A Quasi-Linear Time Algorithm Deciding Whether Weak B\"uchi Automata Reading Vectors of Reals Recognize Saturated Languages

This work considers weak deterministic B\"uchi 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 B\"uchi 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 B\"uchi automata accepting set of reals. Finally, the same problem is also solved for parallel encoding of automata reading vectors of relative reals

(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

Spectra and satisfiability for logics with successor and a unary func- tion

International audienceWe investigate the expressive power of two logics, both with the successor function: first-order logic with an un-interpreted function, and existential monadic second order logic — that is first-order logic over words —, with multiplication by a constant b. We prove that all b-recognizable sets are spectra of those logics. Furthermore, it is proven that some encoding of the set of halting times of a non-deterministic 2-counter automaton is also a spectrum. This yields undecidability of the finite satisfiability problem for those logics. Finally, it is shown that first-order logic with one uninterpreted function and successor can encode quickly increasing functions, such as the Knuth's up-arrows

Büchi Automata Recognizing Sets of Reals Definable in First-Order Logic with Addition and Order

Proceedings of the International Conference on Theory and Applications of Models of Computation, Bern, 20-22 April 2017International audienceThis work considers encodings of non-negative reals in a fixed base, and their encoding by weak deterministic Büchi automata. A Real Number Automaton is an automaton which recognizes all encodings of elements of a set of reals. We explain in this paper how to decide in linear time whether a set of reals recognized by a given minimal weak determin-istic RNA is FO[IR; +, <, 1]-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an exis-tential (respectively, existential-universal) FO[IR; +, <, 1]-formula which defines the set of reals recognized by the automaton. As an additional contribution, the techniques used for obtaining our main result lead to a characterization of minimal deterministic Büchi automata accepting FO[IR; +, <, 1]-definable set

Undecidability of Satisfiability of Expansions of FO[<] over Words with a Semilinear set

International audienceWe provide two new characterizations of FO[<, mod]-definable sets, i.e. sets of integers definable in first-order logic with the order relation and modular relations, and use these characterizations to prove that satisfiability of first-order logic over words with an order relation and a semilinear set (i.e. a set definable in first-order logic with addition) that is not FO[<, mod]-definable is undecidable

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 $\nu_{R}(x)$ 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 $\nu(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $\nu(x)$ 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

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 $\nu_{R}(x)$ 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 $\nu(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $\nu(x)$ 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

Büchi Automata Recognizing Sets of Reals Definable in First-Order Logic with Addition and Order

Proceedings of the International Conference on Theory and Applications of Models of Computation, Bern, 20-22 April 2017International audienceThis work considers encodings of non-negative reals in a fixed base, and their encoding by weak deterministic Büchi automata. A Real Number Automaton is an automaton which recognizes all encodings of elements of a set of reals. We explain in this paper how to decide in linear time whether a set of reals recognized by a given minimal weak determin-istic RNA is FO[IR; +, <, 1]-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an exis-tential (respectively, existential-universal) FO[IR; +, <, 1]-formula which defines the set of reals recognized by the automaton. As an additional contribution, the techniques used for obtaining our main result lead to a characterization of minimal deterministic Büchi automata accepting FO[IR; +, <, 1]-definable set

Efficient algorithms and tools for MITL model-checking and synthesis

info:eu-repo/semantics/publishe