# Automata on infinite trees with counting constraints

## Abstract

We investigate finite automata on infinite trees with the usual Muller criterion for the success of an infinite computation path, but with the acceptance paradigm modified in that not all the computation paths need to be successful. Instead, it is required that the number of successful paths must belong to a specified set of cardinals Γ. We show that Muller automata with the acceptance constraint of the form “there are at least *γ* accepting paths” can be always simulated by tree automata with a weaker criterion for successful paths, namely Büchi acceptance condition. We also show that this is the most general class of constraints for which a simulation by Büchi automata is always possible. Next, we characterize the maximal class of constraints which can be simulated by classical Muller automata (known to be more powerful than Büchi automata). The condition requiered of the set Γ there, is that the intersection with natural numbers forms a recognizable set. Finally, we exhibit a set of trees which is recognized by a classical Büchi automaton but fails to be recognized by any Muller automaton with a non trivial cardinality constraint (i.e., except for Γ = 0).

## Keywords

Finite Automaton Cardinal Number Computation Path Acceptance Condition Cardinality Constraint## References

- [D. Beauquier, M. Nivat and D. Niwiński(1992)] The Effect of the Number of Successful Paths in a Büchi Tree Automaton, to appear in
*Int. Jour. of Alg. and Comp.*Google Scholar - [E.A. Emerson(1990)] The Role of Büchi's Automata in Computing Science,
*in*: MacLane and Siefkes, eds., The Collected Works of J.R.Büchi, Springer Verlag, Berlin, 18–22.Google Scholar - [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 - [F. Gecseg and M. Steinby(1984)] Tree Automata, Akademiai Kiado, Budapest.Google Scholar
- [D.Perrin(1990)] Finite Automata,
*in*: Handbook of Theoretical Computer Science, vol.B (J.van Leeuven,ed.), 1–57.Google Scholar - [M.O. Rabin(1969)] Decidability of second-order theories and automata on infinite trees,
*Trans.Amer.Soc.*141, 1–35.Google Scholar - [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 - [M.O.Rabin(1972)] Automata on infinite objects and Church 's problem,
*Amer. Math.Soc.*, 1–22.Google Scholar - [M.O. Rabin(1977)] Decidable theories,
*in*: Handbook of Mathematical Logic (J.Barwise, ed.).Google Scholar - [W. Thomas (1990)] Automata on infinite objects
*in*: Handbook of Theoretical Computer Science, vol.B (J. van Leeuven, ed.), 133–191.Google Scholar - [M.Y. Vardi and P.L. Wolper(1986)] Automata-theoretic techniques for modal logics of programs,
*J. Comput. System Sci.*32, 183–221.CrossRefGoogle Scholar