theory of formal languages

Note: This entry is very rough at the moment, and requires work. I mainly wrote it to help motivate other entries and to organize entries on this topic and point out holes in our coverage. Right now, it is mainly a list of entries, many of which have not been written yet. Under the first heading, there is short paragraph. Eventually, there should be such a paragraph under each entry and a bibliography at the end. However, this is a lot of work for one person, so this entry is world editable in the hope that others who are knowledgable in this topic will contribute their expertise.

1 Basic concepts and terminology

Loosely speaking, a formal languageMathworldPlanetmath is a languagePlanetmathPlanetmath whose structrure can be specified with mathematical precision. The study of formal languages is not only interesting as a mathematical discipline in its own right, but also because of its relevance to the foundations of mathematics, its applications, and surprising connections with other branches of mathematics.

2 Classification of languages

3 Regular (type 3) languages

4 Context-free (type 2) languages

5 Context-sensitive (type 1) languages

  • length-increasing grammar

  • a language is context-sensitive iff it can be generated by a length-increasing grammar

  • a language is context-sensitive iff it can be recognized by a linear bounded automaton

6 Phrase-structure (type 0) languages

  • recursively enumerable language, co-recursively enumerable language

  • language that is neither recursively enumerablePlanetmathPlanetmath, nor co-recursively enumerable

  • every phrase-structure language is recursively enumerable

7 Other types of languages and automata that describe them

  • star-free language versus aperiodic finite automaton

  • a star-free language is regularPlanetmathPlanetmathPlanetmath, but not conversely

  • mildly context-sensitive language versus embedded pushdown automaton

  • tree-adjoining grammar

  • languages generated by tree-adjoining grammars are exactly the mildly context-sensitive languages

  • a context-free language is mildly context-sensitive, but not conversely

  • indexed language versus nested stack automaton

  • a mildly context-sensitive language is indexed, but not conversely

  • an indexed language is context-sensitive, but not conversely

8 Connection to group and semigroup theory

9 Decidability

  • membership problem

  • emptiness problem

  • recursively enumerable language

  • recursive language

10 Special languages

Title theory of formal languages
Canonical name TheoryOfFormalLanguages
Date of creation 2013-03-22 15:06:37
Last modified on 2013-03-22 15:06:37
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 16
Author rspuzio (6075)
Entry type Topic
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Classification msc 68Q70
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