Randomized Computation Nabil Mustafa Computational Complexity 1 / 85 Definition: ZPP The complexity class ZPP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) M(x) = 1 or M(x) = ‘Don’t know’ x 2= L =) M(x) = 0 or M(x) = ‘Don’t know’ 8x; Prob[M(x) = ‘Don’t know’] ≤ 1=2 Whenever M answers with a 0 or a 1, it answers correctly. If M is not sure, it’ll output a ‘Don’t know’. On any input x, it outputs ‘Don’t know’ with probability at most 1=2. Zero-error Probabilistic Polynomial time 2 / 85 x 2= L =) M(x) = 0 or M(x) = ‘Don’t know’ 8x; Prob[M(x) = ‘Don’t know’] ≤ 1=2 Whenever M answers with a 0 or a 1, it answers correctly. If M is not sure, it’ll output a ‘Don’t know’. On any input x, it outputs ‘Don’t know’ with probability at most 1=2. Zero-error Probabilistic Polynomial time Definition: ZPP The complexity class ZPP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) M(x) = 1 or M(x) = ‘Don’t know’ 3 / 85 8x; Prob[M(x) = ‘Don’t know’] ≤ 1=2 Whenever M answers with a 0 or a 1, it answers correctly. If M is not sure, it’ll output a ‘Don’t know’. On any input x, it outputs ‘Don’t know’ with probability at most 1=2. Zero-error Probabilistic Polynomial time Definition: ZPP The complexity class ZPP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) M(x) = 1 or M(x) = ‘Don’t know’ x 2= L =) M(x) = 0 or M(x) = ‘Don’t know’ 4 / 85 Whenever M answers with a 0 or a 1, it answers correctly. If M is not sure, it’ll output a ‘Don’t know’. On any input x, it outputs ‘Don’t know’ with probability at most 1=2. Zero-error Probabilistic Polynomial time Definition: ZPP The complexity class ZPP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) M(x) = 1 or M(x) = ‘Don’t know’ x 2= L =) M(x) = 0 or M(x) = ‘Don’t know’ 8x; Prob[M(x) = ‘Don’t know’] ≤ 1=2 5 / 85 Zero-error Probabilistic Polynomial time Definition: ZPP The complexity class ZPP is the class of all languages L for which there exists a polynomial PTM M such that x 2 L =) M(x) = 1 or M(x) = ‘Don’t know’ x 2= L =) M(x) = 0 or M(x) = ‘Don’t know’ 8x; Prob[M(x) = ‘Don’t know’] ≤ 1=2 Whenever M answers with a 0 or a 1, it answers correctly. If M is not sure, it’ll output a ‘Don’t know’. On any input x, it outputs ‘Don’t know’ with probability at most 1=2. 6 / 85 Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP 7 / 85 Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP 8 / 85 We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP 9 / 85 Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar 10 / 85 Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. 11 / 85 0 I x 2 L: M(x) = 1 or M(x) =‘Don’t know’, so M (x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). 12 / 85 M(x) = 1 or M(x) =‘Don’t know’, so M0(x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). I x 2 L: 13 / 85 or M(x) =‘Don’t know’, so M0(x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’. Let M0 be the Turing Machine that on input x, returns 1 if M(x) =‘Don’t know’ and otherwise returns M(x). I x 2 L: M(x) = 1 14 / 85 , so M0(x) = 1. I x 2= L: M(x) = 0, which is correct, or M(x) =‘Don’t know’ where M0(x) returns incorrectly with probability ≤ 1/2. So L 2 coRP, and ZPP ⊆ coRP A Theorem on ZPP Theorem ZPP = RP \ coRP Claim: ZPP ⊆ RP \ coRP Show that ZPP ⊆ RP and ZPP ⊆ coRP We prove ZPP ⊆ coRP , the other proof is similar Let L 2 ZPP . Then 9 PTM M that, for all x, either correctly decides x 2 L or outputs ‘Don’t know’.