) every word has at most one accepting path [ 35 ,Definition I.1.11]. For trim automata (i) and (ii) are equivalent and one may be omitted. ...
Preprint
Full-text available
The (left) linear hull of a weighted automaton over a field is a topological invariant. If the automaton is minimal, the linear hull can be used to determine whether or not the automaton is equivalent to a sequential (deterministic) automaton, respectively, an unambiguous automaton. We show how to compute the linear hull, and thus prove that it is decidable whether or not a given automaton over a (finitely generated) field is equivalent to a sequential [unambiguous] one. In these cases we are also able to compute an equivalent sequential [unambiguous] automaton. The results extend to some commutative domains, in particular PIDs. This resolves a problem posed in a 2006 survey by Lombardy and Sakarovitch.
... A binary relation τ ⊆ M × N is regular (or equivalently, rational) if, and only if, it belongs to the regular closure of the finite subsets of M ×N . Equivalently there is some finite M ×N -automaton (or equivalently, transducer), say R, with behavior |R| = τ [7,24]. The family of regular relations is closed under inverse and composition. ...
Preprint
Given a finite alphabet $A$ and a binary relation $\tau\subseteq A^*\times A^*$, a set $X$ is $\tau$-{\it independent} if $\tau(X)\cap X=\emptyset$. Given a quasi-metric $d$ over $A^*$ (in the meaning of \cite{W31}) and $k\ge 1$, we associate the relation $\tau_{d,k}$ defined by $(x,y)\in\tau_{d,k}$ if, and only if, $d(x,y)\le k$ \cite{CP02}.In the spirit of \cite{JK97,N21}, the error detection-correction capability of variable-length codes can be expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and every quasi-metric associated to (anti-)automorphisms of the free monoid, we examine whether those conditions are decidable for a given regular code.
... A language over a finite alphabet A is said to be rational if it can be obtained from the (finitely many) letters in A using the operators of union, product and star (i.e., submonoid generated by a language) finitely many times. A direct consequence of a fundamental theorem of Kleene (see for example [Sak09]) is that the set of rational A-languages is closed under finite intersection and complement. ...
Article
Full-text available
This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.
... A sequential transducer T defined in [14] is a septuple ...
Preprint
Full-text available
With the two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. In this work, we consider a Fibonacci-equivalent of the two's complement notation. A theorem of Zeckendorf says that every nonnegative integer can be represented as the sum of distinct nonconsecutive Fibonacci numbers thus leading to a binary representation of nonnegative integers where powers of 2 are replaced by Fibonacci numbers in its evaluation. A transducer provided by Berstel computes the sum of the Zeckendorf binary representation of two nonnegative integers. In this work, we consider a numeration system also based on Fibonacci numbers but representing all integers. As for the two's complement notation, we show that addition of integers represented in this numeration system can be computed with the Berstel transducer with three additional transitions. Whether this can be done more generally is an open question raised by the current work.
... Diese I/O-TS werden in der Literatur auch Transducer und im deterministischen Fall auchMealy-Automat genannt[46]. ...
