# About the effect of the number of successful paths in an infinite tree on the recognizability by a finite automaton with Buchi conditions

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## Abstract

We modify an acceptance condition of Büchi automaton on infinite trees: rather than to require that each computation path is successful, we impose various restrictions on the number of successful paths in a run of the automaton on a tree. All these modifications alter the recognizing power of Büchi automata. We examine the classes induced by the acceptance conditions that require ≤α, ≥α, =α successful paths, where α is a cardinal number. It turns out that, except some trivial cases, the “≤” classes are uncomparable with the class *Bü* of Büchi acceptable tree languages, while the classes “≥” are strictly included in *Bü*.

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## References

- [1]Büchi,J.R. (1962)
*On a decision method in restricted second order arithmetic*, in “Logic, Methodology and Philosophy of Science. Proc.1960 Intern. Congr. (E.Nagel et al. eds.),pp. 1–11.Google Scholar - [2]Emerson, E.A. and Jutla, C. (1988)
*The Complexity of Tree Automata and Logics of Programs*, Proc. 29th IEEE FOCS,pp. 328–337.Google Scholar - [3]Rabin, M.O. (1969)
*Decidability of second-order theories and automata on infinite trees*, Trans. AMS 141, pp. 1–35.Google Scholar - [4]Rabin, M.O. (1970)
*Weakly definable relations and special automata*, in “Mathematical Logic and Foundations of Set Theory” (Y.Bar-Hillel, ed.), pp. 1–23.Google Scholar - [5]Street R.S. (1982),
*Propositional dynamic logic of looping and converse*, Inform. Contr. 54, pp. 121–141.Google Scholar - [6]Thomas, W. (1990),
*Automata on Infinite Objects*, in “Handbook of Theoretical Computer Science” (J.v.Leeuwen,ed.).Google Scholar - [7]Vardi, M.Y. and Stockmeyer, L. (1985),
*Improved Upper and Lower Bounds for Modal Logics of Programs*, in Proc.26th FOCS.Google Scholar - [8]Vardi, M.Y. and Wolper, P. (1984),
*Automata-Theoretic Technics for Modal Logics of Programs*in Proc. 16th ACM STOC, pp. 446–456.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1991