Inconsistency on A. Graves' original Connectionist Temporal Classification (CTC) paper? Here's the link to the paper.

For the forward-backward algorithm, they introduce $\alpha_t(s)$ as a definition in eq. (5).
Then they give a recursive formula for it (eq. (6)), and its initialization. However, the two formulae are not equivalent.

For example, let's say we have a labelling $\mathbf{l}=[\text{c},\text{a},\text{t}]$.

$\alpha_1(1)$ using eq. (5):

$\mathcal{B}(\pi_1)=l_1=\text{c}$, so $\pi: \pi_1=\text{c}$ is the only possible path. Therefore $\alpha_1(1)=y_{l_1}^1=y_{\text{c}}^1$ and not $y_\varepsilon^1$ as they state in the initialization (where $\varepsilon$ is the blank label).

Another example. $\alpha_2(1)$ using eq. (5):

$\mathcal{B}(\pi_{1:2})=l_1=\text{c}$, so the possible paths are $\pi: \pi_{1:2}\in\{[\text{c},\text{c}],[\text{c},\varepsilon],[\varepsilon,\text{c}]\}$. Therefore $\alpha_2(1)=y_{\text{c}}^1y_{\text{c}}^2+y_{\text{c}}^1y_{\varepsilon}^2+y_{\varepsilon}^1y_{\text{c}}^2$. However, using equation (6) we get a different result:

We have $l'_1=\varepsilon$ so we are in the first case. Then $\alpha_2(1)=\bar{\alpha}_2(1)y_{l'_1}^2=(\alpha_1(1)+\alpha_1(0))y_\varepsilon^2=(y_\varepsilon^1+0)y_\varepsilon^2=y_\varepsilon^1y_\varepsilon^2$.

I tried many other cases as well, and it seems like, the way they are written, the two formulae are completely different. Am I missing something?
The lecture I take from the paper is not that equation (6) is a recursive form of (5). In (5), the equation refers to label sequence $l$, while (6) refers to $l^\prime$, which is a modified version of $l$ with blanks added to the
beginning and the end and inserted between every pair of labels. See the paragraph between equations (5) and (6):


  To allow for blanks in the output paths, we consider a modified label sequence l, with blanks added to the beginning and the end and inserted between every pair
  of labels. The length of l is therefore 2|l| + 1. In calculating the probabilities of prefixes of l we allow all transitions between blank and non-blank labels, and also those between any pair of distinct non-blank labels.