Problem Detail: According to Wikipedia, for any regular language $L$ there exist constants $lambda_1,ldots,lambda_k$ and polynomials $p_1(x),ldots,p_k(x)$ such that for every $n$ the number $s_L(n)$ of words of length $n$ in $L$ satisfies the equation $qquad displaystyle s_L(n)=p_1(n)lambda_1^n+dots+p_k(n)lambda_k^n$. The language $L ={ 0^{2n} mid n inmathbb{N} }$ is regular ($(00)^*$ matches it). $s_L(n) = 1$ iff n is even, and $s_L(n) = 0$ otherwise. However, I can not find the $lambda_i$ and $p_i$ (that have to exist by the above). As $s_L(n)$ has to be differentiable and is not constant, it must somehow behave like a wave, and I can’t see how you can possibly do that with polynomials and exponential functions without ending up with an infinite number of summands like in a Taylor expansion. Can anyone enlighten me?
Asked By : Alex ten Brink
Answered By : Patrick87
For your language, can you take $p_0(x) = 1/2$, $lambda_0 = 1$, $p_1(x) = 1/2$, $lambda_1 = -1$, and $p_i(x) = lambda_i = 0$ for $i > 1$? The Wikipedia article doesn’t say anything about the coefficients being either positive or integral. The sum for my choices is $qquad displaystyle 1/2 + 1/2(-1)^n = 1/2 (1 + (-1)^n)$ which seems to be 1 for even $n$, and 0 for odd $n$. Indeed, a proof by induction seems straightforward.
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Question Source : http://cs.stackexchange.com/questions/1039