Preprint
Full-text available
Dieser Artikel ist als Beitrag zu der Diskussion gedacht, das Begriffsnetzwerk, das den Kern der Informatik ausmacht, zu bestimmen. Dieses Begriffsnetzwerk steht in dem Spannungsfeld zum einen weitgehend in unserem alltagssprachlichen Verständnis verankert zu sein, aber zum anderen mittels des spezifischen informatischen Ansatz, sich auf das Unterscheidbare zu fokussieren, auch auf genuin informatischen Konzepten aufzubauen. Diese wechselseitige Beziehung birgt einerseits die Chance, einen entscheidenden Beitrag zu einem differenzierteren Verständnis unserer Welt zu leisten aber andererseits das Risiko unsere Alltagswelt mit falsch verstandenen informatischen Konzepten unnötigerweise erheblich zu verwirren. Für beides gibt es mittlerweile gute Beispiele. Die zentralen Begriffe sind meiner Meinung nach die der Zustandsfunktion, des Systems, der Unterscheidbarkeit und damit der Information, der Berechenbarkeit sowie eine ganze Reihe mathematischer Konzepte, allen voran das der Relation, der Funktion, der mathematischen Struktur und der Komposition. Der grundlegende Bezug zum Informationskonzept als Abstraktion vom Ununterscheidbaren macht die Informatik zu einer Strukturwissenschaft und bindet damit die Klarheit ihrer Begrifflichkeit letztlich an deren Bezug zur Mathematik. Was scheint verwirrend in der Informatik und was klärt diese Arbeit auf? Einerseits basiert alles in der Informatik auf dem Austausch von Informationen -- andererseits scheint dieser aber bei der Beschreibung von Operationen, wie sie in imperativen Programmiersprachen verwendet werden, gar keine Rolle zu spielen. Einerseits ist Informatik geprägt durch das Konzept der Berechenbarkeit, in dem Determinismus herrscht und in dem Zustand nur passager notwendig ist -- andererseits sind die Interaktionen zwischen komplexen Systemen ''auf gleicher Augenhöhe'' in der Regel nichtdeterministisch und zustandsbehaftet. Einerseits komponieren Systeme durch Interaktion zu Supersystemen -- andererseits auch wieder nicht. Einerseits scheint in der Informatik ein Modell etwas zu beschreiben -- andererseits scheint in der Informatik ein Modell etwas Beschriebenes. Einerseits scheint das Konzept der Semantik im Bereich der Berechenbarkeit überflüssig -- andererseits scheint die Informatik zu der Aufklärung von Semantik einen wesentlichen Beitrag liefern zu können. Neben der Aufklärung dieser und weiterer Verwirrungen ist es mir wichtig, das Bild zu vermitteln, dass eine wohlverstandene Informatik tatsächlich menschlicher wird. Mit ihr können wir aufzeigen, dass eben nicht die Berechenbarkeit die Welt mit ihrem Determinismus dominiert, sondern wir permanent große Vorteile aus ihrer Unberechenbarkeit, aus ihrer Nichtdeterminiertheit ziehen. Ganz konkret lässt die enge Verbindung des von ihr erarbeiteten Protokollkonzepts mit dem des Spiels den Entscheidungsbegriff näher bestimmen und darauf aufbauend ein Konzept der alltagssprachlichen Semantik im Sinne einer Interaktionssemantik entwickeln. Damit leistet eine solche Informatik einen essentiell wichtigen Beitrag für unser Verständnis solch wichtiger alltagssprachlicher Begriffe wie der unserer persönlichen Freiheit -- was vor den aktuellen gesellschaftlichen, wesentlich durch die Informatik möglich gemachten Entwicklungen der Megainteraktionsnetzwerke, auch zu erwarten wäre und meiner Ansicht nach auch dringend erforderlich ist.
... Note that σ is a so-called sequential function [12,59], not a morphism. We observe however that, if χ : B * → S is the morphism given by χ(t, a) = t · s a , then ϕ(a 1 · · · a n ) = (χσ(a 1 · · · a n ), ψ(a 1 · · · a n )) . ...
Book
... In this section, we introduce briefly the notion of finite automata. For further details on formal aspects of finite automata theory, we particularly recommend reading classical books [14,15]. ...
Article
Full-text available
This paper describes a fast algorithm for constructing directly the equation automaton from the well-known Thompson automaton associated with a regular expression. Allauzen and Mohri have presented a unified construction of small automata and gave a construction of the equation automaton with time and space complexity in O(mlogm+m2), where m denotes the number of Thompson automaton transitions. It is based on two classical automata operations, namely epsilon-removal and Hopcroft’s algorithm for deterministic Finite Automata (DFA) minimization. Using the notion of c-continuation, Ziadi et al. presented a fast computation of the equation automaton in O(m2) time complexity. In this paper, we design an output-sensitive algorithm combining advantages of the previous algorithms and show that its computational complexity can be reduced to O(m×|Q≡e|), where |Q≡e| denotes the number of states of the equation automaton, by an epsilon-removal and Bubenzer minimization algorithm of an Acyclic Deterministic Finite Automata (ADFA).
... The Schützenberger covering in particular has already been employed in a number of decidability results for max-plus automata [10,25,26,34,42]. For more background on the Schützenberger covering and coverings in general, see [46]. ...
