A context-sensitive language is a type of formal language that can be recognized by a context-sensitive grammar. In the Chomsky hierarchy of formal languages, context-sensitive languages are more powerful than regular languages but less powerful than recursively enumerable languages. They are characterized by rules that allow for the manipulation of symbols in a context-dependent manner, taking into account the surrounding symbols and the current state of the derivation.
To understand how a context-sensitive language can be recognized by a context-sensitive grammar, let us first define what a context-sensitive grammar is. A context-sensitive grammar is a formal grammar consisting of a set of production rules that describe how to rewrite symbols in a given context. The context is typically defined by the symbols to the left and right of the symbol being rewritten.
An example of a context-sensitive language is the language of balanced parentheses. This language consists of strings of parentheses, such as "()()", "(())", or "((()))", where the parentheses are properly balanced. In other words, for every open parenthesis, there must be a corresponding closing parenthesis, and they must appear in the correct order.
To recognize this language using a context-sensitive grammar, we can define a set of production rules that enforce the balanced parentheses property. Let us denote the start symbol as S, and the terminal symbols as '(' and ')'.
1. S -> SS: This rule allows for the concatenation of two strings of balanced parentheses.
2. S -> (S): This rule allows for the addition of a pair of parentheses around a string of balanced parentheses.
3. S -> ε: This rule allows for the derivation of the empty string, representing the case where there are no parentheses.
By applying these production rules in a context-sensitive grammar, we can generate strings that represent balanced parentheses. For example, starting with the start symbol S and applying the rules, we can derive the following strings:
S -> SS -> (S)S -> (S)(S) -> ((S))S -> ((S))(S) -> ((S))()
The derivation can be seen as a step-by-step process of rewriting symbols according to the production rules, taking into account the context in which the symbols appear.
A context-sensitive language is a type of formal language that can be recognized by a context-sensitive grammar. The grammar consists of production rules that allow for the manipulation of symbols in a context-dependent manner. An example of a context-sensitive language is the language of balanced parentheses, which can be recognized by a context-sensitive grammar through the use of production rules that enforce the balanced parentheses property.
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