The Complexity of Mathematical Problems

Transcription

1 The Complexity of Mathematical Problems C. S. Calude (UoA) and E. Calude (Massey U) In Honour of Eric Goles 60th Birhday Valparaiso, November / 199

2 2 / 199 Do the following statements

3 Do the following statements the four colour theorem, 2 / 199

4 Do the following statements the four colour theorem, Fermat s great theorem, 2 / 199

5 Do the following statements the four colour theorem, Fermat s great theorem, the Riemann hypothesis, 2 / 199

6 Do the following statements the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture? 2 / 199

7 Do the following statements the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture? share a common mathematical property? 2 / 199

8 Do the following statements the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture? share a common mathematical property? And, if there is such a property, how can we use it for a better understanding of these statements? 2 / 199

9 Computability and Complexity 1 Universality theorem. There exists (and can be constructed) a (Turing) machine U called universal such that for every machine V there exists a constant c = c U,V such that for every program σ there exists a σ for which the following two conditions hold: U(σ ) = V (σ), σ σ + c. 3 / 199

10 Computability and Complexity 2 The halting problem for a machine V is the function Λ V defined by { 1, if V (σ) =, Λ V (σ) = 0, otherwise. 4 / 199

11 Computability and Complexity 2 The halting problem for a machine V is the function Λ V defined by { 1, if V (σ) =, Λ V (σ) = 0, otherwise. Undecidability theorem. If U is universal, then Λ U is incomputable, i.e. the halting problem for a universal machine is undecidable. 4 / 199

12 Π 1 problems A problem π of the form σp(σ), where P is a computable predicate is called a Π 1 problem. 5 / 199

13 Π 1 problems A problem π of the form σp(σ), where P is a computable predicate is called a Π 1 problem. Any Π 1 problem is finitely refutable. 5 / 199

14 Π 1 problems A problem π of the form σp(σ), where P is a computable predicate is called a Π 1 problem. Any Π 1 problem is finitely refutable. For every Π 1 problem π = σp(σ) we associate the program σ π = inf{n : P(n) = false} which satisfies: π is true iff U(σ π ) =. 5 / 199

15 Π 1 problems A problem π of the form σp(σ), where P is a computable predicate is called a Π 1 problem. Any Π 1 problem is finitely refutable. For every Π 1 problem π = σp(σ) we associate the program σ π = inf{n : P(n) = false} which satisfies: π is true iff U(σ π ) =. Solving the halting problem for U solves all Π 1 problems. 5 / 199

16 Examples The problems the four colour theorem, 6 / 199

17 Examples The problems the four colour theorem, Fermat s great theorem, 6 / 199

18 Examples The problems the four colour theorem, Fermat s great theorem, the Riemann hypothesis, 6 / 199

19 Examples The problems the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture 6 / 199

20 Examples The problems the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture are all Π 1 problems. 6 / 199

21 Examples The problems the four colour theorem, Fermat s great theorem, the Riemann hypothesis, the Collatz s conjecture are all Π 1 problems. Of course, not all problems are Π 1 problems. For example, the twin prime conjecture. 6 / 199

22 Complexity Complexity C U (π) = min{ Π P : π = np(n)}. 7 / 199

23 Complexity Complexity C U (π) = min{ Π P : π = np(n)}. Invariance theorem. If U, U are universal, then there exists a constant c = c U,U such that for all π = np(n), P computable: C U (π) C U (π) c. 7 / 199

24 Complexity Complexity C U (π) = min{ Π P : π = np(n)}. Invariance theorem. If U, U are universal, then there exists a constant c = c U,U such that for all π = np(n), P computable: C U (π) C U (π) c. Incomputability theorem. If U is universal, then C U is incomputable. 7 / 199

25 Complexity Classes Because of the incomputability theorem, we work with upper bounds for C U. As the exact value of C U is not important, we classify Π 1 problems into the following classes: C U,n = {π : π is a Π 1 problem, C U (π) n kbit}. 8 / 199

26 Some Results C U,1 : Legendre s conjecture (there is a prime number between n 2 and (n + 1) 2, for every positive integer n), Fermat s last theorem (there are no positive integers x, y, z satisfying the equation x n + y n = z n, for any integer value n > 2) and Goldbach s conjecture (every even integer greater than 2 can be expressed as the sum of two primes) 9 / 199

27 Some Results C U,1 : Legendre s conjecture (there is a prime number between n 2 and (n + 1) 2, for every positive integer n), Fermat s last theorem (there are no positive integers x, y, z satisfying the equation x n + y n = z n, for any integer value n > 2) and Goldbach s conjecture (every even integer greater than 2 can be expressed as the sum of two primes) C U,2 : Dyson s conjecture (the reverse of a power of two is never a power of five) 9 / 199