Article
Full-text available
We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.
... Before discussing how a dSRA can be described by a Markov model, we first discuss a useful result, which bears on the importance of being able to use deterministic automata. It can be shown that a dSRA always has an equivalent deterministic classical automaton, through a simple isomorphic mapping, retaining the exact same structure for the automaton and simply changing the conditions on the transitions with symbols [50]. This result is important for two reasons: a) it allows us to use methods developed for classical automata without having to always prove that they are indeed applicable to symbolic automata as well, and b) it will help us in simplifying our notation, since we can use the standard notation of symbols instead of predicates. ...
Preprint
We propose an automaton model which is a combination of symbolic and register automata, i.e., we enrich symbolic automata with memory. We call such automata Symbolic Register Automata (SRA). SRA extend the expressive power of symbolic automata, by allowing Boolean formulas to be applied not only to the last element read from the input string, but to multiple elements, stored in their registers. SRA also extend register automata, by allowing arbitrary Boolean formulas, besides equality predicates. We study the closure properties of SRA under union, intersection, concatenation, Kleene closure, complement and determinization and show that SRA, contrary to symbolic automata, are not in general closed under complement and they are not determinizable. However, they are closed under these operations when a window operator, quintessential in Complex Event Recognition, is used. We show how SRA can be used in Complex Event Recognition in order to detect patterns upon streams of events, using our framework that provides declarative and compositional semantics, and that allows for a systematic treatment of such automata. We also show how the behavior of SRA, as they consume streams of events, can be given a probabilistic description with the help of prediction suffix trees. This allows us to go one step beyond Complex Event Recognition to Complex Event Forecasting, where, besides detecting complex patterns, we can also efficiently forecast their occurrence.
... A language over a finite alphabet A is said to be rational if it can be obtained from the (finitely many) letters in A using the operators of union, product and star (i.e., submonoid generated by a language) finitely many times. A direct consequence of a fundamental theorem of Kleene (see for example [Sak09]) is that the set of rational A-languages is closed under finite intersection and complement. ...
Preprint
Full-text available
This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.
... [Eil74,Ch. VI.6] and [SS78,KS86,Sak09,DKV09]). More precisely, there exists a wsa such that there is no equivalent deterministic wsa (see, e.g., [BV03,Lm. ...
Article
We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.
Preprint
Full-text available
We give a new characterization of the class of rational string functions from formal language theory using order-preserving interpretations with respect to a very weak monadic programming language. This refines the known characterization of rational functions by order-preserving MSO interpretations.
Article
We consider automata in which transitions are labelled with arbitrary permutations. The language of such an automaton consists of compositions of permutations for all possible admissible computation paths. The membership problem for finite automata over symmetric groups is the following decision problem: does a given permutation belong to the language of a given automaton? We show that this problem is NP-complete. We also propose an efficient algorithm for the case of strongly connected automata.
Article
Several notions of synchronisation in concurrent systems can be modelled by regular shuffle operators. In this paper we consider regular expressions extended with three operators corresponding respectively to strong, arbitrary, and weak synchronisation. For these expressions, we define a location based position automaton. Furthermore, we show that the partial derivative automaton is still a quotient of the position automaton.
Article
We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid F. It is also decidable whether or not a rational subset of F is recognizable. We prove that a submonoid of F is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite J-above monoid is decidable, covering the case of free inverse monoids.
Article
Full-text available
Every univariate rational series over an algebraically closed field is shown to be realised by some polynomially ambiguous unary weighted automaton. Unary weighted automata over algebraically closed fields thus always admit polynomially ambiguous equivalents. On the other hand, it is shown that this property does not hold over any other field of characteristic zero, generalising a recent observation about unary weighted automata over the field of rational numbers.
Chapter
A deterministic finite automaton is called bideterministic if its transpose is deterministic as well. The study of such automata in a weighted setting is initiated. All trim bideterministic weighted automata over integral domains and positive semirings are proved to be minimal. On the contrary, it is observed that this property does not hold over finite commutative rings in general. Moreover, it is shown that the problem of determining whether a given rational series is realised by a bideterministic automaton is decidable over fields as well as over tropical semirings.
