theory of formal languages

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 language

is a

language

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.

•

alphabet

•

automaton

•

concatenation

•

derivation

•

grammar

•

homomorphism of languages

•

isomorphism

of languages

•

language

•

abstract family of languages

(AFL)

•

reversal

or mirror image

•

initial symbol

•

production

•

rewriting rule

•

semantics

•

syntax

•

terminal symbol

•

non-terminal symbol

•

word

2

Classification of languages

•

Chomsky Hierarchy

•

regular language

•

context-free language

•

context-sensitive language

•

phrase-structure language

•

type-3 language

•

type-2 language

•

type-1 language

•

type-0 language

3

Regular (type 3) languages

•

regular expression

•

Kleene star

•

Kleene algebra

•

left-linear

grammar

•

right-linear grammar

•

finite automaton

4

Context-free (type 2) languages

•

Chomsky normal form

•

derivation tree

•

intersection

of context-free and regular languages is context-free

•

leftmost derivation

•

rightmost derivation

•

Greibach normal form

•

pumping lemma

•

pushdown automaton

•

deterministic pushdown automaton

•

a language is context-free iff it can be recognized by a pushdown automaton

•

Earley’s

algorithm

•

ambiguous grammar

•

L

⁢

L

⁢

(

k

)

(

http://planetmath.org/LLk

) grammar

•

left-factored grammar

•

L

⁢

R

⁢

(

k

)

(

http://planetmath.org/LRk

) grammar

•

every

L

⁢

R

⁢

(

k

)

grammar can be recognized by a deterministic pushdown
automaton

•

every language which can be recognized by a deterministic pushdown
automaton can be described by an

L

⁢

R

⁢

(

1

)

grammar

5

Context-sensitive (type 1) languages

•

Kuroda normal form

•

length-increasing grammar

•

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

•

linear bounded automaton

•

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

6

Phrase-structure (type 0) languages

•

recursive language

•

recursively enumerable language, co-recursively enumerable language

•

language that is neither

recursively enumerable

, 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

regular

, 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

•

finitely presented group

•

automatic group

•

semi-Thue system (

http://planetmath.org/SemiThueSystem

)

•

Post system

•

word problem

•

Post correspondence problem

•

conjugacy problem

9

Decidability

•

membership problem

•

emptiness problem

•

recursively enumerable language

•

recursive language

10

Special languages

•

Dyck language

•

derivation language

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

Classification

msc 68R15

Classification

msc 68Q70

Classification

msc 68Q45

Classification

msc 68Q42

Classification

msc 20M05

Classification

msc 20F10

Classification

msc 08A50

Classification

msc 03D40

Classification

msc 03D05

Classification

msc 03C07

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