Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for NP and NQP
Cody D. Murray, Ryan Williams · 2018
We prove that if every problem in NP has nk-size circuits for a fixed constant k, then for every NP-verifier and every yes-instance x of length n for that verifier, the verifier’s search space has an nO(k3)-size witness circuit: a witness for x that can be encoded with a circuit of only nO(k3) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., NQP = NTIME[nlogO(1) n]. This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [JCSS’02] which only held for larger nondeterministic classes such as NEXP.