 Certificate complexity of a Boolean function

A query certificate fixes some input coordinates to their values in a chosen input and must force the output for every completion. The smallest such size is $C(f,x)$; maximizing over accepted or rejected inputs gives $C_1(f)$ or $C_0(f)$, and maximizing both gives $C(f)$. A one-sided maximum over an empty output class is defined as zero. This is query certificate complexity, distinct from polynomial verification of a witness in NP.