Question

What does this general grammar do?

S -> LR 
L -> L0Y 
L -> LX 
X1 -> 1X 
X0 -> 0X 
X0 -> 1Y 
Y1 -> 0Y 
YR -> R     
L  -> epsilon 
R  -> epsilon

the start symbol is S. I tried to generate string from this grammar and I got every binary numbers. but I think it does something specific.

Answer 1

S -> LR 
L -> L0Y 
L -> LX 
X1 -> 1X 
X0 -> 0X 
X0 -> 1Y 
Y1 -> 0Y 
YR -> R     
L  -> epsilon 
R  -> epsilon

terminals: 0,1
start: S

Let’s split the grammar:

S -> LR 
L -> L0Y 
L -> LX

This will generate a string in the form L, string of X and 0Y, R.

X1 -> 1X 
X0 -> 0X 
X0 -> 1Y 
Y1 -> 0Y 
YR -> R

Treat X and Y as acting on the binary string: X will propagate to the right, then change a 0 to 1 and all subsequent 1s to 0s. In effect, a single X increments the binary number without changing its string length (or gets stuck).

A leading Y will rewrite the string of all 1s to all 0s (or gets stuck).

Treat the rules for L as the possible actions on the right part of the string. L => L0Y will reset the string from all ones to all zeroes and increase its length by one. L => LX will increment any other number, but fails if the value is at the maximum.

These two actions together are sufficient to generate (inefficiently) all strings of zeroes and ones (including the empty string).

L  -> epsilon 
R  -> epsilon

will only clean up the sentinels.

one possible description of the language within four words:

set of all strings