# Why do a pair of interactive machines have a common input?

I know that an interactive proof system $(P,V)$ is a pair of interactive machines for a language $L$ if $V$ is polynomial-time and the following two conditions hold:

1) Completeness: For every $x \in L$, $$\Pr [\langle P,V \rangle (x) = 1] \geq \frac{2}{3}$$

2) Soundness: For every $x \not\in L$ and every interactive machine $B$, $$\Pr [\langle B,V \rangle (x) = 1] \leq \frac{1}{3}$$

Actually, in an interactive proof system $(P,V)$ for language $L$, $P$ proves the fact whether $x \in L$ to $V$.

However, for a general pair of interactive machines, what is the meaning of the common input? Can we construct a system $\langle A,B \rangle$ such that $A$ wants to send $x$ to $B$ and prove that $x$ is not malicious before sending it. Or can we construct other communication systems instead of a proof system?

• You say "But the verifier will not be able to verify that the proof is true before even receiving $x$ (that would make no sense); rather, he will be able to check the proof as soon as he receives the statement." So, without loss of generality, we always assume that the common input contains $x$, right? The verifier always needs the statement when he checks the proof, no matter how the verifier gets the statement. Aug 1 '18 at 5:42