137 research outputs found
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
Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities
Some relations between physics and finitary and infinitary mathematics are
explored in the context of a many-minds interpretation of quantum theory. The
analogy between mathematical ``existence'' and physical ``existence'' is
considered from the point of view of philosophical idealism. Some of the ways
in which infinitary mathematics arises in modern mathematical physics are
discussed. Empirical science has led to the mathematics of quantum theory. This
in turn can be taken to suggest a picture of reality involving possible minds
and the physical laws which determine their probabilities. In this picture,
finitary and infinitary mathematics play separate roles. It is argued that
mind, language, and finitary mathematics have similar prerequisites, in that
each depends on the possibility of possibilities. The infinite, on the other
hand, can be described but never experienced, and yet it seems that sets of
possibilities and the physical laws which define their probabilities can be
described most simply in terms of infinitary mathematics.Comment: 21 pages, plain TeX, related papers from
http://www.poco.phy.cam.ac.uk/~mjd101
On the Accepting Power of 2-Tape BĂŒchi Automata
International audienceWe show that, from a topological point of view, 2-tape BĂŒchi automata have the same accepting power than Turing machines equipped with a BĂŒchi 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 BĂŒchi 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 ThĂ©orie Descriptive, Ph. D. Thesis, UniversitĂ© 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
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
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