PCP classes with completeness and soundness

Let $r,q:\mathbb{N}\to \mathbb{N}$ be resource bounds and let $0\le s<c\le 1$ be constants independent of input length. A language $K\subseteq \{0,1{\}}^{\ast }$ belongs to $\mathrm{PCP}(r,q;c,s)$ exactly when there are a verifier $V$ of the type in PCP verifier resources and deterministic proof strings, with randomness bound $r(n)$ and query bound $q(n)$, a fixed finite proof alphabet, and a polynomial $p$ such that its addressable proof length $L_V(n)$ is at most $p(n)$ and the following hold for every input $x$ of length $n$:

• If $x\in K$, there is one fixed proof $\pi$ with $\mathrm{Pr}[V^{\pi }(x)\text{ accepts}]\ge c$.
• If $x\mathrm{\notin }K$, every fixed proof $\pi$ satisfies $\mathrm{Pr}[V^{\pi }(x)\text{ accepts}]\le s$.

Both probabilities are over the verifier's coins; the same deterministic proof is used for every coin string. In the shorthand $\mathrm{PCP}(\log n,O(1))$, the proof alphabet is $\{0,1\}$, the randomness is $O(\log n)$, the number of bit queries is bounded by a constant, completeness is perfect ($c=1$), and soundness is at most some fixed constant $s<1$. Here $O(1)$ means a constant query bound, not exactly one query.