1 . 3 Varieties of languages and varieties of finite monoids

نویسنده

  • Jean-Éric Pin
چکیده

In 1980, 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 the IFIP Congress in 1980, 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 dot-depth 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]. 1 Terminology, notation and background This goal of this section is to fix notation and terminology. We define in particular the notions of syntactic monoid, class of languages, variety, profinite word, profinite identity, semiring and weighted automaton. In the sequel, A denotes a finite alphabet and 1 denotes the empty word. A semigroup S divides a semigroup T if S is a quotient of a subsemigroup of T . ∗The author was funded from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 670624).

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Minimal Noncommutative Varieties and Power Varieties

A variety of finite monoids is a class of finite monoids closed under taking submonoids, quotients and finite direct products. A language L is a subset of a finitely generated free monoid. The variety theorem of Eilenberg sets up a one to one correspondence between varieties of finite monoids and classes of languages called, appropriately, varieties of languages. Recent work in variety theory h...

متن کامل

On the Varieties of Languages Associated with Some Varieties of Finite Monoids with Commuting Idempotents

Eilenberg has shown that there is a one-to-one correspondence between varieties of finite monoids and varieties of recognizable languages. In this paper, we give a description of a variety of languages close to the class of piecewise testable languages considered by I. Simon. The corresponding variety of monoids is the variety of J -trivial monoids with commuting idempotents. This result is the...

متن کامل

On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)

Eilenberg correspondence, based on the concept of syntactic monoids, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence related more general classes of regular languages with classes of more complex algebraic objects. Such generalized varieties also have natural counterparts formed by classes of finite automata equipped wi...

متن کامل

Commutative Positive Varieties of Languages

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets. Most monoids considered in this paper are finite. In particular, we use the term variety of monoids for variety of finite monoids. Similarly, all languages considered in this paper are regular languages and hence their syntactic monoid is finite.

متن کامل

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

We present an extension of Eilenberg’s variety theorem, a wellknown result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the orig...

متن کامل

Author: Saeed Salehi (Saeed @ Math.Net) Title: Varieties of Tree Languages

Trees are among the most fundamental and ubiquitous structures in mathematics. Tree languages and automata on trees have been studied extensively since the 1960s from both a purely mathematical and application point of view. When trees are defined as terms, universal algebra becomes directly applicable to tree automata and, on the other hand, the theory of tree automata suggests new notions and...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2016