## Tree Automata, Mu-Calculus and Determinacy (Extended Abstract) (1991)

Venue: | IN PROCEEDINGS OF THE 32ND ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, FOCS ’91 |

Citations: | 231 - 4 self |

### BibTeX

@INPROCEEDINGS{Emerson91treeautomata,,

author = {E. A. Emerson and C. S. Jutla},

title = {Tree Automata, Mu-Calculus and Determinacy (Extended Abstract)},

booktitle = {IN PROCEEDINGS OF THE 32ND ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, FOCS ’91},

year = {1991},

pages = {368--377},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that the propositional Mu-Calculus is equivalent in expressive power to finite automata on infinite trees. Since complementation is trivial in the Mu-Calculus, our equivalence provides a radically simplified, alternative proof of Rabin's complementation lemma for tree automata, which is the heart of one of the deepest decidability results. We also show how Mu-Calculus can be used to establish determinacy of infinite games used in earlier proofs of complementation lemma, and certain games used in the theory of on-line algorithms.

### Citations

271 |
Results on the propositional mucalculus
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(Show Context)
Citation Context ...esent a new, enormously simpli ed proof of the Complementation Lemma. We would argue that the new proof is straightforward and natural. To obtain our proof, we showthat the propositional Mu-Calculus (=-=[Ko83]-=-) is expressively equivalent to (nondeterministic) tree automata. Since the Mu-Calculus is trivially closed under complementation, the equivalence immediately implies that tree automata are also close... |

213 |
Notes on Set Theory
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(Show Context)
Citation Context ...s a simple proof of (2). Finally, wealsoshowthat the above technique of extremal xpoint (Mu-calculus) characterization gives a general method for proving various determinacy results. Wolfe (1955) (cf =-=[Mo80]-=-) proved the determinacy of in nite games in the second level of the Borel hierarchy (F ). We showthatitispossibleto view such determinacy results as implicitly constructing extremal xpoints. Mu-Calcu... |

120 |
Decidability of second order theories and automata on infinite trees
- Rabin
- 1969
(Show Context)
Citation Context ... in our translation.) That proves the equivalence of nondeterministic Tree Automata and the Mu-Calculus, and hence the Complementation Lemma. Almost all published proofs of the Complementation Lemma (=-=[Ra69]-=-,[Bu77,83], [GH82]) can be seen upon re ection to have taken the following approach, best brought outby [MS87]: Alternating Tree Automata are easy to complement, given the fact that (1) certain in nit... |

88 |
Memory versus randomization in online algorithms
- Raghavan, Snir
- 1987
(Show Context)
Citation Context ...licitly constructing extremal xpoints. Mu-Calculus characterization also gives a clear and simple proof of determinacy of specialized F games used in de-randomizing (existentially) On-line algorithms =-=[RS90]-=-. This makes theresult in [RS90] only slightly more complicated (being more general) than a determinacy result in [B90]. Detailed proofs of Theorems in section 2 and 3 appear in the second author's d... |

67 |
On the complexity of !-automata
- Safra
- 1988
(Show Context)
Citation Context ...dependent of the history of the di erent \forall"-runs). We showhow these history-free, alternating Tree Automata can be reduced to nondeterministic Tree Automata using constructions for determinizing=-=[Sa88]-=- and complementing [EJ89] automata on in nite strings. (For general alternating tree automata, we can still do the translation into nondeterministic tree automata, but now wemust use Gurevich and Harr... |

42 | Propositional dynamic logic of looping and converse - Streett - 1982 |

38 |
On fixed-point clones
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- 1986
(Show Context)
Citation Context ...mma. The relationship of the Mu-Calculus and tree automata has also been investigated previously by Niwinski. In [Ni88] it is shown that tree automata can be translated into the Mu-Calculus, while in =-=[Ni86]-=- it is shown that a restricted Mu-Calculus, in which conjunctions are not allowed, can be translated into tree automata. As we shall see, in our translation of the full Mu-Calculus into tree automata,... |

24 |
Automata on in nite objects, in
- Thomas
- 1990
(Show Context)
Citation Context ...nd tree automata is a new result of some independent interest, since it shows that there indeed exists a natural modal logic of programs precisely equivalent to tree automata in expressive power (cf. =-=[Th88]-=-). Actually, a brief argument establishing this equivalence can be given, by appealing to known lengthy arguments translating through SnS. However, a careful examination of this argumentreveals that r... |

9 |
On simultaneously determinizing and complementing !-automata
- Emerson, Jutla
- 1989
(Show Context)
Citation Context ...of the di erent \forall"-runs). We showhow these history-free, alternating Tree Automata can be reduced to nondeterministic Tree Automata using constructions for determinizing[Sa88] and complementing =-=[EJ89]-=- automata on in nite strings. (For general alternating tree automata, we can still do the translation into nondeterministic tree automata, but now wemust use Gurevich and Harrington's forgetful strate... |

8 | Complexity of tree automata and modal logics of programs - EMERSON, JUTLA - 1988 |