Chapter
Given a finite alphabet A and a binary relation τ⊆A∗×A∗, a set X is τ-independent if τ(X)∩X=∅. Given a quasi-metric d over A∗ (in the meaning of [27]) and k≥1, we associate the relation τd,k defined by (x,y)∈τd,k if, and only if, d(x,y)≤k [3]. In the spirit of [10, 20], the error detection-correction capability of variable-length codes can be expressed in term of conditions over τd,k. With respect to the prefix metric, the factor one, and every quasi-metric associated to (anti-)automorphisms of the free monoid, we examine whether those conditions are decidable for a given regular code.
Article
Full-text available
Rationally controlled one-rule insertion systems are one-rule string rewriting systems for which the rule, that is to insert a given word, may be applied in a word only behind a prefix that must belong to a given rational language called the control language. As for general string rewriting systems, these controlled insertion systems induce a transformation over languages: from a starting word, one can associate all its descendants. In this paper, we investigate the behavior of these systems in terms of preserving the classes of languages: finite, rational and context-free languages. We show that, even for very simple such systems, the images of finite or rational languages need not be context-free. In the case when the control language is in the form u * for some word u , we characterize one-rule insertion systems that induce a rational transduction and we prove that for these systems, the image of any context-free language is always context-free.
Article
We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.
Chapter
Infinite hierarchies of rational series realised by finitely ambiguous and finitely sequential weighted automata over fields, classifying them according to the ambiguity or sequentiality degree of realising automata, are examined. It is shown that both these hierarchies are strict if and only if the field under consideration is not locally finite; in that case, the hierarchies are strict already for series over a unary alphabet. Relations between finitely ambiguous and finitely sequential unary weighted automata are explored. It is also readily observed that polynomially ambiguous weighted automata over a field of characteristic zero are more powerful than finitely ambiguous weighted automata over the same field, again already over a unary alphabet. On the other hand, it is proved that unary alphabets are insufficient to separate the series realised by polynomially and finitely ambiguous weighted automata over algebraically closed fields of positive characteristic.
Article
A well-known construction of a weighted automaton over the integers, assigning zero to precisely the nonempty words from the equality set of a given Post's correspondence problem instance, is extended to the case of weights taken from an arbitrary integral domain that is not locally finite. Undecidability of problems for rational series such as universality or rationality of support thus generalises to such domains as well. In spite of being fairly simple, these findings imply that the support rationality problem can be undecidable over rings of positive characteristic, answering an open question of D. Kirsten and K. Quaas. In addition, a more general undecidability result for rational series over other than locally finite integral domains is proved, which subsumes, e.g., the undecidability of support rationality or context-freeness.
Chapter
We consider finite state non-deterministic but unambiguous transducers with infinite inputs and infinite outputs, and we consider the property of Borel normality of sequences of symbols. When these transducers are strongly connected, and when the input is a Borel normal sequence, the output is a sequence in which every block has a frequency given by a weighted automaton over the rationals. We provide an algorithm that decides in cubic time whether an unambiguous transducer preserves normality.KeywordsFunctional transducersWeighted automataNormal sequences
Article
We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language 〚A〛 has finite image; moreover, A has the preimage property if the preimage under 〚A〛 of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A's structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata.
Chapter
Full-text available
We explore the suitability of mod 2 multiplicity automata (M2MAs) as a representation for regular languages of infinite words. M2MAs are a deterministic representation that is known to be learnable in polynomial time with membership and equivalence queries, in contrast to many other representations. Another advantage of M2MAs compared to non-deterministic automata is that their equivalence can be decided in polynomial time and complementation incurs only an additive constant size increase. Because learning time is parameterized by the size of the representation, particular attention is focused on the relative succinctness of alternate representations, in particular, LTL formulas and Büchi automata of the types: deterministic, non-deterministic and strongly unambiguous. We supplement the theoretical results of worst case upper and lower bounds with experimental results computed for randomly generated automata and specific families of LTL formulas.
Article
We address the following decision problem. Given a numeration system [Formula: see text] and a [Formula: see text]-recognizable set [Formula: see text], i.e. the set of its greedy [Formula: see text]-representations is recognized by a finite automaton, decide whether or not [Formula: see text] is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on [Formula: see text]-adic methods, the DFA given as input provides a bound on the admissible periods to test.
