Noncontracting grammar

Chomsky (1959) introduced the Chomsky hierarchy, in which context-sensitive grammars occur as "type 1" grammars; general noncontracting grammars do not occur.Chomsky, N. 1959a. On certain formal properties of grammars. Information and Control 2: 137–67. (141–42 for the definitions)

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 .{{cite book | author=Noam Chomsky | contribution=Formal properties of grammar | pages=[https://archive.org/details/handbookofmathem017893mbp/page/n333 323]–418 | url=https://archive.org/details/handbookofmathem017893mbp | editor=R.D. Luce and R.R. Bush and E. Galanter | title=Handbook of Mathematical Psychology | location=New York | publisher=Wiley | volume=II | year=1963 }} Here: pp. 360–363 and 367

Kuroda (1964) introduced Kuroda normal form, into which all noncontracting grammars can be converted.{{cite journal | author=Sige-Yuki Kuroda | title=Classes of languages and linear-bounded automata | journal=Information and Control | volume=7 | number=2 | pages=207–223 | date=June 1964 | doi=10.1016/s0019-9958(64)90120-2| doi-access=free }}

This grammar, with the start symbol S, generates the language

{{nowrap|1={ aⁿbⁿcⁿ : n ≥ 1 } }},{{harvtxt|Mateescu|Salomaa|1997}}, Example 2.1, p. 188

which is not context-free due to the pumping lemma.

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

Every context-sensitive grammar is a noncontracting grammar.

There are easy procedures for

• bringing any noncontracting grammar into Kuroda normal form,{{harvtxt|Mateescu|Salomaa|1997}}, Theorem 2.2, p. 190 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 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 {{nowrap|X₁...X_m}} → {{nowrap|Y₁...Y_n}} for nonterminals X_i, Y_j and m≤n.
3. Replace each rule {{nowrap|X₁...X_m}} → {{nowrap|Y₁...Y_n}} with m>1 by 2m rules:For convenience, the non-context part of left and right hand side is shown in boldface.

::

::where each Z_i ∈ N’ is a new nonterminal not occurring elsewhere.{{harvtxt|Mateescu|Salomaa|1997}}, Theorem 2.1, p. 187{{cite book| author=John E. Hopcroft, Jeffrey D. Ullman| title=Introduction to Automata Theory, Languages, and Computation| year=1979| publisher=Addison-Wesley| isbn=0-201-02988-X| url-access=registration| url=https://archive.org/details/introductiontoau00hopc}} Exercise 9.9, p.230. In the 2003 edition, the chapter on noncontracting / context-sensitive languages has been omitted.

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:

• Context-sensitive grammar
• Growing context-sensitive grammar
• Kuroda normal form

{{reflist|group=note}}

{{reflist}}

• {{cite journal | last1 = Book | first1 = R. V. | title = On the structure of context-sensitive grammars | doi = 10.1007/BF00976059 | journal = International Journal of Computer & Information Sciences | volume = 2 | issue = 2 | pages = 129–139 | year = 1973 | hdl = 2060/19710024701 | s2cid = 31699138 | hdl-access = free }}
• {{cite book |last1=Mateescu | first1=Alexandru |last2=Salomaa|first2=Arto |editor1-first=Grzegorz| editor1-last=Rozenberg|editor2-first=Arto| editor2-last=Salomaa |title=Handbook of Formal Languages. Volume I: Word, language, grammar |publisher=Springer-Verlag |year=1997 |pages=175–252 |chapter=Chapter 4: Aspects of Classical Language Theory |isbn=3-540-61486-9}}

{{Formal languages and grammars|state=collapsed}}

Category:Formal languages