4 |
on infinite trees and automata with dead-ends: a new proof for the decidability of the monadic second order theory of two successors
- Games
- 1992
(Show Context)
Citation Context ...inarily complex and di - cult. Because of the importance of the Complementation Lemma, a number of authors have endeavored (and continue to endeavor) to simplify the argument ([HR72], [GH82], [MS84], =-=[Mu84]-=-.) Perhaps the best known of these is the important work of Gurevich and Harrington [GH82] which attacks the problem from the standpoint of determinacy of in nite games. While the presentation is brie... |

3 | Borel Determinacy", Annals of Mathematics - Martin - 1975 |

3 | Testing and generating in nite sequences by a nite automaton - McNaughton - 1966 |

2 |
Using Determinacy to eliminate quanti ers
- Buchi
(Show Context)
Citation Context ... translation.) That proves the equivalence of nondeterministic Tree Automata and the Mu-Calculus, and hence the Complementation Lemma. Almost all published proofs of the Complementation Lemma ([Ra69],=-=[Bu77,83]-=-, [GH82]) can be seen upon re ection to have taken the following approach, best brought outby [MS87]: Alternating Tree Automata are easy to complement, given the fact that (1) certain in nite games ar... |

2 | The emptiness problem for automata on in nite trees - Hossley, Racko - 1972 |

2 |
Alternating Automata on
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- 1985
(Show Context)
Citation Context ...ence the Complementation Lemma. Almost all published proofs of the Complementation Lemma ([Ra69],[Bu77,83], [GH82]) can be seen upon re ection to have taken the following approach, best brought outby =-=[MS87]-=-: Alternating Tree Automata are easy to complement, given the fact that (1) certain in nite games are determined- i.e. at least one of the players has a winning strategy. Nondeterministic Tree Automat... |

2 |
Complexity of automata on in nite objects
- Safra
- 1989
(Show Context)
Citation Context .... Moreover, ParityTree Automata are trivially convertable to Pairs Tree Automata. Also, a simple conversion from Pairs Tree Automata to Parity Tree Automata can be obtained by a slight modi cation of =-=[Sa89]-=-, which converts a deterministic Pairs Automata to deterministic Streett Automata. This is also the essence of the LAR argument in [GH82]. Similarly, in nite games with Muller or Pairs condition easil... |

2 |
An elementary decision procedure for the mu-calculus
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- 1984
(Show Context)
Citation Context ...n A. It can be shown that T ( I) is a pre-model of f0 extending the labelling of t i I is a winning strategy for Player I in ;(A�t). Moreover, the winning strategy (also called the choice function in =-=[StE84]-=-) determines a derivation relation between occurences of sentences in the premodel so obtained [StE84]. We would like tosaythat a pre-model is in fact a model when there is no in nite derivation seque... |

2 |
The strict determinateness of certain in nite games
- Wolfe
- 1955
(Show Context)
Citation Context ...II = y8i x((AXEXx _ colorn�n i) ^ y). This simpli es the existential proof (in [RS90]) of de-randomization of competitive on-line algorithms for task systems, where [RS90] was invoking Wolfe's proof (=-=[W55]-=-) of determinacy of F games. Infact, a similar Mu-Calculus characterization can be given for F (countable union of closed sets) games, elucidating Wolfe's proof. We use terminology from [Mo80]. Let A ... |

1 | State-Strategies for Games in - Buchi - 1983 |

1 |
Automata on In nite Objects and Modal Logics of Programs
- Jutla
- 1990
(Show Context)
Citation Context ... theresult in [RS90] only slightly more complicated (being more general) than a determinacy result in [B90]. Detailed proofs of Theorems in section 2 and 3 appear in the second author's dissertation =-=[J90]-=-. 2 Alternating Tree Automata to Nondeterministic Tree Automata We deal here only with nite state Automata. De nition 2.1: A nondeterministic binary Tree Automaton is a tuple N = ( � OR� AND� � �L�s0�... |

1 |
unpublished notes
- Monk
(Show Context)
Citation Context ...standpoint of determinacy of in nite games. While the presentation is brief, the argument is still extremely di cult, and is probably best appreciated when accompanied by the 40 page supplementofMonk =-=[Mon]-=-. In this paper, we present a new, enormously simpli ed proof of the Complementation Lemma. We would argue that the new proof is straightforward and natural. To obtain our proof, we showthat the propo... |

1 |
Alternating Automata on In nite Objects, Determinacy and Rabin's Theorem", Automata on
- Muller, Schupp
(Show Context)
Citation Context ...extraordinarily complex and di - cult. Because of the importance of the Complementation Lemma, a number of authors have endeavored (and continue to endeavor) to simplify the argument ([HR72], [GH82], =-=[MS84]-=-, [Mu84].) Perhaps the best known of these is the important work of Gurevich and Harrington [GH82] which attacks the problem from the standpoint of determinacy of in nite games. While the presentation... |

1 | The Propositional -Calculus is more expressive than the propositional dynamic logic of looping", manuscript - Niwinski |

1 |
Fixed Points vs
- Niwinski
- 1988
(Show Context)
Citation Context ...Thus this argument is not at all suitable for use in a proof of the Complementation Lemma. The relationship of the Mu-Calculus and tree automata has also been investigated previously by Niwinski. In =-=[Ni88]-=- it is shown that tree automata can be translated into the Mu-Calculus, while in [Ni86] it is shown that a restricted Mu-Calculus, in which conjunctions are not allowed, can be translated into tree au... |