Download Complexity

Complexity and Gödel
Incomplete theorem
電機三 B90901144
劉峰豪
Outline
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Introduction to the idea of “complexity”
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Complexity of some basic Operation
Problems
P and NP class
Gödel Incomplete Theorem
Introduction
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Big O, small O…….are too trivial
Ln(r;c) for approximation of subexponential
time
P and NP class
Complexity of some Basic Operation
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Given a,b
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
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
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n is the input of a problem
O(e
c ((ln n ) r (ln ln n )1r )
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Ln(r;c)=
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Ln(0;c)：linear O((ln n)c)
Ln(1;c)：exponential O( nc)
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)
Problems
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Problem instance: a particular case of the
task
Search problem: it may have several correct
answers
Decision problem: answer yes or no
Some examples
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Given N, and factor it
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6=2*3
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TSP
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Does 91 have a factor between 2 and 63?
P and NP class
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P
NP
NP-complete
Reduction of problems
Some applications in cryptography
P class
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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
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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
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P is in NP
Examples
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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}
Reducing one problem to another
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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
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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
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A problem p is NP-complete
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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
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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
RSA-576 Factored
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December 3, 2003
Number field sieve
Gödel Incomplete Theorem
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Some Terms
Theorem
Effect
Some terms in Gödel Incomplete
Theorem
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Consistent
Undecidable
Peano's Axioms
Answer Hilbert's 2nd Problem
Consistency
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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
undecidable
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Not decidable as a result of being neither
formally provable nor unprovable.
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A：”What B said is true.“
B：”What A said is false.“
Peano's Axioms
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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
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All consistent axiomatic formulations of
number theory include undecidable
propositions
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Any formal system that is interesting enough
to formulate its own consistency can prove its
own consistency iff it is inconsistent.
Conclusions
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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
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Turing： general recursive functions
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John Von Neumann
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