Complexity and Gödel Incomplete theorem

電機三 B90901144

劉峰豪

Outline

Introduction to the idea of “complexity” Complexity of some basic Operation Problems P and NP class

Gödel Incomplete Theorem

Introduction

Big O, small O…….are too trivial Ln(r;c) for approximation of subexponential ti

me P and NP class

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)

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)

)( ))ln(ln)((ln 1 rr nnceO

Problems

Problem instance: a particular case of the task

Search problem: it may have several correct answers

Decision problem: answer yes or no

Some examples

Given N, and factor it 6=2*3

TSP

Does 91 have a factor between 2 and 63?

P and NP class

P NP NP-complete Reduction of problems Some applications in cryptography

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)

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

Examples

Consider a graph G, is there a k-clique?

4-cliquegraph

but no 5-clique

CLIQUE = {<G,k> | graph G has a k-clique}

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

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

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??

Complexity and security of some cryptosystem DES ： linear or differential RSA ： factorization ( quadratic sieve and nu

mber field sieve ) quadratic sieve ： (Ln(1/2;c)) number field sieve ： (Ln(1/3;c)) ECC ： exponential time

RSA-576 Factored

December 3, 2003 Number field sieve

Gödel Incomplete Theorem

Some Terms Theorem Effect

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

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.

A B

true

false

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.“

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.

Gödel Incomplete Theorem All consistent axiomatic formulations of numb

er 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.

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.

Effect

Turing ： general recursive functions

John Von Neumann

AI