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On the cardinality of sets of infinite trees recognizable by finite automata

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Mathematical Foundations of Computer Science 1991 (MFCS 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 520))

Abstract

We show that a Rabin recognizable set of trees is uncountable iff it is of the cardinality continuum iff it contains a non-regular tree. If a Rabin recognizable set L is countable, it can be represented as

$$L = M\left[ {{{t_1 } \mathord{\left/{\vphantom {{t_1 } {x_1 , \ldots ,{{t_n } \mathord{\left/{\vphantom {{t_n } {x_n }}} \right.\kern-\nulldelimiterspace} {x_n }}}}} \right.\kern-\nulldelimiterspace} {x_1 , \ldots ,{{t_n } \mathord{\left/{\vphantom {{t_n } {x_n }}} \right.\kern-\nulldelimiterspace} {x_n }}}}} \right]$$

where M is a regular set of finite terms and t 1, ..., t n are regular trees. We also design an algorithm which, given a Rabin automaton A, computes the cardinality of the set of trees recognized by A.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, ul.Banacha 2, 00-913, Warszawa 59, Poland

    Damian NiwiƄski

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  1. Damian NiwiƄski
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Andrzej Tarlecki

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© 1991 Springer-Verlag Berlin Heidelberg

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NiwiƄski, D. (1991). On the cardinality of sets of infinite trees recognizable by finite automata. In: Tarlecki, A. (eds) Mathematical Foundations of Computer Science 1991. MFCS 1991. Lecture Notes in Computer Science, vol 520. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54345-7_80

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  • DOI: https://doi.org/10.1007/3-540-54345-7_80

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54345-9

  • Online ISBN: 978-3-540-47579-8

  • eBook Packages: Springer Book Archive

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