where A is a nonterminal and α, β, and γ are arbitrary words with γ nonempty. If γ was allowed to be empty then any type 0 (equivalently, recursively enumerable) language of the Chomsky hierarchy could be generated. To derive the empty word, a production S → Λ
must also be included, with S not occurring in the right-hand side of any production. The term context-sensitive refers to the fact that A can be rewritten to γ only in the “context” α…β.
In a length-increasing grammar each production has a right-hand side at least as long as its left-hand side (apart possibly from S → Λ). Clearly any context-sensitive grammar is length-increasing, but it can also be shown that any length-increasing grammar is equivalent to a context-sensitive one. Context-sensitive grammars are a class of phrase-structure grammar.
Compare context-free grammar.
"context-sensitive grammar." A Dictionary of Computing. . Encyclopedia.com. (December 13, 2018). https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/context-sensitive-grammar
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