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.

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

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

Regular (type 3) languages

• regular expression
• Kleene star
• Kleene algebra
• left-linear grammar
• right-linear grammar
• finite automaton

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
• $LL(k)$ grammar
• left-factored grammar
• $LR(k)$ grammar
• every $LR(k)$ 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 $LR(1)$ grammar

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

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

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

Connection to group and semigroup theory

• finitely presented group
• automatic group
• semi-Thue system
• Post system
• word problem
• Post correspondence problem
• conjugacy problem

Decidability

• membership problem
• emptiness problem
• recursively enumerable language
• recursive language

Special languages

• Dyck language
• derivation language