MA protocol is one of the most basic models of interactive proofs.
Merlin is a prover sending a witness $w$ for given input string $x$, and Arthur is a verifier who verifies if $w$ is a positive witness of $x$ using random bits $r$. We say that $L$ is decided by MA protocol $\mathcal{P}$, if the following two conditions hold.
- $x\in L\Rightarrow \exists w\,\Pr_r[\mathcal{P}(x,w,r)=1]>2/3$,
- $x\notin L\Rightarrow \forall w\,\Pr_r[\mathcal{P}(x,w,r)=1]<1/3$.
My question is: What is the formal description of Turing Machine exactly simulating MA protocol with space $S(n)$?