Some classifications of context-free languages (original) (raw)
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On a classification of context-free languages
Kybernetika (Praha), 1967
The set E of strings is said to be definable (strongly definable) if there is a context-free grammar G such that E is the set of all terminal strings generated from the initial symbol (from all non terminal symbols) of G. The classification of definable and strongly definable sets in dependence on minimal number of nonterminal symbols needed for their generation is given.
Journal of the ACM, 1966
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 st...
Oxford Handbooks Online, 2005
This article introduces the preliminaries of classical formal language theory. It outlines the main classes of grammars as language-generating devices and automata as language-recognizing devices. It offers a number of definitions and examples and presents the basic results. It classifies grammar according to several criteria. The most widespread one is the form of their productions. This article presents a systematic study of the common properties of language families has led to the theory of abstract families of languages. It shows that a context-free grammar generates not only a set of strings, but a set of trees too: each one of the trees is associated with a string and illustrates the way this string is derived in the grammar.
On the size of context-free grammars
Kybernetika, 1972
In papers [2] and [3] four criteria of complexity of context-free grammars (CFG's), denoted by Var, Lev, Lev„, and Depth, have been studied. These criteria reflect the intrinsic complexity of CFG's and they induce the criteria of complexity of contextfree languages (CFL's) which reflect the intrinsic complexity of the description of CFL's by CFG's. The criterion Prod (G) = the number of rules of a CFG G, studied in [3] represents the size of CFG's. In the present paper one more criterion of complexity of CFG's, namely Symb (G) = = the number of all occurrences of all symbols in the rules of G, is defined and some results concerning the criteria Prod and Symb are derived.
Complexity and unambiguity of context-free grammars and languages
Information and Control, 1971
Four of the criteria of complexity of the description of context-free languages by context-free grammars are considered. The unsolvability of the basic problems is proved for each of these criteria. For instance, it is unsolvable to determine the complexity of the language generated by a given grammar, or to find out the simplest grammar, or to decide whether a given grammar is the simplest one and so on. Next, it is shown that in some cases one can obtain unambiguity only by increasing complexity. Namely, for each of the four criteria, in any complexity class there are unambiguous languages, all simplest grammars of which are ambiguous. As one would expect, it is unsolvable whether for an arbitrary grammar G there are unambiguous grammars within the simplest grammars for the language generated by G.