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The language is context-free; however, it can be proved that it is not regular. If the productions S → a, S → b, are added, a context-free grammar for the set of all palindromes over the alphabet { a, b} is obtained. [8]
The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing.Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
The bulk of the text discusses examples of teaching English as a foreign language in various contexts. Example contexts studied in the book include national settings like Senegal, Egypt, Argentina Turkey, Ukraine, Estonia and Vietnam. [2] [3] It ends with conclusions around "English as a global language", teaching, teachers, and other topics. [4]
Ogden's lemma is often stated in the following form, which can be obtained by "forgetting about" the grammar, and concentrating on the language itself: If a language L is context-free, then there exists some number (where p may or may not be a pumping length) such that for any string s of length at least p in L and every way of "marking" p or more of the positions in s, s can be written as
To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory. [4]: 87–94 [5] [6] [7] The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).
Deterministic context-free languages can be recognized by a deterministic Turing machine in polynomial time and O(log 2 n) space; as a corollary, DCFL is a subset of the complexity class SC. [3] The set of deterministic context-free languages is closed under the following operations: [4] complement; inverse homomorphism; right quotient with a ...
The grammar doesn't cover all language rules, such as the size of numbers, or the consistent use of names and their definitions in the context of the whole program. LR parsers use a context-free grammar that deals just with local patterns of symbols. The example grammar used here is a tiny subset of the Java or C language: r0: Goal → Sums eof
DECLARE ARRAY S; function INIT (words) S ← CREATE_ARRAY (LENGTH (words) + 1) for k ← from 0 to LENGTH (words) do S [k] ← EMPTY_ORDERED_SET function EARLEY_PARSE (words, grammar) INIT (words) ADD_TO_SET ((γ → • S, 0), S [0]) for k ← from 0 to LENGTH (words) do for each state in S [k] do // S[k] can expand during this loop if not FINISHED (state) then if NEXT_ELEMENT_OF (state) is a ...