# Jacques Sakarovitch's research while affiliated with Télécom ParisTech and other places

## Publications (112)

Chapter
In this paper, we present a new construction of a finite automaton associated with a rational (or regular) expression. It is very similar to the one of the so-called Thompson automaton, but it overcomes the failure of the extension of that construction to the case of weighted rational expressions. At the same time, it preserves all (or almost all)...
Preprint
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...
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 refine...
Preprint
We present here a construction for the derived term automaton (aka partial derivative, or Antimirov, automaton) of a rational (or regular) expression based on a sole induction on the depth of the expression and without making reference to an operation of derivation of the expression. It is particularly well-suited to the case of weighted rational e...
Article
We present here a construction for the derived term automaton (aka partial derivative, or Antimirov, automaton) of a rational (or regular) expression based on a sole induction on the depth of the expression and without making reference to an operation of derivation of the expression. It is particularly well-suited to the case of weighted rational e...
Article
Given any numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists. In the case of the usual base p...
Preprint
Given any numeration system, we call carry propagation at a number $N$ the number of digits that are changed when going from the representation of $N$ to the one of $N+1$, and amortized carry propagation the limit of the mean of the carry propagations at the first $N$ integers, when $N$ tends to infinity, and if this limit exists. In the case of th...
Chapter
This paper reports on the work done for the implementation of the algorithms for the computation of the minimal quotient of an automaton in the Awali platform. In the case of non-deterministic or of weighted automata, the minimal quotient of an automaton is obtained by merging all states in bisimulation. Two strategies are explored for the computat...
Chapter
This invited talk presents the work conducted on the problems that arise when dealing with weighted automata containing $$\varepsilon$$-transitions: how to define the behaviour of such automata in which the presence of $$\varepsilon$$-circuits results in infinite summations, and second how to eliminate the $$\varepsilon$$-transitions in an autom...
Article
Full-text available
Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are...
Article
The signature of a labelled tree (and hence of its prefix-closed branch language) is the sequence of the degrees of the nodes of the tree in the breadth-first traversal. In a previous work, we have characterised the signatures of the regular languages. Here, the trees and languages that have the simplest possible signatures, namely the periodic one...
Article
Full-text available
We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the...
Conference Paper
Full-text available
The signature of a labelled tree (and hence of its prefix-closed branch language) is the sequence of the degrees of the nodes of the tree in the breadth-first traversal. In a previous work, we have characterised the signatures of the regular languages. Here, the trees and languages that have the simplest possible signatures, namely the periodic one...
Article
This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin.
Conference Paper
We present a type system for automata and rational expressions, expressive enough to encompass weighted automata and transducers in a single coherent formalism. The system allows to express useful properties about the applicability of operations including binary heterogeneous functions over automata. We apply the type system to the design of the Va...
Conference Paper
We present here the notion of breadth-first signature and its relationship with numeration system theory. It is the serialisation into an infinite word of an ordered infinite tree of finite degree. We study which class of languages corresponds to which class of words and,more specifically, using a known construction from numeration system theory, w...
Article
Full-text available
For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n + 1 over the...
Article
Full-text available
We present here the notion of breadth-first signature and its relationship with numeration system theory. It is the serialisation into an infinite word of an ordered infinite tree of finite degree. We study which class of languages corresponds to which class of words and,more specifically, using a known construction from numeration system theory, w...
Article
This work introduces the idea of breadth-first generation of infinite trees and languages. It is orthogonal to usual descriptions by classical objects such as automata and grammars which refer more naturally to a depth-first approach. This idea is brought into play with periodic inputs, a case that comes from the study of rational base number syste...
Conference Paper
Full-text available
Vaucanson is an open source C++ platform dedicated to the computation with finite weighted automata. It is generic: it allows to write algorithms that apply on a wide set of mathematical objects. Initiated ten years ago, several shortcomings were discovered along the years, especially problems related to code complexity and obfuscation as well as p...
