arXiv subjects

Jacques Sakarovitch

Publications and source records attributed to Jacques Sakarovitch.

13 recordsLinked to original sources

arXiv2022

Morphisms and minimisation of weighted automata

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.

cs.DM↗

arXiv2021

Derived terms without derivation a shifted perspective on the derived-term automaton

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 expressions.

cs.FL↗

arXiv2020

The carry propagation of the successor function

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 $p$ numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those numeration systems we consider and for which the limit exists. We address the problem of the existence of the amortized carry propagation in non-standard numeration systems of various kinds: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. We tackle the problem by three different types of techniques: combinatorial, algebraic, and ergodic. For each kind of numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.

math.CO↗

arXiv2018

On subtrees of the representation tree in rational base numeration systems

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 the integers, its edges bear labels taken in {0,1,...,p-1}, and its subtrees are all distinct. We associate with each subtree (or with its root n) three infinite words. The bottom word of n is the lexicographically smallest word that is the label of a branch of the subtree. The top word of n is defined similarly. The span-word of n is the digitwise difference between the latter and the former. First, we show that the set of all the span-words is accepted by an infinite automaton whose underlying graph is essentially the same as the tree itself. Second, we study the function that computes for all n the bottom word associated with n+1 from the one associated with n, and show that it is realised by an infinite sequential transducer whose underlying graph is once again essentially the same as the tree itself. An infinite word may be interpreted as an expansion in base p/q after the radix point, hence evaluated to a real number. If T is a subtree whose root is n, then the evaluations of the labels of the branches of T form an interval of $\mathbb{R}$. The length of this interval is called the span of n and is equal to the evaluation of the span-word of n. The set of all spans is then a subset of R and we use the preceding construction to study its topological closure. We show that it is an interval when p is greater than or equal to 2q-1, and a Cantor set of measure zero otherwise.

cs.DM↗

arXiv2015

Automata and rational expressions

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.

cs.FL↗

arXiv2014

Rhythmic generation of infinite trees and languages

This work builds on the notion of breadth-first signature of infinite trees and (prefix-closed) languages introduced by the authors in a previous work. We focus here on periodic signatures, a case coming from the study of rational base numeration systems; the language of integer representations in base~$\frac{p}{q}$ has a purely periodic signature whose period is derived from the Christoffel word of slope~$\frac{p}{q}$. Conversely, we characterise languages whose signature are purely periodic as representations of integers in such number systems with non-canonical alphabets of digits.

cs.FL↗

arXiv2014

Breadth-first serialisation of trees and rational languages

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, we prove that the signature of rational languages are substitutive sequences.

cs.FL↗

arXiv2013

On sets of numbers rationally represented in a rational base number system

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 property in usual formal language theory (as it contradicts any kind of pumping lemma) and an ideal fit to the languages defined through rational base number systems.

cs.FL↗

arXiv2013

Ultimate periodicity of b-recognisable sets : a quasilinear procedure

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 this description. This thus yields a O(n log(n)) procedure for deciding whether a general deterministic automaton accepts an ultimately periodic set of numbers.

cs.FL↗

arXiv2013

Auto-similarity in rational base number systems

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 word. The main result is that a sequential transducer which computes for all n the minimal word associated with n+1 from the one associated with n, has essentially the same underlying graph as the tree itself. These infinite words are then interpreted as representations of real numbers; the difference between the numbers represented by these two consecutive minimal words is the called the span of a node of the tree. The preceding construction allows to characterise the topological closure of the set of spans.

cs.FL↗

arXiv2011

On the enumerating series of an abstract numeration system

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 sets of numbers. It is also shown that conversely it is decidable whether an N-rational series corresponds to a rational abstract numeration system.

cs.DM↗

arXiv2010

On the Minimal Uncompletable Word Problem

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 maximal length of words in S.

cs.FL↗

arXiv2008

On the decomposition of k-valued rational relations

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 complexity of the construction, measured as the number of states of the transducers involved in the decomposition, improves the original one by one exponential. Moreover, this method allows further generalisation that solves the problem of decomposition of rational relations with bounded length-degree, which was left open in Weber's paper.

cs.IT↗