Tree Automata, Mu-Calculus and Determinacy (Extended Abstract) (original) (raw)

Tree automata, mu-calculus and determinacy

[1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science

We show that the propositional Mu-Calculus is equivalent in expressive p o wer to nite automata on innite trees. Since complementation is trivial in the Mu-Calculus, our equivalence provides a radically simpli ed, alternative proof of Rabin's complementation lemma for tree automata, which is the heart of one of the deepest decidability results. We also show h o w Mu-Calculus can be used to establish determinacy of in nite games used in earlier proofs of complementation lemma, and certain games used in the theory of on-line algorithms.

An automata theoretic decision procedure for the propositional mu-calculus

Information and Computation, 1989

The propositional mu-calculus is a propositional logic of programs which incorporates a least fixpoint operator and subsumes the propositional dynamic logic of Fischer and Ladner, the infinite looping construct of Streett, and the game logic of Parikh. We give an elementary time decision procedure, using a reduction to the emptiness problem for automata on infinite trees. A small model theorem is obtained as a corollary.

Two-way cost automata and cost logics over infinite trees

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2014

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixedpoint-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

The Complexity of Tree Automata and Logics of Programs

SIAM Journal on Computing, 1999

The complexity of testing nonemptiness of finite state automata on infinite trees is investigated. It is shown that for tree automata with the pairs (or complemented pairs) acceptance condition having m states and n pairs, nonemptiness can be tested in deterministic time (mn) O(n) ; however, it is shown that the problem is in general NP-complete (or co-NP-complete, respectively). The new nonemptiness algorithm yields exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it follows that satisfiability for the full branching time logic CTL * can be tested in deterministic double exponential time. Another consequence is that satisfiability for propositional dynamic logic (PDL) with a repetition construct (PDL-delta) and for the propositional Mu-calculus (Lµ) can be tested in deterministic single exponential time.

Guarded fixed point logics and the monadic theory of countable trees

Theoretical Computer Science, 2002

Di erent variants of guarded logics (a powerful generalization of modal logics) are surveyed and an elementary proof for the decidability of guarded ÿxed point logics is presented. In a joint paper with Igor Walukiewicz, we proved that the satisÿability problems for guarded ÿxed point logics are decidable and complete for deterministic double exponential time (E. Gr adel and I. Walukiewicz, Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45 -54). That proof relies on alternating automata on trees and on a forgetful determinacy theorem for games on graphs with unbounded branching. The exposition given here emphasizes the tree model property of guarded logics: every satisÿable sentence has a model of bounded tree width. Based on the tree model property, we show that the satisÿability problem for guarded ÿxed point formulae can be reduced to the monadic theory of countable trees (S!S), or to the -calculus with backwards modalities.

Linear Game Automata: Decidable Hierarchy Problems for Stripped-Down Alternating Tree Automata

Lecture Notes in Computer Science, 2009

For deterministic tree automata, classical hierarchies, like Mostowski-Rabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to non-deterministic tree automata, none of these hierarchies is even close to be understood. Here we make an attempt in paving the way towards a clear understanding of tree automata. We concentrate on the class of linear game automata (LGA), and prove within this new context, that all corresponding hierarchies mentioned above-Mostowski-Rabin, Borel, and Wadge-are decidable. The class LGA is obtained by taking linear tree automata with alternation restricted to the choice of path in the input tree. Despite their simplicity, LGA recognize sets of arbitrary high Borel rank. The actual richness of LGA is revealed by the height of their Wadge hierarchy: (ω ω) ω .

The Emptiness Problem for Tree Automata with Global Constraints

2010 25th Annual IEEE Symposium on Logic in Computer Science, 2010

We define tree automata with global constraints (TAGC), generalizing the class of tree automata with global equality and disequality constraints [1] (TAGED). TAGC can test for equality and disequality between subterms whose positions are defined by the states reached during a computation. In particular, TAGC can check that all the subterms reaching a given state are distinct. This constraint is related to monadic key constraints for XML documents, meaning that every two distinct positions of a given type have different values. We prove decidability of the emptiness problem for TAGC. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extend our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different subterms reaching some state during a computation. We also allow local equality and disequality tests between sibling positions and the extension to unranked ordered trees. As a consequence of our results for TAGC, we prove the decidability of a fragment of the monadic second order logic on trees extended with predicates for equality and disequality between subtrees, and cardinality.

Tree Automata with Global Constraints Tree Automata with Global Constraints

2008

A tree automaton with global tree equality and disequality constraints, TAGED for short, is an automaton on trees which allows to test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, it is equipped with an (dis)equality relation on states, so that whenever two subtrees t and t ′ evaluate (in an accepting run) to two states which are in the (dis)equality relation, they must be (dis)equal. We study several properties of TAGEDs, and prove decidability of emptiness of several classes. We give two applications of TAGEDs: decidability of an extension of Monadic Second Order Logic with tree isomorphism tests and of unification with membership constraints. These results significantly improve the results of [10].

On the finite degree of ambiguity of finite tree automata

Acta Informatica, 1989

The degree of ambiguity of a finite tree automaton A, da(A), is the maximal number of different accepting computations of A for any possible input tree. We show: it can be decided in polynomial time whether or not da(A)<~. We give two criteria characterizing an infinite degree of ambiguity and derive the following fundamental properties of an finite tree automaton A with n states and rank L> i having a finite degree of ambiguity: for every input tree t there is a input tree tl of depth less than 22".n! having the same number of accepting computations; the degree of ambiguity of A is bounded by 22 .... geL+ ,).,

Decidability of MSO Theories of Tree Structures

Lecture Notes in Computer Science, 2004

In this paper we provide an automaton-based solution to the decision problem for a large set of monadic second-order theories of deterministic tree structures. We achieve it in two steps: first, we reduce the considered problem to the problem of determining, for any Rabin tree automaton, whether it accepts a given tree; then, we exploit a suitable notion of tree equivalence to reduce (a number of instances of) the latter problem to the decidable case of regular trees. We prove that such a reduction works for a large class of trees, that we call residually regular trees. We conclude the paper with a short discussion of related work.