Article
This paper addresses the problem of the validity of weighted automata in which the presence of ε-circuits results in infinite summations. Earlier works either rule out such automata or characterize the semirings in which these infinite sums are all well-defined. By means of a topological approach, we take here a definition of validity that is stron...
Article
Full-text available
This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the integers and whose subtrees are all distinct. With every node of that tree is then associated a minimal infinite w...
Conference Paper
Full-text available
It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can verified in linear time if a given minimal automaton meets...
Conference Paper
In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition of a strong combinatorial property on languages : the bounded left iteration property. It is both an unnatural...
Conference Paper
The removal of ε-transitions in weighted automata leads to infinite summation when cycles of such transitions are allowed. This paper presents both an algorithm for that purpose, and a framework in which the algorithm is correct.
Article
It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple computation of the value of words in this system and easy constructions for the recognition of recognisable se...
Book
Full-text available
For the past two decades, specialised events on finite-state methods have been successful in presenting interesting studies on natural language processing to the public through journals and collections. The FSMNLP workshops have become well-known among researchers and are now the main forum of the Association for Computational Linguistics' (ACL) Sp...
Article
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...
Article
We give a new, and hopefully more easily understandable, structural proof of the decomposition of a k-valued transducer into k unambiguous functional ones, aresult established by A.Weber in 1996. Our construction is based on a lexicographic ordering of computations of automata and on two coverings that can be build by means of this ordering. The co...
Article
This collaborative volume presents trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Top...
Article
In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is base...
Conference Paper
We prove that the radix cross-section of a rational set for a length morphism, and more generally for a rational function from a free monoid into ℕ, is rational. This property no longer holds if the image of the function is a subset of a free monoid with two or more generators. The proof is based on several results on finite automata, such as the l...
Article
Full-text available
Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested in finding bounds on the minimal length of words in Sigma* which are not elements of Fact(S*) in terms of the...
Article
We prove that the function that maps a word of a rational language onto its successor for the radix order in this language is a finite union of co-sequential functions.
Chapter
This chapter presents the theory of weighted automata over graded monoids and with weights taken in arbitrary semirings. The first benefit of broadening the scope beyond free monoids is that it makes clearer the distinction between the rational and the recognisable series. As the topological machinery is set anyway, the star of series is defined in...
Article
We investigate weighted automata with discounting and their behaviors over semirings and finitely generated graded monoids. We characterize the discounted behaviors of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product This extends a classical result of Kleene-Schutzenberger. Here we show that t...
Conference Paper
We give a new and conceptually different proof for the de- cidability of k-valuedness of transducers (a result due to Gurari and Ibarra), without resorting to any other kind of machines than transduc- ers. In contrast with the previous proof, our algorithm takes into account the structure of the analysed transducers and yields better complexity bou...
Article
We define and study here the class of rational functions that are finite union of sequential functions. These functions can be realized by cascades of sequential transducers. After showing that cascades of any height are equiv-alent to cascades of height at most two and that this class strictly contains sequential functions and is strictly containe...
Article
We give a new, and hopefully more easily understandable, structural proof of the decomposition of a k-valued transducer into k unambiguous functional ones, a result established by A. Weber in 1996. Our construction is based on a lexicographic ordering of computations of automata and on two coverings that can be build by means of this ordering. The...
Conference Paper
Full-text available
We present an XML format that allows to describe a large class of finite weighted automata and transducers. Our design choices stem from our policy of making the implementation as simple as possible. This format has been tested for the communication between the modules of our automata manipulation platform Vaucanson, but this document is less an ex...
Conference Paper
This paper is a survey on the universal automaton, which is an automaton canonically associated with every language. In the last forty years, many objects have been defined or studied, that are indeed closely related to the universal automaton. We first show that every automaton that accepts a given language has a morphic image which is a subautoma...
Article
We give new, conceptually different, and hopefully more easily understandable proofs for two results on transducers: the decidability of k-valuedness (Gurari-Ibarra's theorem) and the decidability of bounded valuedness (Weber's theorem). Our constructions yield algorithms that closely depend on the structure of the transducers, and better complexit...
