Consider the words

w_{1} = "`(`",

w_{2} = "`((`",

...,

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

...

(w

_{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.