# Name for grammars with rules $A \to uA$

Recall that a right-linear grammar is a grammar that consists of rules of the form $A\to uB$, where $A$ and $B$ are non-terminals and $u$ is a (possibly empty) word of terminals. Similarly for left-linear grammars. What are the proper name (if any) for the following kinds of grammars:

1. Grammars that consist of rules of the form $A\to uA$, for various non-terminals $A$ and terminal words $u$ (notice that the non-terminal in the "antecedent" and the "succedent" of each rule is the same; but of course, different rules may have different non-terminals). Uniform? Identical?
2. Grammars that consist of rules of the form $A\to uA$ and $B\to Bv$, for various non-terminals $A,B$ and terminal words $u,v$ (i.e., it is a mix of left and right rules, but each rule of the kind mentioned in item 1 above).

Note that any grammar of each of the above two kinds produces a regular language — in the sense that the set of all words (in the alphabet of all symbols, both terminal and non-terminal) derivable from a given non-terminal symbol is a regular language.

• Unless you allow multiple initial non-terminals, how are all these rules reached? If you always start from the same non-terminal, all but one rule in (1) and (2) seem superfluous...
– fgp
Mar 28, 2014 at 23:42
• You are right, if we consider grammars per se. However, I'm particularly interested in modal logics induced by grammars (so called grammar logics). For logics, this aspect is not so important. So these are more like semi-Thue systems (but with distinction between terminals and non-terminals, which is absent in semi-Thue systems). The question can be reformulated as: how do we call rules of the form $A\to uA$? Mar 29, 2014 at 8:31

$A \rightarrow uA$
is called right-recursive. They are commonly used in top-down parsers (eg, $LL(1)$ parsers). If the terminal is on the right, the grammar is called left-recursive, and is frequently used in bottom-up parsers (eg, $LR(1)$ parsers). More precisely, as discussed in comments above, any such production has that name, and grammars with such productions exhibit left-(right-) recursion. This is in the context of compilers at least - see the Dragon Book or Programming Languages Pragmatics -, but probably the same applies to the general theory of pushdown automata.