Lecture 9 : NL = co - NL & Boolean Circuits

Proof Since PATH is co-NL-complete, it suffices to prove that PATH ∈ NL. We’ll use the the read-once certificate definition of NSPACE(·). Let Ci def = {vertices reachable from s with ≤ i steps}. Clearly, C0 = {s}. To prove “vertex x ∈ V can be reached in at most i steps starting from s” (x ∈ Ci), it’s easy by providing the path as a certificate: s, v1, · · · , v`, x where ` < i. The verifier simply checks the vertex chains: (s, v1), (v1, v2), · · · , (v`, x) ∈ G. Intuitive proof idea: We build a “long” (still poly-size) certificate by concatenating a series of n + 1 sub-certificates, each of which depends on its predecessor sub-certificate excepting for the first one: