Determine whether the given grammar is contextsensitive,...

Determine whether the given grammar is contextsensitive, context-free, regular, or none of these. Give all characterizations that apply. $T=\{a, b, c\}, N=\{\sigma, A, B\},$ with productions $$ \begin{array}{rcc} \sigma \rightarrow A B, & A B ,\rightarrow B A, & A ,\rightarrow a A ,\\ B \rightarrow B b, & A \rightarrow a, & B, \rightarrow b, \end{array} $$ and starting symbol $\sigma$.

Determine whether the given grammar is contextsensitive, context-free, regular, or none of these. Give all characterizations that apply.
$T=\{a, b, c\}, N=\{\sigma, A, B\},$ with productions $$ \begin{array}{rcc} \sigma \rightarrow A B, & A B ,\rightarrow B A, & A ,\rightarrow a A ,\\ B \rightarrow B b, & A \rightarrow a, & B, \rightarrow b, \end{array} $$ and starting symbol $\sigma$.

Key Concepts

Chomsky Hierarchy

The Chomsky Hierarchy is a classification of formal grammars into types (regular, context?free, context?sensitive, and recursively enumerable) based on restrictions on their production rules. Each level in the hierarchy includes the grammars of the lower levels, with more general grammars having fewer restrictions on their production forms.

Context-Sensitive Grammar

A context-sensitive grammar (CSG) allows productions where the length of the output is never shorter than the input, and the productions can have contexts surrounding the nonterminal being replaced. This makes CSGs more powerful than context-free grammars, and plays a crucial role when productions involve more than a single nonterminal on the left-hand side, as seen in rules that rewrite pairs of symbols.

Context-Free Grammar

A context-free grammar (CFG) restricts productions to have a single nonterminal on the left-hand side. While many familiar languages can be generated by CFGs, any rule that involves more than one symbol (or additional context) on the left-hand side falls outside this category, indicating that not all grammars can be classified as context-free.

Regular Grammar

Regular grammars are the most restricted form within the Chomsky Hierarchy, typically requiring productions to have a very simple form (usually a single terminal possibly followed or preceded by a single nonterminal). Because of these restrictions, any rule that involves multiple symbols or more complex rewriting, such as replacing a pair of nonterminals, disqualifies a grammar from being regular.

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