Interactive-proof transcripts, rounds, and prover strategies

An interactive protocol on input $x$ is an alternating finite exchange between a probabilistic polynomial-time verifier $V$ (Probabilistic polynomial-time Turing machines) and a prover $P$. A round consists of one verifier message and one prover message; the verifier uses at most polynomially many rounds, communicated bits, and random bits in $\mathrm{\mid }x\mathrm{\mid }$.

A public transcript is exactly the sequence of messages exchanged so far, together with any random bits that the verifier has revealed. The verifier's unrevealed random bits and private work-tape state belong to its private history, not to the public transcript. A prover strategy is an arbitrary function assigning a legal next message to every public transcript at a prover turn; it need not be computable or efficient. Consequently two verifier histories that give the prover the same public view must receive the same prover response.