Complementation of rational sets on countable scattered linear orderings

@article{Rispal2005ComplementationOR,
  title={Complementation of rational sets on countable scattered linear orderings},
  author={C. Rispal and Olivier Carton},
  journal={Int. J. Found. Comput. Sci.},
  year={2005},
  volume={16},
  pages={767-786}
}
In a preceding paper (Bruyere and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Buchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach. 
A Kleene Theorem for Languages of Words Indexed by Linear Orderings
TLDR
In a preceding paper, Bruyere and the second author introduced automata, as well as rational expressions, which allow to deal with words indexed by linear orderings, and this paper extends this result to languages of Words indexed by all linear ordering. Expand
Automata on linear orderings
TLDR
It is proved that for countable scattered linear orderings, the two notions of finite automata and rational expressions are equivalent, which extends Kleene's theorem. Expand
Logic and Bounded-Width Rational Languages of Posets over Countable Scattered Linear Orderings
  • N. Bedon
  • Mathematics, Computer Science
  • LFCS
  • 2009
TLDR
It is proved that a language of labelled N-free posets is series-rational if and only if it is recognizable by a finite depth-nilpotent algebra and bounded-width and monadic second-order definable. Expand
Regular Languages of Words over Countable Linear Orderings
TLDR
An algebraic model for recognizing languages of words indexed by countable linear orderings that is effectively equivalent to definability in monadic second-order (MSO) logic is developed. Expand
Automata and semigroups recognizing infinite words
TLDR
The various acceptance modes of automata, and two algebraic proofs of McNaughton's theorem on the equivalence between Buchi and Muller automata are given. Expand
Algebraic Characterization of FO for Scattered Linear Orderings
We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with markedExpand
AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS
TLDR
An algebraic notion of recognizability for languages of words indexed by countable linear orderings is developed and it is proved that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. Expand
Languages recognised by finite semigroups, and their generalisations to objects such as trees and graphs, with an emphasis on definability in monadic second-order logic
TLDR
The purpose of this book is to describe the algebraic approach to regular languages in a way that covers these extensions to structures beyond words, e.g. trees or graphs. Expand
Limited Set quantifiers over Countable Linear Orderings
TLDR
This paper gives decidable algebraic characterizations of several sublogics of monadic second- order logic over countable linear orderings, such as first-order logic, first-orders logic on cuts, weak monadicSecond-order Logic with cuts, as well as fragments of monads in which sets have to be well ordered or scattered. Expand
Schützenberger and Eilenberg theorems for words on linear orderings
TLDR
It is extended that the star-free sets of finite words are exactly the languages recognized by finite aperiodic semigroups, and the variety theorem of Eilenberg for finite words is extended: there is a one-to-one correspondence between varieties of languages of words on countable scattered linear orderings and pseudo-varieties of algebras. Expand
...
1
2
3
...

References

SHOWING 1-10 OF 25 REFERENCES
Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank
TLDR
It is proved that rational sets of words on countable scattered linear ordering are closed under complementation using an algebraic approach. Expand
Automata, Semigroups and Recognizability of Words on Ordinals
  • N. Bedon
  • Mathematics, Computer Science
  • Int. J. Algebra Comput.
  • 1998
TLDR
For a given integer n, ωn-semigroups are defined as a generalization of ω-semIGroups for languages of words of length less than ω n+1, and those algebraic structures define the same sets as those recognized by Choueka automata when they are finite. Expand
Automata on Linear Orderings
TLDR
These extend finite, (bi-)infinite words and words on ordinals to include words indexed by linear orderings and introduce automata and rational expressions for words on linear ordering. Expand
An Eilenberg Theorem for Words on Countable Ordinals
TLDR
It is shown that finite Ω1-semigroups are equivalent to automata, and the proof gives a new algorithm for determinizing automata on countable ordinals. Expand
Unambiguous Büchi Automata
TLDR
The main result of the paper is that any rational set of infinite words is recognized by such an automaton called unambiguous, and it is shown that they are well suitable for boolean operations. Expand
Infinite sequences and finite machines
TLDR
A regular set is a set of possible state sequences of an inputless, nondeterministic machine that may either terminate in some equilibrium condition, or else the machine may pass from one state to another without ever reaching equ~librium. Expand
LINEAR ORDERINGS
We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable, such that for any countable linear ordering B, L does not embed into BExpand
Finite Automata and Ordinals
  • N. Bedon
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1996
TLDR
The domain of Wojciechowski automata is restricted to the domain of Choueka's ones (that is, given n ω , the authors keep only α-sequences for α ω n +1 in the language defined by a WojCiechowsky automaton) in order to prove the equivalence between Choueka automata and Woj ciechowski automata. Expand
Decidability of second-order theories and automata on infinite trees
Introduction. In this paper we solve the decision problem of a certain secondorder mathematical theory and apply it to obtain a large number of decidability results. The method of solution involvesExpand
Star-Free Sets of Words on Ordinals
  • N. Bedon
  • Computer Science, Mathematics
  • Inf. Comput.
  • 2001
Let n be a fixed integer; we extend the theorem of Schutzenberger, McNaughton, and Papert on star-free sets of finite words to languages of words of length less than ?n.
...
1
2
3
...