On Context-Free Languages

ABSTRACT

In this report, certain properties of context-free (CF or type 2) grammars are investigated, like that of Chomsky. In particular, questions regarding structure, possible ambiguity and relationship to finite automata are considered. The following results are presented: The language generated by a context-free grammmar is linear in a sense that is defined precisely. The requirement of unambiguity—that every sentence has a unique phrase structure—weakens the grammar in the sense that there exists a CF language that cannot be generated unambiguously by a CF grammar. The result that not every CF language is a finite automaton (FA) language is improved in the following way. There exists a CF language L such that for any L′ ⊆ L, if L′ is FA, an L″ ⊆ L can be found such that L″ is also FA, L′ ⊆ L″ and L″ contains infinitely many sentences not in L′. A type of grammar is defined that is intermediate between type 1 and type 2 grammars. It is shown that this type of grammar is essentially stronger than type 2 grammars and has the advantage over type 1 grammars that the phrase structure of a grammatical sentence is unique, once the derivation is given.

REFERENCES

• 1: BAR-HILLEL, Y., PRLES, M., AND SHAMIR, E. On formal properties of simple phrase structure grammars. Z. Phonetik, Sprachwissen. Komm. 15 (I961), 143-172; in Y. Bar-Hillel, Language and Information, Addison-Wesley, Reading, Mass., 1965, pp. 116-150.
• 2: CHOMSKY, N. Three models for the description of lmguage. IRE trans, IT, 3 (1956), 113-.124.
• 3: ----and MILIEh G. A. Fiaite stae languages, Inform, :nd Contr. 1 (19h3), 9b-..112.
• 4: ---. On certai formal propertiets of grammars, inforem. and Contr.2 (1956) 137-167,
• 5: SCHEINBERG, S. On Booleal properties of phrase structure grammars Inform. and Contr, S (1960), 72---375.
• 6: SCHUTZENBERGER, M. P, Some remarks on Chomsky's eontext-free lsguges. MIT Quart, Progr. Rep. 63, Oct. 1961, p, 155ff.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.4 MATHEMATICAL LOGIC AND FORMAL LANGUAGES
      F.4.3 Formal Languages
          Subjects: Classes defined by grammars or automata (e.g., context-free languages, regular sets, recursive sets)

General Terms:
Languages, Performance, Theory, Verification