Resolving left-recursion in my grammar

Question

My grammar has a case of left-recursion in the sixth production rule.

I resolved this by replacing Rule 6 and 7 like this:

I couldn't find any indirect left recursions in this grammar.

The only thing that bothers me is the final production rule, which has a terminal surrounded by two non-terminals.

My two questions are:

• Is my resolved left recursion correct?
• Is the final production rule a left recursion? I am not sure how to treat this special case.

Answer 1

Yes, your resolution is correct. You may want to remove the epsilon rule for ease of use, but the accepted strings are correct.

X -> -
X -> -Z
Z -> +
Z -> +Z
Z -> X + Y
... and Y is of the form 0* 1 (no syntax collisions)

As a check, note that you could now replace this final rule with two new rules, one for each expansion of X:

Z -> -  + Y
Z -> -Z + Y

This removes X entirely from the Z rules, and each Z rule would then begin with a terminal.

No, your final production rule is no longer left-recursive. X now must resolve to a string beginning with a non-terminal.

I have to admit, though, I'm curious about what use the language has. :-)