# Regular expressions for infinite trees and a standard form of automata

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## Abstract

For Rabin pair automata [R1] a standard form is defined /def. 2/ i.e. such that an ordered subset {s

_{1},...,s_{2I-1}} of states is distinguished in such a way that a path of a run is accepting /rejecting if for some i even/ odd, 1≤i≤2I-1, the s_{i}appears infinitely often, and all s_{j}, j<i only finitely many times. The class of standard automata is big enough to represent all f.a. representable sets /th.1/ but has many properties similar to special automata defined in [R1]. A standard regular expression is defined /def. 6/ describing a process of forming of an infinite tree, as well as a process of building of an automaton /analysis and synthesis theorems 3,4/. The standard regular expressions are a generalisation of McNaughtons formula v d . [N] /.## Keywords

Regular Expression Finite Automaton Mixed Tree Letter Alphabet Finite Tree
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## 4. Bibiography

- [K]Karpiński M., Decidability of the weak definiability in the s2s theories. 3-d symposium of MFCS, Zaborów, January 21–26 1980, ICS PAS Reports 411 p.47–48.Google Scholar
- [N]McNaughton R., Testing and generating infinite sequences by a finite trees and inequalities between various Rabin pair indices, Information processing letters 15 1983, pl.59–163Google Scholar
- [M2]Mostowski A.W., Classes of automata of a given Rabin pair index. Proceedings of workshop on algorithms and computing theory, September 7–10,1981 ed. by M.Karpiński and Z.Habasiński, Poznań 1981.Google Scholar
- [M3]Mostowski A.W., On differences on automata on infinite trees and those on sequences, Report on the 1-st GTI Workshop, Lutz Prietze ed. Universitat Padderborn, Reiche Teoretische Informatik Marz 1982.Google Scholar
- [R1]Rabin M.O., Decidability and definiability in Second-order Theories, Actes Congress Intern. Math., 1970 Tome 1 p.239–244.Google Scholar
- [R2]Rabin M.O., Weakly definiable relations and special automata. Math.Logic Foundations Set Theory, North Holland 1970 p.1–23.Google Scholar
- [TW]Thatcher J.W. Wright J.B., Generalised Finite Automata Theory with an Application to Second-order Logic. Math. Systems Theory, vol.2, p.57–81.Google Scholar
- [W]Wagner K., On ω-regular Sets. Information and Control 43 1979, p.123–177.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1985