Context-free grammar (CFG) is a formal way of describing the syntax of a language. It is a set of production rules that generate strings of symbols based on a given starting symbol. CFGs are widely used in computer science, especially in the field of compilers and programming languages.
A context-free grammar consists of four components:
1. A set of non-terminal symbols, which represent syntactic categories or placeholders for other symbols.
2. A set of terminal symbols, which represent the basic building blocks of the language, such as keywords, operators, and identifiers.
3. A set of production rules, which specify how non-terminal symbols can be expanded into sequences of terminal and/or non-terminal symbols.
4. A starting symbol, which is the initial non-terminal symbol from which the grammar generates all valid strings in the language.
Closure properties of CFG:
1. Union: If L1 and L2 are two context-free languages, then their union L1 ∪ L2 is also a context-free language.
Example: Consider the following two context-free grammars:
G1: S -> aSb | epsilon G2: S -> cSc | d
The union of these two grammars can be defined as:
G: S -> aSb | cSc | d | epsilon
2. Concatenation: If L1 and L2 are two context-free languages, then their concatenation L1.L2 is also a context-free language.
Example: Consider the following two context-free grammars:
G1: S -> aSb | epsilon G2: T -> cTd | e
The concatenation of these two grammars can be defined as:
G: S -> aTB | epsilon T -> cTd | e B -> cTd | e
3. Kleene Closure: If L is a context-free language, then its Kleene closure L* is also a context-free language.
Example: Consider the following context-free grammar:
G: S -> aSb | epsilon
The Kleene closure of this grammar can be defined as:
G*: S -> aSb | epsilon
4. Homomorphism: If L is a context-free language and h is a homomorphism, then h(L) is also a context-free language.
Example: Consider the following context-free grammar:
G: S -> aSb | c
The homomorphism can be defined as:
h(a) = x h(b) = y h(c) = z
The homomorphic image of G can be defined as:
G': S' -> xS'y | z
In conclusion, context-free grammars are a powerful tool for describing the syntax of a language. They consist of production rules that specify how non-terminal symbols can be expanded into sequences of terminal and/or non-terminal symbols. The closure properties of context-free grammars allow us to combine and manipulate context-free languages in various ways, which is useful for developing compilers and programming languages.
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