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The original formulation of the 3 restricted grammar types of Chomsky all included the restriction that the right-hand side of a replacement cannot be $\epsilon$ (non-contracting). This, however, can be (and usually is) lifted for regular and context-free grammars (i.e. they are allowed to have productions of the form $X \to \epsilon$) without altering the class of languages generated.

The rule, however, remains for context-sensitive grammars.

My question is, given a grammar with productions $\alpha X \beta \to \alpha \xi \beta$ where $\alpha, \beta, \xi\in (N \cup T)^*$ (i.e. a context-sensitive grammar with $\epsilon$-rules), what class of languages does that describe? The recursively enumerable ones (same as unrestricted grammars) or something else?

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It can be shown that the context sensitive grammars you describe are equivalent to just asking for productions $\alpha \to \beta$ with $\lvert \alpha \rvert \le \lvert \beta \rvert$. So allowing productions $A \to \varepsilon$ just removes the restriction on lengths, and you have landed on unrestricted grammars.

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  • $\begingroup$ Appreciate the answer. Do you know if there's a book or article or something similar that has this proven? $\endgroup$ – John Doe the Righteous Sep 3 at 13:32

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