I'm reading an NLP paper. In section 4.2, there is the following summation:
$$\alpha[t][k] = \sum^{t-k}_{j=1}p(c^t_{t-k+1} | c^{t-k}_{t-k-j+1}) \cdot \alpha[t-k][j]$$
where $\alpha[0][0] = 1$.
$\alpha[t][k]$ is the probability of a string $c_1...c_t$ with the final $k$ characters being a word.
However, I simply don't see when the equation will ever reach $\alpha[0][0]$, since the summation lower bound is $j=1$. Shouldn't the whole summation result in 0 for being an empty sum, when the lower bound is larger than upper bound, according to this answer?
As a result, I have trouble understanding what happens in the case where $t=k$, for example $\alpha[1][1]$ for the word "a" or $\alpha[3][3]$ for the word "she". In that case we arrive at $\alpha[0][j]$ at the right part of the equation, while $j$ must be always larger than 0. If this results in an empty sum, then the possibility of such words must always be 0. It doesn't seem to make much sense.
Did the authors make an error in saying $\alpha[0][0] = 1$ (if yes, what should it actually be. Should it be $\alpha[0][n]=1$?), or did I simply not understand the equation?