Outline The complexity class coNP Piotr Wojciechowski1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Wojciechowski coNP Outline Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP Outline Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP Outline Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P coNP as related to NP Definition (coNP) coNP is the complexity class which contains the complements of problems found in NP. Another way of looking at coNP Just as NP can be considered to be the set of problems with succinct ”yes” certificates, coNP can be considered to be the set of problems with succinct ”no” certificates. This means that a ”no” instance of a problem in coNP has a short proof of it being a ”no” instance. Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P coNP as related to NP Definition (coNP) coNP is the complexity class which contains the complements of problems found in NP. Another way of looking at coNP Just as NP can be considered to be the set of problems with succinct ”yes” certificates, coNP can be considered to be the set of problems with succinct ”no” certificates. This means that a ”no” instance of a problem in coNP has a short proof of it being a ”no” instance. Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P Examples 1 coSAT = fhbi : b is a boolean expression with no satisfying assignmentsg 2 PRIMES = fhpi : p is a prime numberg Wojciechowski coNP coNP What is coNP NP \ coNP Examples of problems in coNP NP, coNP, and P Examples 1 coSAT = fhbi : b is a boolean expression with no satisfying assignmentsg 2 PRIMES = fhpi : p is a prime numberg Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P Properties Problems have both succinct ”yes” and succinct ”no” certificates. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P Outline 1 Description of coNP and examples of problems What is coNP Examples of problems in coNP 2 The NP \ coNP complexity class Properties of NP \ coNP Problems in NP \ coNP 3 NP, coNP, and P The P, NP, coNP Hierarchy Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P Examples 1 PRIMES 2 All problems in P Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P Examples 1 PRIMES 2 All problems in P Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P PRIMES is in NP \ coNP Goal We first want to develop a different way of determining primality. Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p p−1 such that r p−1 = 1 mod p and r q 6= 1 mod p for all prime divisors q of p − 1. Definition (Relative Primality) Two numbers a and b are relatively prime iff their greatest common divisor, (a; b), is 1. Examples 5 and 234 are relatively prime, 57 and 95 are not, 19 is a common factor. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P PRIMES is in NP \ coNP Goal We first want to develop a different way of determining primality. Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p p−1 such that r p−1 = 1 mod p and r q 6= 1 mod p for all prime divisors q of p − 1. Definition (Relative Primality) Two numbers a and b are relatively prime iff their greatest common divisor, (a; b), is 1. Examples 5 and 234 are relatively prime, 57 and 95 are not, 19 is a common factor. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P PRIMES is in NP \ coNP Goal We first want to develop a different way of determining primality. Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p p−1 such that r p−1 = 1 mod p and r q 6= 1 mod p for all prime divisors q of p − 1. Definition (Relative Primality) Two numbers a and b are relatively prime iff their greatest common divisor, (a; b), is 1. Examples 5 and 234 are relatively prime, 57 and 95 are not, 19 is a common factor. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P PRIMES is in NP \ coNP Goal We first want to develop a different way of determining primality. Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p p−1 such that r p−1 = 1 mod p and r q 6= 1 mod p for all prime divisors q of p − 1. Definition (Relative Primality) Two numbers a and b are relatively prime iff their greatest common divisor, (a; b), is 1. Examples 5 and 234 are relatively prime, 57 and 95 are not, 19 is a common factor. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P An alternate look at primality Definition (Φ(n)) Φ(n) = fm : 1 ≤ m < n; (m; n) = 1g. Definition (Euler φ function) φ(n) = jΦ(n)j and φ(1) = 1. In other words, φ(n) is the number of numbers between 1 and n − 1 which are relatively prime to n Lemma (1) Q 1 φ(n) = n pjn(1 − p ) where p is a prime. Proof. Assume that p1; p2 :::; pk are the prime divisors of n. Observe that each pi knocks off 1 one in every pi candidates for φ(n), leaving n · (1 − ) candidates for φ(n). It therefore pi Q 1 follows that φ(n) = n pjn(1 − p ) where p is a prime. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P An alternate look at primality Definition (Φ(n)) Φ(n) = fm : 1 ≤ m < n; (m; n) = 1g. Definition (Euler φ function) φ(n) = jΦ(n)j and φ(1) = 1. In other words, φ(n) is the number of numbers between 1 and n − 1 which are relatively prime to n Lemma (1) Q 1 φ(n) = n pjn(1 − p ) where p is a prime. Proof. Assume that p1; p2 :::; pk are the prime divisors of n. Observe that each pi knocks off 1 one in every pi candidates for φ(n), leaving n · (1 − ) candidates for φ(n). It therefore pi Q 1 follows that φ(n) = n pjn(1 − p ) where p is a prime. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P An alternate look at primality Definition (Φ(n)) Φ(n) = fm : 1 ≤ m < n; (m; n) = 1g. Definition (Euler φ function) φ(n) = jΦ(n)j and φ(1) = 1. In other words, φ(n) is the number of numbers between 1 and n − 1 which are relatively prime to n Lemma (1) Q 1 φ(n) = n pjn(1 − p ) where p is a prime. Proof. Assume that p1; p2 :::; pk are the prime divisors of n. Observe that each pi knocks off 1 one in every pi candidates for φ(n), leaving n · (1 − ) candidates for φ(n). It therefore pi Q 1 follows that φ(n) = n pjn(1 − p ) where p is a prime. Wojciechowski coNP coNP Properties of NP \ coNP NP \ coNP Problems in NP \ coNP NP, coNP, and P An alternate look at primality Definition (Φ(n)) Φ(n) = fm : 1 ≤ m < n; (m; n) = 1g.