Preprint
This article fits in the area of research that investigates the application of topological duality methods to problems that appear in theoretical computer science. One of the eventual goals of this approach is to derive results in computational complexity theory by studying appropriate topological objects which characterize them. The link which relates these two seemingly separated fields is logic, more precisely a subdomain of finite model theory known as logic on words. It allows for a description of complexity classes as certain families of languages, possibly non-regular, on a finite alphabet. Very few is known about the duality theory relative to fragments of first-order logic on words which lie outside of the scope of regular languages. The contribution of our work is a detailed study of such a fragment. Fixing an integer $k \geq 1$, we consider the Boolean algebra $\mathcal{B}\Sigma_1[\mathcal{N}^{u}_k]$. It corresponds to the fragment of logic on words consisting in Boolean combinations of sentences defined by using a block of at most $k$ existential quantifiers, letter predicates and uniform numerical predicates of arity $l \in \{1,...,k\}$. We give a detailed study of the dual space of this Boolean algebra, for any $k \geq 1$, and provide several characterizations of its points. In the particular case where $k=1$, we are able to construct a family of ultrafilter equations which characterize the Boolean algebra $\mathcal{B} \Sigma_1[\mathcal{N}^{u}_1]$. We use topological methods in order to prove that these equations are sound and complete with respect to the Boolean algebra we mentioned.
Article
Weighted finite automata over the field of rational numbers and unary alphabets are considered. The notion of a characteristic polynomial is introduced for such automata as a means to provide a decidable necessary and sufficient condition, under which a unary weighted automaton admits a deterministic, i.e., sequential equivalent. The sequentiality problem for univariate rational series is thus proved to be decidable both over the rational numbers and over the integers, confirming a conjecture of S. Lombardy and J. Sakarovitch; its decidability over the nonnegative integers is observed as well. The decision algorithm proposed for these tasks is shown to run in polynomial time. A determinisation algorithm for determinisable unary weighted automata over the rational numbers is also described.
Chapter
Regular model checking is an exploration technique for infinite state systems where state spaces are represented as regular languages and transition relations are expressed using rational relations over infinite (or finite) strings. We extend the regular model checking paradigm to permit the use of more powerful transition relations: the class of regular relations, of which the rational relations are a strict subset. We use the language of monadic second-order logic (MSO) on infinite strings to specify such relations and adopt streaming string transducers (SSTs) as a suitable computational model. We introduce nondeterministic SSTs over infinite strings ($$\omega$$-NSSTs) and show that they precisely capture the relations definable in MSO. We further explore theoretical properties of $$\omega$$-NSSTs required to effectively carry out regular model checking. In particular, we establish that the regular type checking problem for $$\omega$$-NSSTs is decidable in Pspace. Since the post-image of a regular language under a regular relation may not be regular (or even context-free), approaches that iteratively compute the image can not be effectively carried out in this setting. Instead, we utilize the fact that regular relations are closed under composition, which, together with our decidability result, provides a foundation for regular model checking with regular relations.
Article
Full-text available
Whether we analyze phonological processes using a system of rules or constraints, the resulting map from underlying representations to surface pronunciations can be characterized as a function. Viewing processes as mathematical objects in this way allows us to study properties of phonology that hold no matter how it is implemented. Work in this vein has found that a majority of phonological processes only consider information within a finite window, placing them in the highly restrictive class of Strictly Local (SL) functions (Chandlee 2014; Chandlee et al. 2014;2015). Long-distance phonological processes, however, lie outside the capabilities of the SL functions since they consider information that can be arbitrarily distant. The more powerful class of subsequential functions has been offered as a potential upper bound on the complexity of long-distance phonology (Heinz and Lai 2013; Luo 2017; Payne 2017), but we will argue that an even tighter bound is possible. Specifically, by incorporating an autosegmental tier (e.g., Goldsmith 1976) into the structure of an SL function, the non-local information crucial for applying long-distance processes can be rendered local. In addition to assessing the typological coverage of these Tier-based Strictly Local (TSL) functions (Burness and McMullin 2019; Hao and Andersson 2019; Hao and Bowers 2019), we show that they fail to generate a number of pathological patterns that can be characterized as subsequential functions. We therefore conclude that the TSL functions are a better hypothesized upper bound on phonological complexity.
