Notes on Complexity Theory Last updated : September , 2011 Lecture 8 Jonathan

MAX-IND-SET def = {(G, k) : the largest independent set in G has size exactly k} . This language does not appear to be in NP: we can certify that some graph G has an independent set of size k, but how do we certify that this is the largest independent set in G? The language does not appear to be in coNP, either: although we could prove that (G, k) 6∈ MAX-IND-SET if G happened to have an independent set of size larger than k, there is no easy way to prove that (G, k) 6∈ MAX-IND-SET in case its largest independent set has size smaller than k. As another example, consider the problem of CNF-formula minimization. A CNF formula φ on n variables naturally defines a function fφ : {0, 1}n → {0, 1}, where fφ(x) = 1 iff the given assignment x satisfies φ. Can we tell when a given formula is minimal? Consider the language