Include challenge in input MLE
question 1: the challenge in logup denominators are really necessary to be in MLE?
yes, this challenge is irrelevant to the GKR protocol (don’t mess it with the challenges in sum-check and challenges for later random linear combination of MLE evaluations), \(\frac{1}{\alpha +x}\) \(\alpha\) guarantees the prover can not manipulate the table values, it is needed to be include in input MLE
question 2: if denominator includes challenge, what is the representation of the input MLE then? and how it affect the trace commitment?
if no challenge the input layer looks like \(\frac{1}{x_0},\frac{1}{x_1},\frac{1}{x_2},…\)\
the MLE evaluations on hypercube is \(x_0,x_1,x_2,…\)
The MLE commitment is directly the trace itself
add challenge:
include challenge the input layer looks like \(\frac{1}{\alpha+x_0},\frac{1}{\alpha + x_1},\frac{1}{\alpha+x_2},…\)\
the MLE evaluations on hypercube is \(\alpha+x_0,\alpha+x_1,\alpha+x_2,…\)
The MLE commitment is \(alpha+\) trace
the gkr_trace is not affected, the evaluation of MLE is \(\alpha+\) trace mle claim
so the important part is when get mle evaluation oracle from trace component, the result should plus \(alpha\)