Name for terminals on the left-hand side of grammar rules? [closed]

Consider rules as they are used for context-sensitive languages:

$\alpha A \beta \rightarrow \alpha \gamma \beta$

If $\alpha$ is always empty, we have right-context sensitive grammars:

$A \beta \rightarrow \gamma \beta$

So $\beta$ is here the context. But now consider rules of the form

$A \beta \rightarrow \gamma$

(We are now beyond context-sensitive languages.) Is there a name for this $\beta$? It is not a context, but almost. I am particularly interested if $\beta$ is restricted to zero, one or more terminals.

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closed as not a real question by KavehNov 1 '12 at 20:50

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If the β part changes by applying this rewriting rule, what do you mean by saying that it consists of terminals? It does not look like a context to me at all. I would just call β “the string obtained by dropping the first symbol from the left-hand side of a rewriting rule.” –  Tsuyoshi Ito Oct 19 '12 at 22:13
I did not say “dropping β” at all. β is just one of the substrings of LHS of a rewriting rule, and it does not have any special meaning. –  Tsuyoshi Ito Oct 20 '12 at 14:00
Unless I'm missing something, if you allow β to be the empty string, your rules are equivalent to the rules of an unrestricted grammar. –  Antonio Valerio Miceli-Barone Oct 24 '12 at 1:39
@Antonio Valerio Miceli-Barone: No. If β is empty, then the rule is of the form A→γ, that is, context-free. If β is nonempty, then the rule can be essentially anything. –  Tsuyoshi Ito Oct 26 '12 at 17:30
Perhaps, undrstanding your intention of seeking a name for this could help better in giving a sensible answer. So, what's the context of your question? Perhaps, more info about "Definite Clause Grammar", which you have mentioned, would clarify your question. The correct general answer has already been given by Tsuyoshi Ito. –  imz -- Ivan Zakharyaschev Oct 27 '12 at 1:15