zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Distance desert automata and the star height problem. (English) Zbl 1082.20041

Let Σ be a given alphabet. The ‘star height’ of a rational expression over Σ is defined inductively as follows: 𝐬𝐡():=0; 𝐬𝐡(a):=0 for all aΣ; and if r and s are regular expressions over Σ, then 𝐬𝐡(rs)=𝐬𝐡(rs):=max{𝐬𝐡(r),𝐬𝐡(s)}, and 𝐬𝐡(r * ):=𝐬𝐡(r)+1. A language L is called ‘recognizable’ over Σ if L=L(r) for some regular expression r over Σ.

A natural classification of such languages is obtained as follows: for every nonnegative integer k, define k :={L(r)𝐬𝐡(r)k}.

The star height of a recognizable language L, 𝐬𝐡(L), is the smallest k such that L k . The language classes 0 , 1 , form a hierarchy that was shown by Eggan to be infinite. In other words, for every k, k k+1 but k k+1 . Eggan’s proof, which uses an alphabet of cardinality 2 k+1 -1, got improved by Dejean and Schützenberger who used an alphabet of two letters.

An outstanding problem is whether one can decide if a language has star height k. This is equivalent to the question “Is k decidable?” or “Does there exist an algorithm which enables us to test if a language is or is not in k ?”. This is referred to as the ‘star height problem’ which was raised by Eggan in 1963 and which has received a lot of attention in the past several years. The class 0 consists of the finite languages, and Hashiguchi answered the question positively in 1982 for star height one, and in 1988 for arbitrary star height.

Hashiguchi’s proof of the star height problem is very complicated. In this very interesting paper, the author gives a new proof by reducing the star height problem to the limitedness of his so-called ‘nested distance desert automata’. This notion is introduced as a generalization of both the notion of ‘distance automata’ previously introduced by Hashiguchi and the notion of ‘desert automata’ previously introduced independently by Bala and the author. Such automata compute mappings from the free monoid over Σ to the set of positive integers. They are called ‘limited’ if the range of the computed mapping is finite.

Limitedness of nested distance desert automata turns out to be PSPACE-complete. The author’s construction gives the first upper complexity bound for the star height problem. More precisely, he shows that given a nonnegative integer k, it is decidable in 2 2 𝒪(n) space whether the language accepted by an n-state nondeterministic automaton has star height less than k. (Also submitted to MR.)

20M35Semigroups in automata theory, linguistics, etc.
68Q17Computational difficulty of problems
68Q70Algebraic theory of languages and automata
20M05Free semigroups, generators and relations, word problems