Knowledge soundness is a meaningful notion when a prover claims that there are *no* satisfying assignments to a system of constraints

Knowledge soundness is a meaningful notion for the sumcheck protocol

Knowledge soundness (as defined in lecture 2) implies soundness

Non-interactive implies publicly verifiable

Vector commitments are as expressive as polynomial commitments

Polynomial extensions are distance amplifying encodings

Multivariate polynomial encoding reduces the total degree by an exponential factor compared to univariate polynomial encoding

The verifier in the sum-check protocol is oblivious to the polynomial g whose evaluations are being summed until the last step

The uniqueness of multilinear extensions is crucial for the soundness of the sumcheck protocol

The claim f(x) = g(x) for all k-bit inputs, where f and g are low degree polynomials over a large field, can be reduced to the following sumcheck claim: \sum_{x \in {0,1}^k} (f(x)-g(x)) = 0

The polynomial IOP for SAT (from lecture 4) can be transformed into a SNARK with verifier complexity independent of circuit size

The polynomial IOP for SAT (from lecture 4) has optimal prover complexity

Extending the polynomial IOP for SAT in the natural way to support gates with n inputs will result in a sumcheck protocol over (n+1) * logS variables