CS 21 Decidability and Tractability Lecture 23 March

CS 21 Decidability and Tractability Lecture 23 March 1, 2021 CS 21 Lecture 23 1

Outline • March 1, 2021 CS 21 Lecture 23 2

co. NP • language L is in co. NP iff its complement (co-L) is in NP • it is believed that NP ≠ co. NP • note: P = NP implies NP = co. NP – proving NP ≠ co. NP would prove P ≠ NP – another major open problem… March 1, 2021 CS 21 Lecture 23 3

co. NP • canonical co. NP-complete language: UNSAT = {φ : φ is an unsatisfiable 3 -CNF formula} – proof? March 1, 2021 CS 21 Lecture 23 4

co. NP Disjunctive Normal Form = OR of ANDs • another example 3 -DNF-TAUTOLOGY = {φ : φ is a 3 -DNF formula and for all x, φ(x) =1} – proof? • another example: EQUIV-CIRCUIT = {(C 1, C 2) : C 1 and C 2 are Boolean circuits and for all x, C 1(x) = C 2(x)} – proof? March 1, 2021 CS 21 Lecture 23 5

Quantifier characterization of co. NP • March 1, 2021 CS 21 Lecture 23 6

Proof interpretation of co. NP • “proof” “short” proof “proof verifier” March 1, 2021 CS 21 Lecture 23 7

co. NP • March 1, 2021 CS 21 Lecture 23 8

co. NP • Picture of the way we believe things are: co. NP EXP decidable languages P March 1, 2021 NP CS 21 Lecture 23 9

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PRIMES in NP • March 1, 2021 CS 21 Lecture 23 12

Summary • Picture of the way we believe things are: (decision version of ) FACTORING P March 1, 2021 EXP co. NP decidable languages NP CS 21 Lecture 23 13