Rabin Tree Research Articles

Top-down complementation of automata on finite trees

We present a new complementation construction for nondeterministic automata on finite trees. The traditional complementation involves determinization of the corresponding bottom-up automaton (recall that top-down deterministic automata are less powerful than nondeterministic automata, whereas bottom-up deterministic automata are equally powerful).The construction works directly in a top-down fashion, therefore without determinization. The main advantages of this construction are: (i) in the special case of finite words it boils down to the standard subset construction (which is not the case of the traditional bottom-up complementation construction), and (ii) it illustrates the core argument of the complementation lemma for infinite trees, central in the proof of Rabin's tree theorem, in a simpler setting where issues related to acceptance conditions over infinite words and determinacy of infinite games are not present.

A hierarchy of tree-automatic structures

AbstractWe considerωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of lengthωnfor some integern≥ 1. We show that all these structures areω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation forω2-automatic (resp.ωn-automatic forn> 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem forωn-automatic boolean algebras,n≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a-set nor a-set. We obtain that there exist infinitely manyωn-automatic, hence alsoω-tree-automatic, atomless boolean algebras, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].

The isomorphism relation between tree-automatic Structures

Abstract An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set.

Relating word and tree automata

In the automata-theoretic approach to verification, we translate specifications to automata. Complexity considerations motivate the distinction between different types of automata. Already in the 60s, it was known that deterministic Büchi word automata are less expressive than nondeterministic Büchi word automata. The proof is easy and can be stated in a few lines. In the late 60s, Rabin proved that Büchi tree automata are less expressive than Rabin tree automata. This proof is much harder. In this work we relate the expressiveness gap between deterministic and nondeterministic Büchi word automata and the expressiveness gap between Büchi and Rabin tree automata. We consider tree automata that recognize derived languages. For a word language L , the derived language of L , denoted L △ , is the set of all trees all of whose paths are in L . Since often we want to specify that all the computations of the program satisfy some property, the interest in derived languages is clear. Our main result shows that L is recognizable by a nondeterministic Büchi word automaton but not by a deterministic Büchi word automaton iff L △ is recognizable by a Rabin tree automaton and not by a Büchi tree automaton. Our result provides a simple explanation for the expressiveness gap between Büchi and Rabin tree automata. Since the gap between deterministic and nondeterministic Büchi word automata is well understood, our result also provides a characterization of derived languages that can be recognized by Büchi tree automata. Finally, it also provides an exponential determinization of Büchi tree automata that recognize derived languages.

Decidability of Hierarchies of Regular Aperiodic Languages

A new, logical approach is propounded to resolve the decidability problem for the hierarchies of Straubing and Brzozowski based on preservation theorems in model theory, a theorem of Higman, and the Rabin tree theorem. We thus manage to obtain purely logical, short proofs of some known decidability facts, which definitely may be of methodological interest. The given approach also applies in some other similar situations, for instance, to the hierarchies of formulas modulo a theory of linear orderings with finitely many unary predicates.

Determinization and Memoryless Winning Strategies

We show that Safra's determinization of ω -automata with Streett (strong fairness) acceptance condition also gives memoryless winning strategies in infinite games, for the player whose acceptance condition is the complement of the Streett condition. Both determinization and memorylessness are essential parts of known proofs of Rabin's tree automata complementation lemma. Also, from Safra's determinization construction, along with its memoryless winning strategy extension, a single exponential complementation of Streett tree automata follows. A different single exponential construction and proof first appeared in [N. Klarlund (1992), Progress measures, immediate determinacy, and a subset construction for tree automata, in “Proceedings, 7th IEEE Symposium on Logics in Computer Science”].

Rabin tree automata and finite monoids

We incorporate finite monoids into the theory of recognizability of ω-tree languages by Rabin automata. We define a free monoid of ω-trees and associate with each ω-tree language L a language L̂ of infinite words over this monoid. Using this correspondence we introduce strong monoid recognizability of ω-tree languages (strengthening the standard notion for infinite words) and show it to be equivalent to Rabin recognizability. We also show that there exists an ω-tree language L which is not Rabin recognizable, but its associated language L̂ is monoid recognizable (in the standard sense). Our positive result opens the theory of varieties of ω-tree languages, in extension of the ones for finite and infinite words and finite trees.

Infinite trees and automaton- definable relations over ω-words

We study relations over ω-words using a representation by tree languages. An ω-word over an alphabet with k letters is considered as a path through the k-ary tree, an n-tuple of ω-words as an n-tuple of paths (coded by an appropriate valuation of the k-ary tree using values in {0,1} n ), and a relation over ω-words as a tree language. In the first part of the paper we give a logical characterization of the “Rabin-recognizable relations” (whose associated tree languages are recognized by Rabin tree automata) in terms of “weak chain logic”, a restriction of monadic second-order logic over trees. In the second part of the paper an extended logic is considered, obtained by adjoining the “equal-level predicate” over trees. We describe the class of relations over ω-words which (in the tree language representation) are definable in this logic, and show that the theory of the k-ary tree in this logic is decidable. It covers tree properties which are not expressible in the monadic second-order logic S kS.