On Two Hierarchies of Subregularly Tree Controlled Languages (original) (raw)

A Logical Approach to Decidability of Hierarchies of Regular Star—Free Languages

Lecture Notes in Computer Science, 2001

We propose a new, logical, approach to the decidability problem for the Straubing and Brzozowski hierarchies based on the preservation theorems from model theory, on a theorem of Higman, and on the Rabin tree theorem. In this way, we get purely logical, short proofs for some known facts on decidability, which might be of methodological interest.

The Wadge Hierarchy of Deterministic Tree Languages

Logical Methods in Computer Science, 2008

We provide a complete description of the Wadge hierarchy for deterministically recognisable sets of infinite trees. In particular we give an elementary procedure to decide if one deterministic tree language is continuously reducible to another. This extends Wagner's results on the hierarchy of ω-regular languages of words to the case of trees.

Syntactic monoids in the construction of systolic tree automata

1985

The acceptance of regular languages by systolic tree automata is analyzed in more detail by investigating the structure of the individual processors needed. Since the processors essentially compute the monoid product, our investigation leads to questions concerning syntactic monoids. Hereby certain properties of monoids turn out to be important. These properties, as well as the language families induced by them, are also studied in the paper.

Relating hierarchies of word and tree automata

1998

For an w-word language L, the derived tree language Path(L) is the language of trees having all their paths in L. We consider the hierarchies of deterministic automata on words and nondeterministic automata on trees with P~abin conditions in chain form. We show that L is on some level of the hierarchy of deterministic word automata iff Path(L) is on the same level of the hierarchy of nondeterministic tree automata.

Tree algebras and varieties of tree languages

Theoretical Computer Science, 2007

We consider several aspects of Wilke's [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85-106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields normal form representations of binary trees and contexts, as well as a new completeness proof and a computational decision method for the axiomatization of tree algebras. Varieties of binary tree languages are compared with varieties of tree languages studied earlier in the literature. We also prove a variety theorem thus solving a problem noted by several authors. Syntactic tree algebras are studied and compared with ordinary syntactic algebras. The expressive power of the language of tree algebras is demonstrated by giving equational definitions for some well-known varieties of binary tree languages.

On the separation question for tree languages

2012

We show that the separation property fails for the classes Σ n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Σ 2 (i.e., for co-Büchi sets). The non-separation result is also adapted to the analogous classes induced by weak alternating automata. To prove our main result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Π n and fails for Σ n-classes. The construction invented for words turns out to be useful for trees via a suitable game. It remains open if the separation property holds for all classes Π n of the index hierarchy for tree automata. To give a positive answer it would be enough to show the reduction property of the dual classes-a method well-known in descriptive set theory. We show that it cannot work here, because the reduction property fails for all classes in the index hierarchy. Keywords Alternating tree automata • Separation property • Rabin-Mostowski index H. Michalewski and D. Niwiński supported by the Polish Ministry of Science grant nr. N N206 567840.

Synchronized bottom-up tree automata and L-systems

Lecture Notes in Computer Science, 1985

Napo] i D i p a r t i m e n t o di I n f o r m a t i c a e A p p l i c a z l o n i U n i v e r s i t a ~ dl S a l e r n o 8 4 1 0 0 Salerno, Italy I n t r o d u c t i o n . In t h i s p a p e r a new t y p e o f b o t t o m -u p t r e e a u t o m a t o n , c a l l e d s y n c h r on i z e d b o t t o m -u p t r e e automaton, i s c o n s i d e r e d . T h i s automaton p r o c e s s e s a t r e e i n a b o t t o m -u p way and one l e v e l a t a t i m e . M o r e o v e r , more t h a n one t r a n s i t i o n f u n c t i o n i s a l l o w e d , b u t o n l y one o~ them a t a t i m e can be a p p l i e d t o nodes a t t h e same l e v e l o f a t r e e . The t r e e l a n g u a g e r e c o g n i z e d by t h e s e a u t o m a t a a r e t h e images, u n d e r p r o j e c t i o n , o f t h e s e t o f d e r i v a t i o n t r e e s o f EPTOL l a n g u a g e s . The model i n t r o d u c e d i n t h i s p a p e r i s a g e n e r a l i z a t i o n o f t h e b o ttoni-up tree automaton. Its b e h a v i o u r , r e l a t i v e to E T O L systems, is the same as the b o t t o m -u p t r e e a u t o m a t o n b e h a v i o u r r e l a t i n g to c o n t e x t f r e e g r a m m a r s ( 7 ) .

Ambiguity Hierarchy of Regular Infinite Tree Languages

2020

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k>0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if there is k, such that for every input it has at most k accepting computations. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words, every regular language is accepted by a deterministic automaton. Over finite trees, every regular language is accepted by an unambiguous automaton. Over omega\omegaomega-words every regular language is accepted by an unambiguous Buchi automaton and by a dete...