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 |α| ≤ |β|, that is β has at least as many symbols as α. 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.

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.[1]

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.[2]

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 .[3]

Kuroda (1964) introduced Kuroda normal form, into which all noncontracting grammars can be converted.[4]

Example
S→abc
S→aSBc
cB→Bc
bB→bb

This grammar, with the start symbol S, generates the language { anbncn : n ≥ 1 },[5] 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

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:[note 1]
X1X2...Xm-1 Xm→Z1X2...Xm-1 Xm
Z1X2...Xm-1 Xm→Z1Z2...Xm-1 Xm
 :
Z1Z2...Xm-1 Xm→Z1Z2...Zm-1 Xm
Z1Z2...Zm-1 Xm→Z1Z2...Zm-1 ZmYm+1...Yn
Z1Z2...Zm-1 ZmYm+1...Yn      →      Y1Z2...Zm-1 ZmYm+1...Yn
Y1Z2...Zm-1 ZmYm+1...Yn→Y1Y2...Zm-1 ZmYm+1...Yn
 :
Y1Y2...Zm-1 ZmYm+1...Yn→Y1Y2...Ym-1 ZmYm+1...Yn
Y1Y2...Ym-1 ZmYm+1...Yn→Y1Y2...Ym-1 YmYm+1...Yn
where each Zi ∈ N’ is a new nonterminal not occurring elsewhere.[7][8]

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:

[a]→afrom step 1
[b]→bfrom step 1
[c]→cfrom step 1
S→[a][b][c]from step 2, unchanged
S→[a]SB[c]      from step 2, unchanged
[c]B→B[c]from step 2, further modified below
[c]B→Z1Bmodified from above in step 3
Z1B→Z1Z2modified from above in step 3
Z1Z2      →      BZ2modified from above in step 3
BZ2→B[c]modified from above in step 3
[b]B→[b][b]from step 2, further modified below
[b]B→Z3Bmodified from above in step 3
Z3B→Z3Z4modified from above in step 3
Z3Z4→[b]Z4modified from above in step 3
[b]Z4→[b][b]modified from above in step 3

See also

Notes
  1. For convenience, the non-context part of left and right hand side is shown in boldface.

References
  1. Willem J. M. Levelt (2008). An Introduction to the Theory of Formal Languages and Automata. John Benjamins Publishing. pp. 125–126. ISBN 978-90-272-3250-2.
  2. Chomsky, N. 1959a. On certain formal properties of grammars. Information and Control 2: 137–67. (141–42 for the definitions)
  3. Noam Chomsky (1963). "Formal properties of grammar". In R.D. Luce and R.R. Bush and E. Galanter (ed.). Handbook of Mathematical Psychology. Vol. II. New York: Wiley. pp. 323–418. Here: pp. 360–363 and 367
  4. Sige-Yuki Kuroda (June 1964). "Classes of languages and linear-bounded automata". Information and Control. 7 (2): 207–223. doi:10.1016/s0019-9958(64)90120-2.
  5. Mateescu & Salomaa (1997), Example 2.1, p. 188
  6. Mateescu & Salomaa (1997), Theorem 2.2, p. 190
  7. Mateescu & Salomaa (1997), Theorem 2.1, p. 187
  8. John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Exercise 9.9, p.230. In the 2003 edition, the chapter on noncontracting / context-sensitive languages has been omitted.
  • Book, R. V. (1973). "On the structure of context-sensitive grammars". International Journal of Computer & Information Sciences. 2 (2): 129–139. doi:10.1007/BF00976059. hdl:2060/19710024701. S2CID 31699138.
  • Mateescu, Alexandru; Salomaa, Arto (1997). "Chapter 4: Aspects of Classical Language Theory". In Rozenberg, Grzegorz; Salomaa, Arto (eds.). Handbook of Formal Languages. Volume I: Word, language, grammar. Springer-Verlag. pp. 175–252. ISBN 3-540-61486-9.
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