Article
Full-text available
Article
Full-text available
We show in this article that the most usual finiteness conditions on a subgroup of a finitely generated group all have equivalent formulations in terms of formal language theory. This correspondence gives simple proofs of various theorems concerning intersections of subgroups and the preservation of finiteness conditions in a uniform manner. We then establish easily the theorems of Greibach and of Griffiths by considering free reductions of languages that describe the computations of pushdown automata in one case and of Turing machines in the other, thus making clear that they are essentially the same.
Article
Full-text available
Some results are given in the theory of rational power series over a broad class of semirings. In particular, it is shown that for unambiguous sets the notion of rationality is independent of the semiring over which representations are defined. The undecidability of the rationality of probabilistic word functions is also established.
Article
Full-text available
We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable.
Article
The language of an infinite word is the set of all finite prefixes. Several characterizations of the language of an infinite word when the language is regular are given. The regularity of every language of an infinite word belonging to a given set, where the least rational cone contains the commutative languages (those languages that are closed under permutation).
Article
La représentation matricielle minimale attachée à une série ratioñnelle en variables non commutatives permet de généraliser à ces séries des résultats connus en théorie des langages. Les méthodes employées permettent aussi de décrire les parties rationnelles de certains monoïdes comme le groupe libre.
Article
This paper is concerned with the problem of determining whether a set of sequences R\t', obtained by some given rule from a regular set of sequences R, is again a regular set. A number of such problems are solved in this paper and a basic technique is used which is easy to apply to problems of this type. This technique yields an explicit construction of the sequential machine which recognizes the sequences in R\t', if R\t' is a regular set. Among other things it is shown that: (i) the derivative of a regular set R with respect to any set of sequences W is a regular set, although a regular expression designating this set cannot in general be computed; (ii) the set of sequences obtained from a regular set R by removing “arbitrary halves” of the sequences in R and the set of these removed “halves” are both regular sets; (iii) the set of sequences, obtained from a regular set R by making no more than k changes in any m consecutive digits in sequences from R, is regular; (iv) the set of sequences which can be concatenated in one and only one way from sequences in R is regular.
Article
On applique la théorie de S. Eilenberg au moyen de la méthode des transducteurs de M. Nivat pour obtenir quelques indications sur la vitesse de croissance en fonction de la longueur du mot du nombre de mots de l'image d'une relation rationnelle.
Article
We shall provide a ‘simple’ algorithm allowing, with formal power series, to decide whether a rational subset of k, given by an ambiguous rational expression, is recognizable.
Article
This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations: Generators: . Defining relations: where k m = a m b m a m ⁻¹ b m ⁻¹ . We shall denote this group by F(p; n 1 , … , n d ; r) . In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.
Chapter
Let us recall the following elementary result in the theory of analytic functions in one variable.
Article
We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway's conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities.
Article
The decidability of the equivalence problem and the disjointness problem for regular trace languages is considered. By describing the structure of the independence relations involved, precise characterizations are given of those concurrency alphabets for which these problems are decidable. In fact, the first problem is decidable if and only if the independence relation is transitive, while the second problem is decidable if and only if the independence relation is a so-called transitive forest.
Article
We give an algorithm which computes the set of descendants of a regular set R, for Thue systems of certain type. The complexity of the algorithm is O(m3) where m is the number of states of an automaton recognising R. This allows to improve the known complexity bounds for some extended word problems defined by cancellation rules.
Article
Two properties of languages which are supports of rational power series are proved: (i) if two supports are complementary, then they are regular languages; (ii) the Ehrenfeucht conjecture is true for these languages.
Article
We show how a construction on matrix representations of two tape automata proposed by Schützenberger to prove that rational functions are unambiguous can be given a central rôle in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as the “Cross-Section Theorem”, its dual the theorem of rational uniformisation, or the decomposition theorem of rational functions into sequential functions, appear as direct and formal consequences of it.RésuméNous montrons comment une construction sur la représentation matricielle des automates à deux bandes proposée par Schützenberger pour prouver que toute fonction rationnelle est non ambiguë est en fait au cœur de la théorie des relations et fonctions réalisées par automates finis et permet d'établir naturellement les autres résultats fondamentaux de la théorie comme le “Cross-Section Theorem”, son dual, le théorème d'uniformisation rationnelle ou celui de décomposition des fonctions rationnelles en fonctions séquentielles.
