15 research outputs found
On the Topological Complexity of Infinitary Rational Relations
International audienceWe prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992]
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
On the Continuity Set of an omega Rational Function
In this paper, we study the continuity of rational functions realized by
B\"uchi finite state transducers. It has been shown by Prieur that it can be
decided whether such a function is continuous. We prove here that surprisingly,
it cannot be decided whether such a function F has at least one point of
continuity and that its continuity set C(F) cannot be computed. In the case of
a synchronous rational function, we show that its continuity set is rational
and that it can be computed. Furthermore we prove that any rational
Pi^0_2-subset of X^omega for some alphabet X is the continuity set C(F) of an
omega-rational synchronous function F defined on X^omega.Comment: Dedicated to Serge Grigorieff on the occasion of his 60th Birthda
An Example of Pi^0_3-complete Infinitary Rational Relation
We give in this paper an example of infinitary rational relation, accepted by
a 2-tape B\"{u}chi automaton, which is Pi^0_3-complete in the Borel hierarchy.
Moreover the example of infinitary rational relation given in this paper has a
very simple structure and can be easily described by its sections
On the Accepting Power of 2-Tape Büchi Automata
International audienceWe show that, from a topological point of view, 2-tape Büchi automata have the same accepting power than Turing machines equipped with a Büchi acceptance condition. In particular, we show that for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete infinitary rational relations accepted by 2-tape Büchi automata. This very surprising result gives answers to questions of W. Thomas [Automata and Quantifier Hierarchies, in: Formal Properties of Finite automata and Applications, Ramatuelle, 1988, LNCS 386, Springer, 1989, p.104-119] , of P. Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992], and of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In: "A Decade of Concurrency", LNCS 803, Springer, 1994, p. 583-621]
Some Problems in Automata Theory Which Depend on the Models of Set Theory
We prove that some fairly basic questions on automata reading infinite words
depend on the models of the axiomatic system ZFC. It is known that there are
only three possibilities for the cardinality of the complement of an
omega-language accepted by a B\"uchi 1-counter automaton . We prove
the following surprising result: there exists a 1-counter B\"uchi automaton
such that the cardinality of the complement of the omega-language
is not determined by ZFC: (1). There is a model of ZFC in which
is countable. (2). There is a model of ZFC in which has
cardinal . (3). There is a model of ZFC in which
has cardinal with . We prove a very
similar result for the complement of an infinitary rational relation accepted
by a 2-tape B\"uchi automaton . As a corollary, this proves that the
Continuum Hypothesis may be not satisfied for complements of 1-counter
omega-languages and for complements of infinitary rational relations accepted
by 2-tape B\"uchi automata. We infer from the proof of the above results that
basic decision problems about 1-counter omega-languages or infinitary rational
relations are actually located at the third level of the analytical hierarchy.
In particular, the problem to determine whether the complement of a 1-counter
omega-language (respectively, infinitary rational relation) is countable is in
. This is rather surprising if
compared to the fact that it is decidable whether an infinitary rational
relation is countable (respectively, uncountable).Comment: To appear in the journal RAIRO-Theoretical Informatics and
Application