Context-free grammar - Formal definition

In linguistics and computer science, a context-free grammar (CFG) is a formal grammar in which every production rule is of the form V → w where V is a non-terminal symbol and w is a string consisting of terminals and/or non-terminals. The term "context-free" comes from the fact that the non-terminal V can always be replaced by w, regardless of the context in which it occurs. A formal language is context ...

Including:

• Context-free grammar - Formal definition
• Context-free grammar - Examples
• Context-free grammar - Example 1
• Context-free grammar - Example 2
• Context-free grammar - Example 3
• Context-free grammar - Example 4
• Context-free grammar - Other examples
• Context-free grammar - Derivations and syntax trees
• Context-free grammar - Normal forms
• Context-free grammar - Undecidable problems
• Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia - Context-free grammar

Just as any formal grammar, a context-free grammar G can be defined as a 4-tuple: G = (Vt,Vn,P,S) where Vt is a finite set of terminals Vn is a finite set of non-terminals P is a finite set of productions rules S is an e ...

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Context-free grammar, Context-free grammar - Formal definition, Context-free grammar - Examples, Context-free grammar - Example 1, Context-free grammar - Example 2, Context-free grammar - Example 3, Context-free grammar - Example 4, Context-free grammar - Other examples, Context-free grammar - Derivations and syntax trees, Context-free grammar - Normal forms, Context-free grammar - Undecidable problems, Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia II - Context-free grammar - Formal definition

There are basically two ways to describe how in a certain grammar a string can be derived from the start symbol. The simplest way is to list the consecutive strings of symbols, beginning with the start symbol and ending with the string, and the rules that have been applied. If we introduce a strategy such as "always replace the left-most nonterminal first" then for context-free grammars the list of applied grammar rules is by itself sufficient. This is called the leftmost derivation of a string. For example, if we take the follow ...

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Context-free grammar, Context-free grammar - Formal definition, Context-free grammar - Examples, Context-free grammar - Example 1, Context-free grammar - Example 2, Context-free grammar - Example 3, Context-free grammar - Example 4, Context-free grammar - Other examples, Context-free grammar - Derivations and syntax trees, Context-free grammar - Normal forms, Context-free grammar - Undecidable problems, Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia II - Context-free grammar - Derivations and syntax trees

Context-free grammar - Example 1. A simple context-free grammar is S → aSb | ε where | is a logical OR, and is used to separate multiple options for the same non-terminal—ε stands for an empty string. This grammar generates the language which is not regular. Context-free grammar - Example 2. Here is a context-free grammar for syntactically correct infix algebraic expressions in the variables x, y and z: S → x | y | z | S + S | S - S | S * S | S/S | (S) This grammar can, for example, generat ...

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Context-free grammar, Context-free grammar - Formal definition, Context-free grammar - Examples, Context-free grammar - Example 1, Context-free grammar - Example 2, Context-free grammar - Example 3, Context-free grammar - Example 4, Context-free grammar - Other examples, Context-free grammar - Derivations and syntax trees, Context-free grammar - Normal forms, Context-free grammar - Undecidable problems, Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia II - Context-free grammar - Examples

Although some operations on context-free grammars are decidable due to their limited power, unlike finite automata CFGs do have interesting undecidable problems. One of the simplest and most cited is the problem of deciding whether a CFG accepts the language of all strings. A reduction can be demonstrated to this problem from the well-known undecidable problem of determining whether a Turing machine accepts a particular input. The reduction uses the concept of a computation history, a string describing an entire computation of a Turin ...

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Context-free grammar, Context-free grammar - Formal definition, Context-free grammar - Examples, Context-free grammar - Example 1, Context-free grammar - Example 2, Context-free grammar - Example 3, Context-free grammar - Example 4, Context-free grammar - Other examples, Context-free grammar - Derivations and syntax trees, Context-free grammar - Normal forms, Context-free grammar - Undecidable problems, Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia II - Context-free grammar - Undecidable problems

Every context-free grammar which does not generate the empty string can be transformed into an equivalent one in Chomsky normal form or Greibach normal form. "Equivalent" here means that the two grammars generate the same language. Because of the especially simple form of production rules in Chomsky Normal Form grammars, this normal form has both theoretical and practical implications. For instance, given a context-free grammar, one can use the Chomsky Normal Form to construct a polynomial-time algorithm which decides whether a given string is in the language re ...

See also:

Context-free grammar, Context-free grammar - Formal definition, Context-free grammar - Examples, Context-free grammar - Example 1, Context-free grammar - Example 2, Context-free grammar - Example 3, Context-free grammar - Example 4, Context-free grammar - Other examples, Context-free grammar - Derivations and syntax trees, Context-free grammar - Normal forms, Context-free grammar - Undecidable problems, Context-free grammar - Properties of context-free languages

Read more here: » Context-free grammar: Encyclopedia II - Context-free grammar - Normal forms