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

• Damian Niwiński
Contributions
Part of the Lecture Notes in Computer Science book series (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 t1, ..., 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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## References

1. B. Courcelle(1983) Fundamental properties of infinite trees, Theor.Comput.Sci. 25, 95–169.Google Scholar
2. E.A. Emerson(1985) Automata, tableaux and temporal logics, in: Proc. Workshop on Logics of Programs, LNCS 193, 79–88.Google Scholar
3. E.A.Emerson and C.Jutla (1988) The complexity of tree automata and logics of programs, Proc. 29th IEEE Symp. on Foundations of Computer Science,, N.Y.,328–337.Google Scholar
4. F.Gecseg and M.Steinby (1984) Tree Automata, Akademiai Kiado, Budapest.Google Scholar
5. K.Kuratowski and A.Mostowski(1976) Set Theory, North-Holland.Google Scholar
6. M.O. Rabin(1969) Decidability of second-order theories and automata on infinite trees, Trans.Amer.Soc.141, 1–35.Google Scholar
7. M.O.Rabin (1970) Weakly definable relations and special automata, in: Mathematical Logic in Foundations of Set Theory (Y.Bar-Hillel,ed.),1–23.Google Scholar
8. M.O.Rabin(1972) Automata on infinite objects and Church's problem, Amer.Math.Soc., 1–22.Google Scholar
9. M.O.Rabin(1977) Decidable theories, in: Handbook of Mathematical Logic (J.Barwise,edGoogle Scholar
10. S. Safra(1988) On The Complexity of ω-Automata, Proc. 29th IEEE Symp. on Foundations of Computer Science, White Plains, N.Y., 319–327.Google Scholar
11. W. Thomas(1990) Automata on infinite objects, in: Handbook of Theoretical Computer Science, vol.B (J.van Leeuven,ed.), 133–191.Google Scholar
12. M.Y. Vardi and P.L. Wolper(1986) Automata-theoretic techniques for modal logics of programs, J. Comput.System Sci. 32, 183–221.Google Scholar