Simulation-based proofs - advice tape of reduction algorithm I am currently reading Yehuda Lindell's amazing tutorial on simulation proof technique and trying to write my own proof for the first time. In page 4 of this tutorial it is mentioned that 


  [...] The value $a$ must therefore be written on the advice tape of the reduction algorithm.


and I also knew that $a$ in context of secure computation represents the parties' inputs (as prof. Lindell said in page 3 of the tutorial).

For proving security of oblivious transfer in semi-honest case (page 13), reduction algorithm $A$ is given $\sigma , b_\sigma$ on its advice tape. Could I also give $b_{1-\sigma}$ (other party's input) to $A$ on its advice tape if needed (which is in my case)?
Yes.

We work under the assumption that the protocol is not secure, which means that for infinitely many $n$, there is a pair of inputs $(x,y)$ on which the simulation fails (i.e., when the inputs are $x$ and $y$ the view generated by the simulator for at least one of the parties can be distinguished from a real one). Thus, when running on security parameter $n$, all non-uniform algorithms can be given $(x,y)$ as advice.

By the way, the reason why adversaries are only given $\sigma$ and $b_\sigma$ in this case (instead of $\sigma$ and $(b_\sigma,b_{1-\sigma})$) is that the given simulator always succeeds when $b_{1-\sigma} = 0$, so since we give an input pair where the simulator fails, it must be that $b_{1-\sigma} = 1$ and there is no need to give it explicitly.