Consider the following extension to context-free grammars that permits rules to have in the left-hand side, one (or more) terminal on the right side of the non-terminal. That is, rules of the form:
A b -> ...
The right-hand side may be anything, like in context-free grammars. In particular, it is not required, that the right-hand side will have exactly the same terminal symbol at the end. In that case, this extension would be context-sensitive. But the terminal is not just a context. Sometimes, this terminal is called "pushback".
Clearly, this is no longer CFG (type-2). It includes type-1. But what is it? Really type-0 already?
This particular extension is permitted in Definite Clause Grammars dcg in Prolog. (To avoid misunderstandings, I do not consider here Prolog's full extensions. I.e. I assume terminals to come from a finite alphabet and not being arbitrary terms, also I do not consider Prolog's additional arguments that are permitted in DCGs, which also go into type-0 already.)
Edit: Here is a simpler way to describe the extension: Add to a CFG rules of the form
A b -> <epsilon>
To answer my question with respect to Prolog's DCG formalism, this extension is now called a semicontext. See N253 DIN Draft for DCGs 2014-04-08 - ISO/IEC WDTR 13211-3:2014-04-08
Given
a1, [b] --> ... .
a2, [b,b] --> ... .
The terminal-sequence [b] is now a semicontext, as well as the terminal-sequence [b,b].
Would the same terminal sequence now appear at the end of the rule, we would have a context:
a3, [b,b] --> ..., [b,b].
So "semi" means here "half" - similar to a semigroup where half of the algebraic properties of a group hold.