I try to understand the advantages of using a probabilistic polynomial-time verifier instead of an determininistic one. I use as literature "Arora, Barak: Computational Complexity", in which the class $dIP$ is defined as languages where a deterministic polynomial-time proof system exists. I also undestand that $dIP = NP$. Now, the class $IP$ is defined as class where the languages have an interactive polynomial-time proof system, where the verifier works probabilistic. Shamir proved, that $IP = PSPACE$, this means, all PSPACE problems can be interactive proved in probabilistic polynomial time
Now, I am still asking myself if there are any advantages of class $IP$ without looking at $PSPACE$ problems? By knowing that $dIP = NP \subseteq IP = PSPACE$, I can use a deterministic polynomial-time interactive proof system and also a probabilistic polynomial-time proof system for a language in $NP$. The deterministic version would take one-round, because the verifier can ask the prover for the whole $NP$ witness. But, $NP$ witness are not "short", so, would be a probabilistic system with multiple rounds but shorter size (thus not the complete $NP$ witness) better?