Article
Let = {R1, …, Rm} be a finite class of regular languages over a finite alphabet Σ. Let Δ = {b1, …, bm} be an alphabet, and δ be the substitution from Δ∗ into Σ∗ such that δ(bi) = Ri for all i (1 ≤ i ≤ m). Let R be a regular language over Σ which can be defined from by a finite number of applications of the operators union, concatenation, and star. Then there exist regular languages over Δ which can be transformed onto R by δ. The relative star height of R w.r.t. is the minimum star height of regular languages over Δ which can be transformed onto R by δ. This paper proves the existence of an algorithm for determining relative star height. This result obviously implies the existence of an algorithm for determining the star height of any regular language.
Article
This paper studies the relationship between the apparent star height of a given regular expression and the structure of its reduced deterministic state graph. Sufficient conditions for the star height of a regular event R to equal the cycle rank of its reduced state graph GR are derived. The cycle rank of GR is also shown to constitute a lower bound to the star height of certain subsets of R. These results are then applied to fully characterize the star height of events consisting of ℰ sets of paths in finite digraphs and two open problems posed by Eggan are answered.
Conference Paper
The cyclic shift of a language L, defined as SHIFT(L)={vu∣uv∈L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov’s pioneering paper on the state complexity of regular language operations [A. N. Maslov, Sov. Math., Dokl. 11, 1373–1375 (1970); translation from Dokl. Akad. Nauk SSSR 194, 1266-1268 (1970; Zbl 0222.94064)], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)!·2 (n-1)(n-2) , which shows that the state complexity of cyclic shift is 2 n 2 +nlogn-O(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove 2 Θ(n 2 ) state complexity. We also establish a tight 2n 2 +1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.
Conference Paper
The paper discusses results on -languages in a recursion theoretic framework which is adapted to the treatment of formal languages. We consider variants of the arithmetical hierarchy which are not based on the recursive sets but on sets defined in terms of finite automata. In particular, it is shown how the theorems of Büchi and McNaughton on regular -languages can be viewed as results on collapsing such quantifier hierarchies. Further automata theoretic hierarchies are outlined.
Conference Paper
It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata.
Conference Paper
Reversible languages occur in many different domains. Although the decision for the membership of reversible languages was solved in 1992 by Pin, an effective construction of a reversible automaton for a reversible language was still unknown. We give in this paper a method to compute a reversible automaton from the minimal automaton of a reversible language. With this intention, we use the universal automaton of the language that can be obtained from the minimal automaton and that contains an equivalent automaton which is quasi-reversible. This quasi-reversible automaton has nearly the same properties as a reversible one and can easily be turned into a reversible automaton.
Article
A decision procedure is described for equivalence of deterministic two-tape (one-way) automata.
Article
Properties of star height of regular events are investigated. It is shown that star height is preserved under such operations as taking quotients, addition or subtraction of a finite event, removal of all words beginning with a given letter, and removal of certain subsets of smaller star height. Next it is shown that there exist events of arbitrarily large star height whose union, concatenation, and star is of star height one. Also, arbitrarily large increases in star height can be obtained by using the intersection or complement operations. In the second part of the paper a technique for establishing the star height of regular events is developed. It is shown that for every regular event R of star height n there exists a nondeterministic state graph G whose states correspond to subsets of the set of states Q of the reduced automaton accepting R and whose cycle rank is precisely n. Unfortunately a given subset Q′ of Q may have to be repeated k times in G and no bound on k is known. Thus it is still not known whether an algorithm for determining star height exists. However, it is felt that the techniques presented here provide new insight into the problem.
Article
A wordw is called recurrent with respect to a substitution if any descendant of it can regeneratew itself by iterations of the substitution. The set of recurrent words with respect to a regular (resp. context free) substitution is a regular (resp. context free) language. The set of recurrent words which are the descendants of a single fixed word with respect to a context free substitution is context free.