Article
We give a new and hopefully more easily understandable structural proof for the decompo-sition of a k-valued transducer into k unambiguous functional transducers, a result established by A. Weber. The number of states of the transducers given by our construction outperforms Weber's one by one exponential. Moreover, we solve a problem left open by W...
Conference Paper
Numbers do exist, independently of the way we represent them, of the way we write them. And there are many ways to write them: integers as finite sequence of digits once a base is fixed, rational numbers as a pair of integer or as an ultimately periodic infinite sequence of digits, or reals as an infinite sequence of digits but also as a continued...
Conference Paper
We described 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.
Article
Full-text available
A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter fini...
Conference Paper
We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, or any (skew) fleld and that the same holds true for functional tranducers as well.
Article
This paper is a survey where we try to organise the known answers to the question whether a given finite automaton with multiplicity in a semiring KK is equivalent to a sequential, or input deterministic, one. We shall see that depending on KK, the question goes from obvious to open, that the answer goes from yes to undecidable. We review results o...
Chapter
This paper presents a survey of recent results in the theory of rational sets in arbitrary monoids. Main topics considered here are : the so-called Kleene monoids (i.e. monoids where Kleene's theorem holds), rational functions and relations, rational sets in partially commutative monoids, and rational sets in free groups.
Conference Paper
This paper presents some features of the Vaucanson platform. We describe some original algorithms on weighted automata and transducers (computation of the quotient, conversion of a regular expression into a weighted automaton, and composition). We explain how complex declarations due to the generic programming are masked from the user and finally w...
Chapter
Full-text available
Without Abstract
Article
We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend the one of covering of automata. We make use of this notion, together with the Schutzenberger construct that we have explained in a previous work and that we briefly recall here, in order to give a direct and constructive proof of a fundamental theorem on...
Conference Paper
We prove that two automata with multiplicity in \mathbb Z{\mathbb Z} are equivalent, i.e. define the same rational series, if and only if there is a sequence of \mathbb Z{\mathbb Z}-coverings, co- \mathbb Z{\mathbb Z}-coverings, and circulations of –1, which transforms one automaton into the other. Moreover, the construction of these transfor...
Conference Paper
This survey paper reviews the means that allow to go from one representation of the languages to the other and how, and to what ex- tend, one can keep them small. Some emphasis is put on the comparison between the expressions that can be computed from a given automaton and on the construction of the derived term automaton of an expression.
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 coefficie...
Conference Paper
This paper presents some features of the VAUCANSON platform. We describe some original algorithms on weighted automata and transducers (computation of the quotient, conversion of a regular expression into a weighted automaton, and composition). We explain how complex declarations due to the generic programming are masked from the user and finally w...
Article
Full-text available
A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique such expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter...
Conference Paper
We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of -1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results:...
Article
This paper reports on a new software platform called VAUCANSON and dedicated to the computation with automata and transducers. Its main feature is the capacity of dealing with automata whose labels may belong to various algebraic structures.The paper successively describes the main features of the VAUCANSON platform, including the fact that the ver...
Conference Paper
In this paper we investigate how it is possible to recover an automaton from a rational expression that has been computed from that automaton. The notion of derived term of an expression, introduced by Antimirov, appears to be instrumental in this problem. The second important ingredient is the co-minimization of an automaton, a dual and generalize...
Conference Paper
In this paper we investigate how,it is possible to recover an automaton,from a rational expression that has been computed,from that automaton. The notion of derived term of an expression, intro- duced by Antimirov, appears to be instrumental in this problem. The second important ingredient is the co-minimization of an automaton, a dual and generali...
Article
Full-text available
Conference Paper
Full-text available
This paper reports on a new software platform called Vaucanson and dedicated to the computation with automata and transducers. Its main feature is the capacity of dealing with automata whose labels may belong to various algebraic structures. The paper successively shows how Vaucanson allows to program algorithms on automata in a way which is very...