28 Some Results C U,1 : Legendre s conjecture (there is a prime number between n 2 and (n + 1) 2, for every positive integer n), Fermat s last theorem (there are no positive integers x, y, z satisfying the equation x n + y n = z n, for any integer value n > 2) and Goldbach s conjecture (every even integer greater than 2 can be expressed as the sum of two primes) C U,2 : Dyson s conjecture (the reverse of a power of two is never a power of five) C U,3 : the Riemann hypothesis (all non-trivial zeros of the Riemann zeta function have real part 1/2) 9 / 199

29 Some Results C U,1 : Legendre s conjecture (there is a prime number between n 2 and (n + 1) 2, for every positive integer n), Fermat s last theorem (there are no positive integers x, y, z satisfying the equation x n + y n = z n, for any integer value n > 2) and Goldbach s conjecture (every even integer greater than 2 can be expressed as the sum of two primes) C U,2 : Dyson s conjecture (the reverse of a power of two is never a power of five) C U,3 : the Riemann hypothesis (all non-trivial zeros of the Riemann zeta function have real part 1/2) C U,4 the four colour theorem (the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour) 9 / 199

30 More Results and Open Questions C U,5 :? 10 / 199

31 More Results and Open Questions C U,5 :? C U,6 :? 10 / 199

32 More Results and Open Questions C U,5 :? C U,6 :? C U,7 : Euler s integer partition theorem (the number of partitions of an integer into odd integers is equal to the number of partitions into distinct integers). 10 / 199

33 More Results and Open Questions C U,5 :? C U,6 :? C U,7 : Euler s integer partition theorem (the number of partitions of an integer into odd integers is equal to the number of partitions into distinct integers). In which class is the Collatz conjecture? ( given any positive integer a 1 there exists a natural N such that a N = 1, where { ) an /2, if a a n+1 = n is even, 3a n + 1, otherwise. 10 / 199

34 Inductive Complexity and Complexity Classes of First Order By transforming each program Π P for U into a program Π ind,1 P for U ind (U working in inductive mode ) we can define the inductive complexity of first order by C ind,1 U (π) = min{ Π ind,1 : π = np(n)}, P 11 / 199

35 Inductive Complexity and Complexity Classes of First Order By transforming each program Π P for U into a program Π ind,1 P for U ind (U working in inductive mode ) we can define the inductive complexity of first order by C ind,1 U (π) = min{ Π ind,1 : π = np(n)}, the inductive complexity classes of order one by C ind,1 U,n P = {π : π is a Π 1 statement, C ind,1 U (π) n kbit}, 11 / 199

36 Inductive Complexity and Complexity Classes of First Order By transforming each program Π P for U into a program Π ind,1 P for U ind (U working in inductive mode ) we can define the inductive complexity of first order by C ind,1 U (π) = min{ Π ind,1 : π = np(n)}, the inductive complexity classes of order one by C ind,1 U,n P = {π : π is a Π 1 statement, C ind,1 U (π) n kbit}, and prove that C U,n = C ind,1 U,n. 11 / 199

37 Inductive Complexity and Complexity Classes of Higher Orders By allowing inductive programs of order 1 as routines we get inductive programs of order 2, so we can define the inductive complexity of second order (for more complex problems) C ind,2 U (ρ) = min{ M ind,2 R : ρ = n ir(n, i)}, 12 / 199

38 Inductive Complexity and Complexity Classes of Higher Orders By allowing inductive programs of order 1 as routines we get inductive programs of order 2, so we can define the inductive complexity of second order (for more complex problems) C ind,2 U (ρ) = min{ M ind,2 R : ρ = n ir(n, i)}, and the inductive complexity class of second order: C ind,2 U,n ind,2 = {ρ : ρ = n ir(n, i), C (ρ) n kbit}. U 12 / 199

39 Inductive Complexity and Complexity Classes of Higher Orders By allowing inductive programs of order 1 as routines we get inductive programs of order 2, so we can define the inductive complexity of second order (for more complex problems) C ind,2 U (ρ) = min{ M ind,2 R : ρ = n ir(n, i)}, and the inductive complexity class of second order: C ind,2 U,n ind,2 = {ρ : ρ = n ir(n, i), C (ρ) n kbit}. The Collatz conjecture is in the class C ind,2 U,3. U 12 / 199

40 Two open problems What is the complexity of 13 / 199

41 Two open problems What is the complexity of P vs NP problem? 13 / 199

42 Two open problems What is the complexity of P vs NP problem? Poincaré s conjecture? 13 / 199

43 Thank you 14 / 199

44 Thank you VIVE ERIC! 14 / 199