case, one can consider a combinational operator $ (S+B)_j $ which regards an **observability parameter** $ \alpha $ for a given environment that is some portion $ \alpha $ of all percepts to be fully observable and the rest of them be partially observable. In other words, a given world percept is visible with the probability of $ \alpha $ and is invisible with the probability of $1 - \alpha $.

$$\alpha \, S_j + ( 1 - \alpha ) \, B_j =  \alpha \, | s_j \rangle \langle s_j | \, + \, ( 1 - \alpha ) \,   |  b_j \rangle \langle b_j | ,$$

$$r^{(t)} = \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*} | \, \alpha \, S_j + ( 1 - \alpha ) \, B_j \, | s_i \rangle ,$$

$$r^{(t)}= \frac{\alpha }{N} \sum_{i=1}^N \, \langle a_{s_i}^{*} | {s_i} \rangle \, + \, \frac{1 - \alpha }{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*} | b_j \rangle \, \langle b_j | s_i \rangle .
\label{kabli}$$

Further, we could include the imaginary space to define our generic belief state $ | (S+B)_{j} \rangle $ as $$| (S+B)_{j} \rangle = \sqrt{\alpha} \, | s_{j} \rangle  + i \, \sqrt{1 - \alpha 