Article
We give an algorithm, its correctness proof, and its proof of execution time bound, for finding the sets of equivalent states in a deterministic finite state automaton. The time bound is K · m · n · log n where K is a constant, m the number of input, symbols, and n the number of states. Hopcroft [3] has already published such an algorithm. The main reason for this paper is to illustrate the use of communicating an algorithm to others using a structured, top-down approach. We have also been able to improve on Hopcroft's algorithm by reducing the size of the algorithm and correspondingly complicating the proof of the running time bound.
Article
We present an algorithm for computing the prefix of an automaton. Automata considered are non-deterministic, labelled on words, and can have epsilon -transitions. The prefix automaton of an automaton A has the following characteristic properties. It has the same graph as A. Each accepting path has the same label as in A. For each state q, the longest common prefix of the labels of all paths going from q to an initial or final state is empty. The interest of the computation of the prefix of an automaton is that it is the first step of the minimization of sequential transducers. The algorithm that we describe has the same worst case time complexity as another algorithm due to Mohri but our algorithm allows automata that have empty labelled cycles. If we denote by P(q) the longest common prefix of labels of paths going from q to an initial or final state, it operates in time O((P + 1) x \E \) where P is the maximal length of all P(q).
Article
We describe here a construction on transducers that give a new conceptual proof for two classical decidability results on transducers: it is decidable whether a finite transducer realizes a functional relation, and whether a finite transducer realizes a sequential relation. A better complexity follows then for the two decision procedures.Zusammenfassunge papier présente une construction sur les transducteurs qui donne une nouvelle preuve conceptuelle pour deux résultats classiques de décidabilité sur les transducteurs: on peut décider si un transducteur fini réalise une relation fonctionnelle et s'il réalise une relation séquentielle. Il en résulte un algorithme polynomial pour les deux procédures de décision.
Article
This paper addresses the problem of turning a rational (ie regular) expres- sion into a finite automaton. We formalize and generalize the idea of "partial deriva- tives" introduced in 1995 by V. Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing a formal power se- ries with coefficients in a semiring. We first define precisely what is such a rational expression with multiplicity and which hypothesis should be put on the semiring of coefficients in order to keepe the usual identities. We then define the derivative of such a rational expression as a linear combination of expressions called derived terms and we show that all derivatives of a given expression are generated by a finite set of derived terms, that yields a finite automaton with multiplicity whose behaviour is the series denoted by the expression. We also prove that this automaton is a quotient of the standard (or Glushkov) automaton of the expression. Finally, we propose and discuss some possible modifications to our definition of derivation.
Article
This paper considers the application of signal flow graph techniques to the problem of characterizing sequential circuits by regular expressions. It is shown that the methods of signal flow graph theory, with the proper interpretation, apply to state diagrams of sequential circuits. The use of these methods leads to a simple algorithm for obtaining a regular expression describing the behavior of a sequential circuit directly from its state diagram. COPYRIGHT © 1963—THE INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC.
Article
The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems. 1 Introduction One of the richest extensions of finite automaton theory is obtained by associating multiplicities to words, edges and states. Perhaps the most intuitive appearence of this concept is obtained by counting for every word the number of successful paths spelling it in a (nondeterministic) finite automaton. This is motivated by the formalization of ambiguity in a finite automaton and leads to the theory of recognizable subsets of a free monoid with multiplicities in the semiring of natural numbers. This...
Article
Symbolic dynamics is a field which was born with the work in topology of Marston Morse at the beginning of the 1920s [44]. It is, according to Morse, an “algebra and geometry of recurrence”. The idea is the following. Divide a surface into regions named by certain symbols. We then study the sequences of symbols obtained by scanning the successive regions while following a trajectory starting from a given point. A further paper by Morse and Hedlund [45] gave the basic results of this theory. Later, the theory was developed by many authors as a branch of ergodic theory (see for example the collected works in [59] or [12]). One of the main directions of research has been the problem of the isomorphism of shifts of finite type (see below the definition of these terms). This problem is not yet completely solved although the latest results of Kim and Roush [35] indicate a counterexample to a long-standing conjecture formulated by F. Williams [61].