arXiv subjects

Arthur Milchior

Publications and source records attributed to Arthur Milchior.

5 recordsLinked to original sources

arXiv2017

A Quasi-Linear Time Algorithm Deciding Whether Weak Büchi Automata Reading Vectors of Reals Recognize Saturated Languages

This work considers weak deterministic Büchi automata reading encodings of non-negative $d$-vectors of reals in a fixed base. A saturated language is a language which contains all encoding of elements belonging to a set of $d$-vectors of reals. A Real Vector Automaton is an automaton which recognizes a saturated language. It is explained how to decide in quasi-linear time whether a minimal weak deterministic Büchi automaton is a Real Vector Automaton. The problem is solved both for the two standard encodings of vectors of numbers: the sequential encoding and the parallel encoding. This algorithm runs in linear time for minimal weak Büchi automata accepting set of reals. Finally, the same problem is also solved for parallel encoding of automata reading vectors of relative reals.

cs.FL↗

arXiv2017

(Quasi-)linear time algorithm to compute LexDFS, LexUP and LexDown orderings

We consider the three graph search algorithm LexDFS, LexUP and LexDOWN. We show that LexUP orderings can be computed in linear time by an algorithm similar to the one which compute LexBFS. Furthermore, LexDOWN orderings and LexDFS orderings can be computed in time $\left(n+m\log m\right)$ where $n$ is the number of vertices and $m$ the number of edges.

cs.DS↗

arXiv2016

Büchi automata recognizing sets of reals definable in first-order logic with addition and order

This work considers weak deterministic Büchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time whether a set of reals recognized by a given minimal weak deterministic RNA is ${FO}[\mathbb R;+,<,1]$-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an existential (respectively, existential-universal) ${FO}[\mathbb R;+,<,1]$-formula which defines the set of reals recognized by the automaton. It is also shown that techniques given by Muchnik and by Honkala for automata over vector of natural numbers also works on vector of real numbers. It implies that some problems such as deciding whether a set of tuples of reals $R\subseteq\mathbb R^{d}$ is a subsemigroup of $(\mathbb R^{d},+)$ or is ${FO}[\mathbb R;+,<,1]$-definable is decidable.

cs.FL↗

arXiv2016

Uniform definition of sets using relations and complement of Presburger Arithmetic

In 1996, Michaux and Villemaire considered integer relations $R$ which are not definable in Presburger Arithmetic. That is, not definable in first-order logic over integers with the addition function and the order relation (FO[N,+,<]-definable relations). They proved that, for each such $R$, there exists a FO[N,+,<,$R$]-formula $ν_{R}(x)$ which defines a set of integers which is not ultimately periodic, i.e. not FO[N,+,<]-definable. It is proven in this paper that the formula $ν(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $ν(x)$ can be chosen such that it defines an expanding set. That is, an infinite set of integers such that the distance between two successive elements is not bounded.

math.LO↗

arXiv2014

A Note on Higher Order and Variable Order Logic over Finite Models

We show that descriptive complexity's result extends in High Order Logic to capture the expressivity of Turing Machine which have a finite number of alternation and whose time or space is bounded by a finite tower of exponential. Hence we have a logical characterisation of ELEMENTARY. We also consider the expressivity of some fixed point operators and of monadic high order logic. Finally, we show that Variable Order logic over finite structures contain the Analytical Hierarchy.

cs.LO↗