Consider the words

w(w_{1}= "(",

w_{2}= "((",

...,

w_{n}= "("^{n},

...

_{n}is n open braces). Then, using the notation of the Myhill / Nerode theorem, S(w

_{n}) is different for each n: it contains "

`)`"

^{n}, but no other string of only close braces). Thus there are infinitely many types for S(w), so by the Myhill / Nerode Theorem the balanced braces language is not a regular language.