RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes 35 / 93 I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) 36 / 93 coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error 37 / 93 I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) 38 / 93 BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error 39 / 93 I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) 40 / 93 ZPP (Zero-error Probabilistic Polynomial time) I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error 41 / 93 I Zero error Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) 42 / 93 Randomized Complexity Classes RP (Randomized Polynomial time) I One-sided error coRP (complement of RP ) I One-sided error BPP (Bounded-error Probabilistic Polynomial time) I Two-sided error ZPP (Zero-error Probabilistic Polynomial time) I Zero error 43 / 93 Definition: RP The complexity class RP is the class of all languages L for which there exists a polynomial PTM M such that 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 Definition: coRP The complexity class coRP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 Randomized Polynomial time 44 / 93 Definition: coRP The complexity class coRP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 Randomized Polynomial time Definition: RP The complexity class RP is the class of all languages L for which there exists a polynomial PTM M such that 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 45 / 93 Randomized Polynomial time Definition: RP The complexity class RP is the class of all languages L for which there exists a polynomial PTM M such that 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 Definition: coRP The complexity class coRP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 46 / 93 Let PTM M decide a language L 2 RP 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 If M(x) = 1, then x 2 L. If M(x) = 0, then x 2 L or x 2= L, we can’t really tell. Let PTM M decide a language L 2 coRP x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 If M(x) = 0, then x 2= L. If M(x) = 1, then x 2 L or x 2= L, we can’t really tell. Randomized Polynomial time 47 / 93 Let PTM M decide a language L 2 coRP x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 If M(x) = 0, then x 2= L. If M(x) = 1, then x 2 L or x 2= L, we can’t really tell. Randomized Polynomial time Let PTM M decide a language L 2 RP 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 If M(x) = 1, then x 2 L. If M(x) = 0, then x 2 L or x 2= L, we can’t really tell. 48 / 93 Randomized Polynomial time Let PTM M decide a language L 2 RP 1 x 2 L =) Prob[M(x) = 1] ≥ 2 x 2= L =) Prob[M(x) = 0] = 1 If M(x) = 1, then x 2 L. If M(x) = 0, then x 2 L or x 2= L, we can’t really tell. Let PTM M decide a language L 2 coRP x 2 L =) Prob[M(x) = 1] = 1 1 x 2= L =) Prob[M(x) = 0] ≥ 2 If M(x) = 0, then x 2= L. If M(x) = 1, then x 2 L or x 2= L, we can’t really tell. 49 / 93 Theorem RP ⊆ NP Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP 50 / 93 Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP Theorem RP ⊆ NP 51 / 93 Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP Theorem RP ⊆ NP Proof 52 / 93 If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP Theorem RP ⊆ NP Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. 53 / 93 So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP Theorem RP ⊆ NP Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. 54 / 93 If x 2= L, then Prob[M(x) = 1] = 0. So there is no certificate. M rejects x. So L 2 NP RP is in NP Theorem RP ⊆ NP Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine. 55 / 93 So L 2 NP RP is in NP Theorem RP ⊆ NP Proof Let L be a language in RP . Let M be a polynomial probabilistic Turing Machine that decides L. If x 2 L, then there exists a sequence of coin tosses y such that M accepts x with y as the random string. So we can consider y as a certificate instead of a random string. y can be verified in polynomial time by the same machine.