Article
We show that the finite power property is decidable for rational sets in the free group. The complexity of the construction involved in the decision procedure may be lowered to O(n3)—where n is the cardinality of the state set of the automaton that defines the rational set.RésuméLa propriété de puissance finie est décidable pour les parties rationn...
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.Zusammenfa...
Conference Paper
This paper introduces a generalization of the partial derivatives of rational expressions, due to Antimirov, to rational expressions with multiplicity. We define the derivation of a rational expression with multiplicity in such a way that the result is a polynomial of expressions. This amounts to interpreting the addition symbol at the upper level...
Conference Paper
The star height of a regular language is an invariant that has been shown to be effectively computable in 1988 by Hashiguchi. But the algorithm that corresponds to his proof leads to impossible computations even for very small instances. Here we solve the problem (of computing star height) for a special class of regular languages, called reversible...
Article
In a previous work, we have investigated an automata-theoretic property of numeration systems associated with quadratic Pisot units that yields, for every such number , a certain group G  .
Article
This paper presents properties of relations between words that are realized by deterministic finite 2-tape automata. It has been made as complete as possible, and is structured by the systematic use of the matrix representation of automata. It is first shown that deterministic 2-tape automata are characterized as those which can be given a prefix m...
Conference Paper
In a previous work, we have investigated an automata-theoretic property of numeration systems associated with quadratic Pisot units that yields, for every such number θ, a certain group Gθ . In this paper, we characterize a cross-section of a congruence γθ of $$\mathbb{Z}$$ 4 that had arisen when constructing Gθ . In spite of the algebraic connec...
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 o...
Article
Full-text available
Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean φ. We show that there exists a letter-to-letter finite two-tape automaton that maps the Fibonacci representation of any positive integer onto its φ-expansion, provided the latter is folde...
Article
It is shown that a subsequential automaton (i.e. an automaton deterministic with respect to the input) with bounded delay is equivalent to an on-line automaton, that is an automaton which is letter-to-letter after an initial period where it reads the input and output nothing, and subsequential. The results on the synchronisation of automaton with b...
Article
An abstract is not available.
Article
It is first shown that deterministic 2-tape automata are characterized as those which can be given a prefix matrix representation. Schuetzenberger construction on representations, the one that gives semi-monomial representations for rational functions of words, is then applied to this prefix representation in order to obtain a new proof of the fact...
Article
The purpose of this paper is a comprehensive study of a family of rational relations, both of finite and infinite words, namely those that are computable by automata where the reading heads move simultaneously on the n input tapes, and that we thus propose to call synchronized rational relations.
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 rec...
Conference Paper
This paper presents a generalization of Eilenberg and Schtzenberger's Theorem on length-preserving relations to rational relations with the property that the difference of lengths of two related words is bounded, and to rational relations of infinite words that are realized by 2-tape automata such that the distance between the two heads during any...
Article
In our first part of this paper we defined and studied the family of rational monoids, which are monoids with an “easy” multiplication (by this we mean indeed that the multiplication can be realized by a finite automaton). In this second part of our work, we consider rational semigroups (instead of monoids) and investigate the properties of two ope...
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 the...
Article
We characterize here the free partially commutative monoids the regular sets of which form a Boolean algebra or are all unambiguous: these are, in both cases, the free products of free commutative monoids. This result has been established independently by other authors but the method used here is original. It is based on the properties of generaliz...
Article
We study here the properties of a family of monoids, which we call the rational monoids, and which are monoids with a multiplication of low complexity. A monoid is rational if its multiplication may be described by a rational function from a free monoid into itself. The main results are that rational monoids, like finite ones, have the properties t...
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.
Conference Paper
The analysis of the famous Kleene's theorem shows that it consists indeed in two different propositions that are better distinguished when one tries to generatize the result. The first one relates rational expressions and a suitable generalization of finite automata. It holds in any monoid or, even better, in the semiring of formal power series on...
Chapter
We characterize the deterministic context-free languages that are unions of equivalence classes in the equivalence generated by the one-sided Dyck reduction over a two-letter alphabet.
