Tree automata, mu-calculus and determinacy

@article{Emerson1991TreeAM,
  title={Tree automata, mu-calculus and determinacy},
  author={E. Allen Emerson and Charanjit S. Jutla},
  journal={[1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science},
  year={1991},
  pages={368-377}
}
  • E. EmersonC. Jutla
  • Published 1 September 1991
  • Mathematics, Computer Science
  • [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
It is shown that the propositional mu-calculus is equivalent in expressive power to finite automata on infinite trees. Since complementation is trivial in the mu-calculus, the equivalence provides a radically simplified, alternative proof of M.O. Rabin's (1989) complementation lemma for tree automata, which is the heart of one of the deepest decidability results. It is also shown how mu-calculus can be used to establish determinacy of infinite games used in earlier proofs of complementation… 

Automata for the mu-calculus and Related Results

It is shown that every formula is semantically equivalent to a disjunctive formula and this kind of formula gives rise to a new notion of finite automaton which characterizes the expressive power of the mu-calculus over all transition systems.

Progress measures, immediate determinacy, and a subset construction for tree automata

  • Nils Klarlund
  • Mathematics
    [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
  • 1992
Using the concept of a progress measure, a simplified proof is given of M.O. Rabin's (1969) fundamental result that the languages defined by tree automata are closed under complementation and a graph-theoretic duality theorem for such acceptance conditions is shown.

On completeness of the mu -calculus

  • I. Walukiewicz
  • Computer Science
    [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science
  • 1993
A finitary complete axiom system for the mu -calculus can be roughly described as a system for propositional modal logic with the addition of a induction rule to reason about least fixpoints.

Automata for the Modal mu-Calculus and related Results

It is shown that every formula is semantically equivalent to a disjunctive formula and this kind of formula gives rise to a new notion of finite automaton which characterizes the expressive power of the Μ-calculus over all transition systems.

Fixpoints for Rabin Tree Automata Make Complementation Easy

Direct fixpoint constructions for Rabin-automata are described, allowing us to translate modal mu-calculus inductively to Rabin, and provide a new proof of the expressive equivalence of the two formalisms.

Automata for the-calculus and Related Results

It is shown that every formula is semantically equivalent to a disjunctive formula and this kind of formula gives rise to a new notion of nite automaton which characterizes the expressive power of the-calculus over all transition systems.

A Complete Axiomatization of MSO on Infinite Trees

We show that an adaptation of Peano's axioms for second-order arithmetic to the language of MSO completely axiomatizes the theory over infinite trees. This continues a line of work begun by Büchi

Satisfiability and Finite Model Property for the Alternating-Time mu-Calculus

It is shown that language emptiness of these automata can be checked in exponential time and it follows that the satisfiability problem is EXPTIME-complete for the alternating-time μ-calculus.

Coalgebraic Automata Theory: Basic Results

The main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.

Weak Automata for the Linear Time µ-Calculus

  • M. Lange
  • Computer Science, Mathematics
    VMCAI
  • 2005
It is shown that the linear time alternation hierarchy collapses at level 0 and not just at level 1 as known so far, and the translation from formulas to automata can be optimised yielding automata whose size is only exponential in the alternation depth of the formula.
...

References

SHOWING 1-10 OF 32 REFERENCES

The complexity of tree automata and logics of programs

  • E. EmersonC. Jutla
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)/sup 3n/), even though the problem is in general NP-complete, and it follows that satisfiability for propositional dynamic logic with a repetition construct and for the propositional mu-calculus can be tests in deterministic single exponential time.

Trees, automata, and games

This work gives here an alternative and transparent proof of Rabin's result on tree automata, which is based on ideas of his predecessors and especially those of B- and-uuml;chi-&-mdash;.

The Propositional Mu-Calculus is Elementary

An elementary time decision procedure is given, using a reduction to the emptiness problem for automata on infinite trees, and a small model theorem is obtained as a corollary.

Propositional Dynamic Logic of looping and converse

Propositional Dynamic Logic can be extended to include both an infinitary iteration construct delta and a backtracking construct converse. The resulting logic does not satisfy the finite model

Automata on Infinite Objects

  • W. Thomas
  • Computer Science
    Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics
  • 1990

On simultaneously determinizing and complementing omega -automata

  • E. EmersonC. Jutla
  • Mathematics, Computer Science
    [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science
  • 1989
The authors give a construction to determine and complement simultaneously a Buchi automaton in infinite strings, with an exponential blowup in states and a linear blow up in the number of pairs, which permits exponentially improved essentially optimal decision procedures for various modal logics of programs.

The Emptiness Problem for Automata on Infinite Trees

The proof reduces the emptiness problem for automata on infinite trees to that for Automata on finite trees, by showing that any automata definable set of infinite trees must contain a finitely-generable tree.

Fixed points vs. infinite generation

  • D. Niwinski
  • Mathematics
    [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science
  • 1988
The author characterizes Rabin definability (see M.O. Rabin, 1969) of properties of infinite trees of fixed-point definitions based on the basic operations of a standard powerset algebra of trees and

Complexity of Automata on Innnite Objects

We investigate in this thesis problems concerning the complexity of translation among, and decision procedure for, di erent types of nite automata on in nite words (!automata). An !-automaton is the

Decidability of second-order theories and automata on infinite trees

Introduction. In this paper we solve the decision problem of a certain secondorder mathematical theory and apply it to obtain a large number of decidability results. The method of solution involves