Decidability of Hierarchies of Regular Aperiodic 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.

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.

Decidability of Monadic Theories

Mathematical Foundations of Computer Science, Praha, Czechoslovakia, September 3–7, 1984. Proceedings. Lecture Notes in Computer Science. V. 176., 1984

In this survey article the author describes recent results on monadic theories. The first part concerns the monadic theory of < N, ≤, f > MTf, where N is the set of nonnegative integers, ≤ is the usual order on N, and f is a function from N into N with a finite range. f can be thought of as an ω-word in some finite alphabet. The author gives a general theorem characterizing the decidability of MTf. In the case where f is almost periodic one gets MTF is decidable if and only if f is computable and effectively almost periodic. Results on the monadic theory of < Z, ≤, f > where Z is the set all integers are also presented and their relation to symbolic dynamics indicated. The notion of a minimal transducer is described and a uniformization theorem for the monadic theory of < N, ≤> is given in terms of these automata. The second part of the paper describes a proof due to Muchnik of Rabin’s theorem on the decidability of the monadic theory of S2S. Muchnik’s proof uses a notion of automaton but avoids transfinite induction. The paper ends with brief remarks on weak monadic theories.

Two collapsing hierarchies of subregularly tree controlled languages

Theoretical Computer Science, 2009

Tree controlled grammars are context-free grammars where the associated language only contains those terminal words which have a derivation where the word of any level of the corresponding derivation tree belongs to a given regular language. In this paper, we consider first as control sets such regular languages which can be represented by finite unions of monoids. We show that the corresponding hierarchy of tree controlled languages collapses already at the second level. Second, we restrict the number of states allowed in the accepting automaton of the regular control language. We prove that the associated hierarchy has at most five levels.

On Two Hierarchies of Subregularly Tree Controlled Languages

2008

Chapter 2 Languages , Decidability , and Complexity

2012

Control problems for discrete-event systems or hybrid systems typically involve manipulation of languages that describe system behaviors. This chapter introduces basic automata and grammar models for generating and analyzing languages of the Chomsky hierarchy, as well as their associated decision problems, which are necessary for the understanding of other parts of this book. Notions of decidability of a problem (that is, is there an algorithm solving the given problem?) and of computational complexity (that is, how many computation steps are necessary to solve the given problem?) are introduced. The basic complexity classes are recalled. This chapter is not intended to replace a course on these topics but merely to provide basic notions that are used further in this book, and to provide references to the literature. In the following, we introduce the basic terminology, notation, definitions, and results concerning the Chomsky hierarchy of formal languages to present the necessary p...

Hierarchies and reducibilities on regular languages related to modulo counting

RAIRO - Theoretical Informatics and Applications, 2009

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

A Predicative Harmonization of the Time and Provable Hierarchies

Corr, 2006

A decidable transfinite hierarchy is defined by assigning ordinals to the programs of an imperative language. It singles out: the classes TIMEF(n c ) and TIMEF(n c ); the finite Grzegorczyk classes at and above the elementary level, and the Σ k -IND fragments of PA. Limited operators, diagonalization, and majorization functions are not used.

On decidability and axiomatizability of some ordered structures

Soft Computing

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.