Noncontracting grammar

In formal language theory, a grammar is noncontracting (or monotonic) if for all of its production rules, α → β (where α and β are strings of nonterminal and terminal symbols), it holds that A grammar is essentially noncontracting if there may be one exception, namely, a rule S → ε where S is the start symbol and ε the empty string, and furthermore, S never occurs in the right-hand side of any rule.
 * α| ≤ |β|, that is β has at least as many symbols as α.

A context-sensitive grammar is a noncontracting grammar in which all rules are of the form αAβ → αγβ, where A is a nonterminal, and γ is a nonempty string of nonterminal and/or terminal symbols.

However, some authors use the term context-sensitive grammar to refer to noncontracting grammars in general.

A noncontracting grammar in which |α| < |β| for all rules is called a growing context-sensitive grammar.

History
Chomsky (1959) introduced the Chomsky hierarchy, in which context-sensitive grammars occur as "type 1" grammars; general noncontracting grammars do not occur.

Chomsky (1963) calls a noncontracting grammar a "type 1 grammar", and a context-sensitive grammar a "type 2 grammar", and by presenting a conversion from the former into the latter, proves the two weakly equivalent.

Kuroda (1964) introduced Kuroda normal form, into which all noncontracting grammars can be converted.

Example
This grammar, with the start symbol S, generates the language { anbncn : n ≥ 1 }, which is not context-free due to the pumping lemma.

A context-sensitive grammar for the same language is shown below.

Expressive power
Every context-sensitive grammar is a noncontracting grammar.

There are easy procedures for
 * bringing any noncontracting grammar into Kuroda normal form, and
 * converting any noncontracting grammar in Kuroda normal form into a context-sensitive grammar.

Hence, these three types of grammar are equal in expressive power, all describing exactly the context-sensitive languages that do not include the empty string; the essentially noncontracting grammars describe exactly the set of context-sensitive languages.

A direct conversion
A direct conversion into context-sensitive grammars, avoiding Kuroda normal form:

For an arbitrary noncontracting grammar (N, Σ, P, S), construct the context-sensitive grammar (N’, Σ, P’, S) as follows:
 * 1) For every terminal symbol a ∈ Σ, introduce a new nonterminal symbol [a] ∈ N’, and a new rule ([a] → a) ∈ P’.
 * 2) In the rules of P, replace every terminal symbol a by its corresponding nonterminal symbol [a]. As a result, all these rules are of the form X1...Xm → Y1...Yn for nonterminals Xi, Yj and m≤n.
 * 3) Replace each rule X1...Xm → Y1...Yn with m>1 by 2m rules:


 * X1 || X2 || ... || Xm-1 || Xm || || || || → || Z1 || X2 || ... || Xm-1 || Xm
 * Z1 || X2 || ... || Xm-1 || Xm || || || || → || Z1 || Z2 || ... || Xm-1 || Xm
 * Z1 || Z2 || ... || Xm-1 || Xm || || || || → || Z1 || Z2 || ... || Zm-1 || Xm
 * Z1 || Z2 || ... || Zm-1 || Xm || || || || → || Z1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn
 * Z1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn      || →        || Y1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Y2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Ym || Ym+1 || ... || Yn
 * }
 * where each Zi ∈ N’ is a new nonterminal not occurring elsewhere.
 * Z1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn      || →        || Y1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Z2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Y2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Ym || Ym+1 || ... || Yn
 * }
 * where each Zi ∈ N’ is a new nonterminal not occurring elsewhere.
 * Y1 || Y2 || ... || Zm-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn
 * Y1 || Y2 || ... || Ym-1 || Zm || Ym+1 || ... || Yn || → || Y1 || Y2 || ... || Ym-1 || Ym || Ym+1 || ... || Yn
 * }
 * where each Zi ∈ N’ is a new nonterminal not occurring elsewhere.
 * }
 * where each Zi ∈ N’ is a new nonterminal not occurring elsewhere.

For example, the above noncontracting grammar for { anbncn | n ≥ 1 } leads to the following context-sensitive grammar (with start symbol S) for the same language: