Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-08T14:02:06.799Z Has data issue: false hasContentIssue false
  • English
  • Français

The μ-calculus alternation-depth hierarchy isstrict on binary trees

Published online by Cambridge University Press:  15 August 2002

André Arnold
André Arnold*
Affiliation:
LaBRI, Université de Bordeaux 1, CNRS, 351 cours de la Libération, 33405 Talence, France.
Get access

Abstract

In this paper we give a simple proof that the alternation-depthhierarchy of the μ-calculus for binary trees is strict.Thewitnesses for this strictness are the automata that determinewhether there is a winning strategy for the parity gameplayed on a tree.

Keywords

Mu-calculalternating automataparity games.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 33 , Issue 4-5 , July 1999 , pp. 329 - 339
Copyright
© EDP Sciences, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arnold, A., Logical definability of fixed points. Theoret. Comput. Sci. 61 (1988) 289-297. CrossRef
Arnold, A. and Nivat, M., The metric space of infinite trees. Algebraic and topological properties. Fund. Inform. 4 (1980) 445-476.
A. Arnold and D. Niwinski, Fixed-point characterization of büchi automata on infinite trees. J. Inf. Process. Cybern. EIK 26 (1990).
A. Arnold and D. Niwinski, Fixed point characterization of weak monadic logic definable sets of trees, M. Nivat and A. Podelski, Eds., Tree automata and Languages. Elsevier (1992) 159-188.
J.C. Bradfield, Fixpoint alternation: Arithmetic, transition systems, and the binary tree, this issue.
Bradfield, J.C., The modal mu-calculus alternation hierarchy is strict, U. Montanari and V. Sassone, Eds., in Proc. CONCUR '96, Lecture Notes in Comput. Sci. 1119 (1996) 233-246. CrossRef
Bradfield, J.C., Simplifying the modal mu-calculus alternation hierarchy, M. Morvan, C. Meinel and D. Krob, Eds., in Proc. STACS '98, Lecture Notes in Comput. Sci. 1373 (1998) 39-49. CrossRef
E.A. Emerson and C.S. Jutla, Tree automata, mu-calculus and determinacy, in Proc. FOCS '91. IEEE Computer Society Press (1991) 368-377.
Y. Gurevitch and L. Harrington, Trees, automata and games, in Proc. 14th ACM Symp. on the Theory of Computing (1982) 60-65.
Lenzi, G., A hierarchy theorem for the mu-calculus, F. Meyer auf der Heide and B. Monien, Eds., in Proc. ICALP '96, Lecture Notes in Comput. Sci. 1099 (1996) 87-109. CrossRef
Mostowski, A.W., Hierarchies of weak automata and weak monadic formulas. Theoret. Comput. Sci. 83 (1991) 323-335. CrossRef
Muller, D.E., Saoudi, A. and Schupp, P.E., Alternating automata, the weak monadic theory of the tree and its complexity. Theoret. Comput. Sci. 97 (1992) 233-244. CrossRef
Muller, D.E. and Schupp, P.E., Alternating automata on infinite trees, Theoret. Comput. Sci. 54 (1987) 267-276. CrossRef
Niwinski, D., On fixed point clones, L. Kott, Ed., in Proc. 13th ICALP, Lecture Notes in Comput. Sci. 226 (1986) 464-473. CrossRef
Niwinski, D., Fixed points characterization of infinite behaviour of finite state systems. Theoret. Comput. Sci. 189 (1997) 1-69. CrossRef
Thomas, W., A hierarchy of sets of infinite trees, A.B. Cremers and H.P. Kriegel, Eds., Theoret. Comput. Sci., Lecture Notes in Comput. Sci. 145 (1983) 335-342. CrossRef
Walukiewicz, I., Monadic second-order logic on tree-like structures, C. Puech and R. Reischuk, Eds., in Proc. STACS '96, Lecture Notes in Comput. Sci. 1046 (1996) 401-414.