Complexity and G ö del Incomplete theorem
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Complexity and Gödel Incomplete theorem
電機三 B90901144
劉峰豪
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Outline
Introduction to the idea of “complexity” Complexity of some basic Operation Problems P and NP class
Gödel Incomplete Theorem
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Introduction
Big O, small O…….are too trivial Ln(r;c) for approximation of subexponential ti
me P and NP class
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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)
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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
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Problems
Problem instance: a particular case of the task
Search problem: it may have several correct answers
Decision problem: answer yes or no
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Some examples
Given N, and factor it 6=2*3
TSP
Does 91 have a factor between 2 and 63?
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P and NP class
P NP NP-complete Reduction of problems Some applications in cryptography
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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)
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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
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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}
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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
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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
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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??
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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
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RSA-576 Factored
December 3, 2003 Number field sieve
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Gödel Incomplete Theorem
Some Terms Theorem Effect
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Some terms in Gödel Incomplete Theorem Consistent Undecidable Peano's Axioms Answer Hilbert's 2nd Problem
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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
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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.“
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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.
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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.
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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.
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Effect
Turing : general recursive functions
John Von Neumann
AI