Proving that language is regular or not

Question

How to prove that the language over the alphabet $\{0, 1, +, =\}$ is regular or not:

$\{a+b=c:a,b,c \text{ are integers in binary for which } a \text{ plus } b\text{ equals } c\}$

I started with the pumping lemma:

1. |$y$| ≥ 1
2. |$xy$| ≤ $p$
3. for all $i$ ≥ 0, $xy^iz$ ∈ $L$

But I don't know what to do next. How can split "$a+b=c$" string to start with the pumping lemma? Or I should apply another method?

Answer 1

The easiest way is to take the intersection of your language with $1^*+0=1^*$, which is $$ \{ 1^n+0=1^n : n \geq 0 \}. $$ A similar option is to intersect with $1^*+1=10^*$, which gives $$ \{ 1^n+1 = 10^n : n \geq 0 \}. $$ I'll let you finish the argument.