Article
We study the following classical problem of formal language theory: let L1,…,Ln be n languages recognized by the monoids M1,…,Mn respectively. Given an operation ϕ, we want to build a monoid M, function of M1,…,Mn, which recognizes the language (L1,…,Ln)ϕ. We show that most of the constructions given in the literature for this kind of problem are p...
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 wi...

## Citations

... In this case, using an appropriate version of the linear form of an expression, the equations (8) and (6) hold and a partial derivative automaton can be defined. Lombardy and Sakarovitch [51] expanded and generalised this approach, showing that the partial derivative automaton is a quotient of position automaton, even considering weighted regular expressions over non free monoids. ...
... This hypothesis is also crucial in order to generalize various properties of integer bases to abstract numeration systems. In particular, it is used in order to be able to represent real numbers [14] or to study the carry propagation of the successor function [6]. Finally, we note that from the proof of the fact that an infinite word is S-automatic for some abstract numeration S if and only if it is morphic, it can be deduced that an infinite word is S-automatic for some abstract numeration S if and only if it is S -automatic for some abstract numeration S having a prefix-closed numeration language. ...
... Motivated by a question of Mahler in number theory, the introduction of rational base numeration systems has brought to light a family of formal languages with a rich combinatorial structure [1]. In particular, the generation of infinite trees with a periodic signature has emerged [20,21,22,23]. Marsault and Sakarovitch very quickly linked the enumeration of the vertices of such trees (called breadth-first serialization) to the concept of abstract numeration system built on the corresponding prefix-closed language: the traversal of the tree is exactly the radix enumeration of the words of the language. ...
... Motivated by a question of Mahler in number theory, the introduction of rational base numeration systems has brought to light a family of formal languages with a rich combinatorial structure [1]. In particular, the generation of infinite trees with a periodic signature has emerged [20,21,22,23]. Marsault and Sakarovitch very quickly linked the enumeration of the vertices of such trees (called breadth-first serialization) to the concept of abstract numeration system built on the corresponding prefix-closed language: the traversal of the tree is exactly the radix enumeration of the words of the language. ...
... We now briefly recall some basic facts about noncommutative formal power series and weighted automata. More information can be found in [8,11,12,32]. ...
Citing article
... Yet an expression such as (a|a + b|b + 1 (1|a + 1|b + a|1 + b|1)) * clearly denotes the behavior of A. 1 Provided that operators such as + can be used below the tupling operator |, an more concise expression is E := (a|a + b|b + 1 (1|(a + b) + (a + b)|1)) * . 2 Similarly E := (a|a + b|b) * ( 1 (1|(a + b) + (a + b)|1)+ 2 (a|b + b|a)) * denotes the behavior of A . 3 Our purpose is to define multitape rational expressions such as E and E (Section 2) and to introduce an algorithm that computes precisely automata A and A from them (Section 4). To this end, we rely on an intermediary structure, expansions, studied in Section 3. ...
... This explains the particular attention we pay in this paper to rational abstract numeration systems. The sequel of this work [10] is in the preparation phase and will hopefully be completed in a not too distant future. ...
... Integers can also be expressed in other numeration systems [6,5]. A typical example uses the Fibonacci numbers instead of the powers of 2. Let (F n ) n≥0 be the Fibonacci sequence defined with the recurrence relation F n = F n−1 + F n−2 , for all n ≥ 2, and the initial conditions F 0 = 1, F 1 = 2. ...
... The polycyclic monoids were introduced by Nivat and Perot as a natural generalisation of the bicyclic monoid [22] and have been repeatedly re-discovered in numerous different settings (notably the 'dynamical algebra' of the logicians [6,7,8], with the equivalence given in [9]), the Cuntz algebras of C * algebra theory and theoretical physics [15], and the 'bracketing language' or 'stack algebra' of theoretical computer science & automata theory [26]). ...
... is a transition in δ ′ -see (6). Moreover, m b · · · m b+r ∈ I(β b )I(r * q,q )I(β b+r ). ...