Techniques for establishing star height of regular sets

Cohen, Rina

doi:10.1007/bf01702866

Cited by 10 publications

(34 citation statements)

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“…Dejean and Schützenberger [20] first proved in 1966 that for each n 0, there exists a language of restricted star height n. Only a few partial results [11,12,13,14,37] were known until Hashiguchi [33] proved in 1982 that restricted star height one is decidable. A few years later, in 1988, Hashiguchi [34] succeeded to prove the general case: restricted star height is decidable.…”

Section: Problem 4 Is There An Algorithm Which Computes the Restrictementioning

confidence: 99%

Open Problems About Regular Languages, 35 Years Later

2017

The Role of Theory in Computer Science

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In December 1979, Janusz A. Brzozowski presented a selection of six open problems about regular languages and mentioned two other problems in the conclusion of his article. These problems have been the source of some of the greatest breakthroughs in automata theory over the past 35 years. This survey article summarizes the state of the art on these questions and the hopes for the next 35 years. Thirty-five years ago, at a conference in Santa Barbara in December 1979, Janusz A. Brzozowski [8] presented a selection of six open problems about regular languages and mentioned two other topics in the conclusion of his article. These six open problems were, in order, star height, restricted star height, group complexity, star removal, regularity of non-counting classes and optimality of prefix codes. The two other topics were the limitedness problem and the dotdepth hierarchy. These problems proved to be very influential in the development of automata theory and were the source of critical breakthroughs. The aim of this paper is to survey these results, to describe their impact on current research and to outline some hopes for the next thirty-five years. Due to the lack of space, the dot-depth hierarchy is treated in a separate article [61].

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Section: Problem 4 Is There An Algorithm Which Computes the Restrictementioning

confidence: 99%

Open Problems About Regular Languages, 35 Years Later

2017

The Role of Theory in Computer Science

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“…restriction of the homomorphism ϕ in the proof of Theorem 2.9. We divide N k into three main sets N (1) , N (2) , N (3) as follows:…”

Section: On the Chomsky-schützenberger Theoremmentioning

confidence: 99%

“…r ∪ N (3) then the pair [ i , ] i should embed a left subtree, i.e., the case of the left embedded subtree in the proof of Theorem 2.8. In this case the bracket [ i may have a left, long distance, placement from its pairwise ] i , in t w,D .…”

Section: On the Chomsky-schützenberger Theoremmentioning

confidence: 99%

“…(2) r ∪N (3) (always placed at the left side of the respective core). The structure of the trace-word of a word w ∈ L(G k ), for a grammar G k in linear-Dyck normal form, is depicted in (1), where by vertical lines we emphasize the image through the homomorphism ϕ of each bracket occurring in t w,D .…”

Section: On the Chomsky-schützenberger Theoremmentioning

confidence: 99%

“…If from v to v there is no other vertex, then the loop is a self-loop. The cycle rank of a graph is a measure of the loop complexity formally defined 6 and studied in [3] and [7]. In [7] it is proved that from each two vertices u and v belonging to a digraph of cycle rank k, there exists a regular expression of star-height 7 at most k that describes the set of paths from u to v. On the other hand, the cycle rank of a digraph with n vertices is upper bounded by n log n [13].…”

Section: On the Chomsky-schützenberger Theoremmentioning

confidence: 99%

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Around Context-Free Grammars - a Normal Form, a Representation Theorem, and a Regular Approximation

Cojocaru¹

2015

Preprint

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We introduce a normal form for context-free grammars, called Dyck normal form. This is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property allows to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism ϕ such that L = ϕ(D K ), where D K ⊆ D K , and D K is the one-sided Dyck language over K letters. Through a transition-like diagram for a context-free grammar in Dyck normal form, we effectively build a regular language R such that D K = R ∩ D K , which leads to the Chomsky-Schützenberger theorem. Using graphical approaches we refine R such that the Chomsky-Schützenberger theorem still holds. Based on this readjustment we sketch a transition diagram for a regular grammar that generates a regular superset approximation for the initial context-free language.

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The solution of two star-height problems for regular trees

Braquelaire

Courcelle

1985

Automata on Infinite Words

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Techniques for establishing star height of regular sets

Cited by 10 publications

References 12 publications

Open Problems About Regular Languages, 35 Years Later

Open Problems About Regular Languages, 35 Years Later

Around Context-Free Grammars - a Normal Form, a Representation Theorem, and a Regular Approximation

The solution of two star-height problems for regular trees

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