Complexity and G ö del Incomplete theorem

1. Complexity and Gödel Incomplete theorem 電機三 B90901144 劉峰豪

2. Outline • Introduction to the idea of “complexity” • Complexity of some basic Operation • Problems • P and NP class • Gödel Incomplete Theorem

3. Introduction • Big O, small O…….are too trivial • Ln(r;c) for approximation of subexponential time • P and NP class

4. Complexity of some Basic Operation • Given a,b • size a = |a| = ln a • +、-： O(max(size a, size b)) • *： O(size a * size b) • /,%： a=bq+r， O(size b * size q)

5. Ln • n is the input of a problem • Ln(r;c)= • Ln(0;c)：linear O((ln n)c) • Ln(1;c)：exponential O( nc)

6. Problems • Problem instance: a particular case of the task • Search problem: it may have several correct answers • Decision problem: answer yes or no

7. Some examples • Given N, and factor it • 6=2*3 • TSP • Does 91 have a factor between 2 and 63?

8. P and NP class • P • NP • NP-complete • Reduction of problems • Some applications in cryptography

9. P class • A decision problem p is in class P if there exists a constant c and an algorithm such that if an instance of p has input length <=n, then the algorithm answers the question in time O(nc)

10. Class NP • A decision problem p is in the class NP, if given any instance of p, a person with unlimited computing power can answer it “yes”, and another person can verify it in time P • P is in NP

11. Examples • Consider a graph G, is there a k-clique? but no 5-clique graph 4-clique CLIQUE = {<G,k> | graph G has a k-clique}

12. Reducing one problem to another • Let p1 and p2 be 2 decision problems. We say that p1 reduces to p2 if there exists an algorithm that is polynomial time as a function of the input length of p1 and that, given any instance P1 of p1, constructs an instance P2 of p2 such that the answer for P1 is the same as the answer in P2

13. Examples • P1： input: a quadratic polynomial p(x) with integer coefficients Questions: does p(x) have two distinct roots? P2： input：an integer N question: is N positive? P1 reduces to P2 p1 < = p2

14. NP completeness • A problem p is NP-complete • if every other problem q in NP can be reduced to P in polynomial time • p is in NP • P = NP ?? • Relation between P and NPC??

15. Complexity and security of some cryptosystem • DES：linear or differential • RSA：factorization ( quadratic sieve and number field sieve ) • quadratic sieve： (Ln(1/2;c)) • number field sieve：(Ln(1/3;c)) • ECC：exponential time

16. RSA-576 Factored • December 3, 2003 • Number field sieve

17. Gödel Incomplete Theorem • Some Terms • Theorem • Effect

18. Some terms in Gödel Incomplete Theorem • Consistent • Undecidable • Peano's Axioms • Answer Hilbert's 2nd Problem

19. Consistency • The absence of contradiction (i.e., the ability to prove that a statement and its negative are both true) in an Axiomatic system is known as consistency. true A B false

20. undecidable • Not decidable as a result of being neither formally provable nor unprovable. • A：”What B said is true.“ • B：”What A said is false.“

21. Peano's Axioms • 1. Zero is a number. • 2. If a is a number, the successor of a is a number. • 3. zero is not the successor of a number. • 4. Two numbers of which the successors are equal are themselves equal. • 5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

22. Gödel Incomplete Theorem • All consistent axiomatic formulations of number theory include undecidable propositions • Any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

23. Conclusions • All formal mathematical systems have only limited power. • We will never be able to have a system that can prove all true statements about {0,1,2,…}, +, . • Note that this result predates that of Turing and the solution of Hilbert’s polynomial problem.

24. Effect • Turing： general recursive functions • John Von Neumann • AI