The traditional definition of PCPs have perfect completeness -- If $x\in L$, then the prover can give a proof on which the verifier (on reading constantly many bits) always accepts. Suppose we modify the definition as follows:
A language $L$ is in $iPCP(r,q)$ such that there exists a verifier who tosses $r$ coins and queries $q$ locations of a proof $\pi$ such that:
- If $x\in L$, then there exists a $\pi$ that makes the verifier accept with probability at least $3/4$.
- If $x\notin L$, then for every proof $\pi$, the verifier accepts with probability at most $1/4$.
There is an easy reduction from $3SAT$ to $\text{Max}2SAT$ (problem 5 in this) that gives a simple $iPCP$ for $3SAT$. And by definition, the class is closed under complement so it includes $coNP$ as well. Therefore, it is unlikely that there is a way to convert an $iPCP$ to a $PCP$ with the same parameters (well, that would at least mean $coNP = NP$). (thanks to Peter Shor for pointing this out)
Have such incomplete PCPs been studied earlier?