Probabilistically Checkable Proofs Haris Angelidakis MPLA February 16, 2012 Outline 1 PCP Theorems 2 PCP's and Hardness of Approximation Haris Angelidakis (MPLA) PCP's February 16, 2012 2 / 27 Outline 1 PCP Theorems 2 PCP's and Hardness of Approximation Haris Angelidakis (MPLA) PCP's February 16, 2012 3 / 27 Introduction to PCP's Question: How easy is to check a proof? Immediate answer: At least you have to read the whole proof, and try to check every step in it. Weird question: Can we do better than that? I mean, can we ignore most part of the proof?? Even weirder answer: Yes! Ok, almost yes....:D Haris Angelidakis (MPLA) PCP's February 16, 2012 4 / 27 Introduction to PCP's Question: How easy is to check a proof? Immediate answer: At least you have to read the whole proof, and try to check every step in it. Weird question: Can we do better than that? I mean, can we ignore most part of the proof?? Even weirder answer: Yes! Ok, almost yes....:D Haris Angelidakis (MPLA) PCP's February 16, 2012 4 / 27 Introduction to PCP's Question: How easy is to check a proof? Immediate answer: At least you have to read the whole proof, and try to check every step in it. Weird question: Can we do better than that? I mean, can we ignore most part of the proof?? Even weirder answer: Yes! Ok, almost yes....:D Haris Angelidakis (MPLA) PCP's February 16, 2012 4 / 27 Introduction to PCP's Question: How easy is to check a proof? Immediate answer: At least you have to read the whole proof, and try to check every step in it. Weird question: Can we do better than that? I mean, can we ignore most part of the proof?? Even weirder answer: Yes! Ok, almost yes....:D Haris Angelidakis (MPLA) PCP's February 16, 2012 4 / 27 Introduction to PCP's Question: How easy is to check a proof? Immediate answer: At least you have to read the whole proof, and try to check every step in it. Weird question: Can we do better than that? I mean, can we ignore most part of the proof?? Even weirder answer: Yes! Ok, almost yes....:D Haris Angelidakis (MPLA) PCP's February 16, 2012 4 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 The PCP Idea So, we want to check a proof faster than usual. How is this done? We first rewrite the proof in a certain format, the PCP format. We then check randomly a constant number of its bits: A correct proof always convinces us. A false proof will convince us with probability ≤ 1=2. Detail: The rewriting is completely mechanical and does not greatly increase its size. But, it requires proofs to be written in a formal axiomatic system (such as ZF Set Theory). Haris Angelidakis (MPLA) PCP's February 16, 2012 5 / 27 A nice analogue is the following: Initial Proof PCP transformation PCP Format The surprising main idea In general, a mathematical proof is invalid if it has even a single error somewhere, which can be very difficult to detect. What PCP theorems tell us is that there is a mechanical way to rewrite the proof so that the error is almost everywhere! Haris Angelidakis (MPLA) PCP's February 16, 2012 6 / 27 A nice analogue is the following: Initial Proof PCP transformation PCP Format The surprising main idea In general, a mathematical proof is invalid if it has even a single error somewhere, which can be very difficult to detect. What PCP theorems tell us is that there is a mechanical way to rewrite the proof so that the error is almost everywhere! Haris Angelidakis (MPLA) PCP's February 16, 2012 6 / 27 PCP transformation PCP Format The surprising main idea In general, a mathematical proof is invalid if it has even a single error somewhere, which can be very difficult to detect. What PCP theorems tell us is that there is a mechanical way to rewrite the proof so that the error is almost everywhere! A nice analogue is the following: Initial Proof Haris Angelidakis (MPLA) PCP's February 16, 2012 6 / 27 PCP Format The surprising main idea In general, a mathematical proof is invalid if it has even a single error somewhere, which can be very difficult to detect. What PCP theorems tell us is that there is a mechanical way to rewrite the proof so that the error is almost everywhere! A nice analogue is the following: Initial Proof PCP transformation Haris Angelidakis (MPLA) PCP's February 16, 2012 6 / 27 The surprising main idea In general, a mathematical proof is invalid if it has even a single error somewhere, which can be very difficult to detect. What PCP theorems tell us is that there is a mechanical way to rewrite the proof so that the error is almost everywhere! A nice analogue is the following: Initial Proof PCP transformation PCP Format Haris Angelidakis (MPLA) PCP's February 16, 2012 6 / 27 Towards a new definition of NP Note: From now on, we shall refer to languages L ⊆ f0; 1g∗. Definition (NP classic definition) k NP = [k2NNTIME(n ) Definition (NP \yes”-certificate definition) A language L is in NP if there exists a polynomial p : N ! N and a deterministic polynomial-time TM M (called the verifier of L) such that for every x 2 f0; 1g∗, x 2 L , 9u 2 f0; 1gp(jxj) such that M(x; u) = 1: If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (with respect to the language L and machine M). Haris Angelidakis (MPLA) PCP's February 16, 2012 7 / 27 Towards a new definition of NP Note: From now on, we shall refer to languages L ⊆ f0; 1g∗. Definition (NP classic definition) k NP = [k2NNTIME(n ) Definition (NP \yes”-certificate definition) A language L is in NP if there exists a polynomial p : N ! N and a deterministic polynomial-time TM M (called the verifier of L) such that for every x 2 f0; 1g∗, x 2 L , 9u 2 f0; 1gp(jxj) such that M(x; u) = 1: If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (with respect to the language L and machine M). Haris Angelidakis (MPLA) PCP's February 16, 2012 7 / 27 Towards a new definition of NP Note: From now on, we shall refer to languages L ⊆ f0; 1g∗. Definition (NP classic definition) k NP = [k2NNTIME(n ) Definition (NP \yes”-certificate definition) A language L is in NP if there exists a polynomial p : N ! N and a deterministic polynomial-time TM M (called the verifier of L) such that for every x 2 f0; 1g∗, x 2 L , 9u 2 f0; 1gp(jxj) such that M(x; u) = 1: If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (with respect to the language L and machine M).