Completeness

Of a proof system. A proof system is complete if, for every TRUE statement, the Prover can always produce an accepting proof.

Completeness is a property of a proof system best understood as: “an honest Prover should always be able to convince an honest Verifier of a valid statement”.

Given a relation $\mathcal{R}$ and the associated language ${\mathcal{L}}_{\mathcal{R}}$, the mathematical expression for completeness looks often like the equation below: $$\forall x\in {\mathcal{L}}_{\mathcal{R}},\mathrm{Pr}\left[⟨P,V⟩(x)=1\right]=1$$ where $P$ and $V$ are the honest Prover and Verifier respectively, and $⟨P,V⟩(x)$ denotes the bit output by $V$ at the end of the interaction with $P$ for the instance $x$.

Perfect completeness vs Completeness.
The definition above imposes that the honest prover always convince the honest verifier (i.e. the probability is 1). We call this property perfect completeness. Sometimes this is not necessary, nor is it achievable. In those cases, we can relax the success probability to be greater than $1-\delta$, where $\delta$ is something small.