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Computational intelligence: an F-matrix view

Qianchuan ZhaoCenter for Intelligent and Networked

SystemsTsinghua University

Beijing 100084, China Presented to: SFI summer school at

QingdaoJuly 8, 2004

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Joint work with

• Prof. Yu-Chi Ho, Dr. David Pepyne, Prof. Da-Zhong Zheng, Prof. Bruce Krogh, Prof. Qiang Lu, Mr. Kai Sun, Dr. Ke Yang, Mr. Qingshan Jia

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Acknowledgement

• National Science Foundation of China

60074012 and 60274011, funding from ministry of education (Chinese) and a Tsinghua University (China) Fundamental Research Funding Grant.

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Computational Intelligence

• Methods inspired by nature intelligence (Genetic Algorithms, Swarm Intelligence, Simulated Annealing, Quantum Computing)

• Methods inspired by human brain structure

(Artificial Neural Networks)

• Methods inspired by how human reasons (Fuzzy Logic)

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Outline

• Optimization

• Modeling strategies

• General search strategies

• General design strategies

• Complexity in behavior of dynamic systems

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Outline

• Optimization

• Modeling strategies

• General search strategy

• General design strategy

• Complexity in behavior of dynamic systems

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Optimization

A optimization problem is to maximum (minimum) performance index in a search space subject to some constraints.

)(max xfx

0)( xgSubject to:

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Complexity in evaluating objective function

An objective function f is complex to evaluate if it can only be evaluated by simulation.

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Outline

• Optimization

• Modeling strategies

• General search strategy

• General design strategy

• Complexity in behavior of dynamic systems

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Modeling of optimization problems

• Encoding

• Filtering

• Surrogate

• Goal soften

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Representing solutions

• EncodingUsing strings or numbers to represent a

solution to the optimization problem as input such that optimization algorithms can proceed.

Solutions should be able to obtained by decoding the outputs of optimization algorithms.

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Example

• TSP (traveling salesman problem):

Find a minimum cost tour of n cities with each city visited once and only once.

1

2

3

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The sequence of nodes x=1234 is a solution.

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Example

• Buffer allocation example:

A solution is a vector of ten buffers.

Alternatively, in observing the constraints, a solution can be defined as a vector of 4 variables (B0,B4,B5,B8).

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Filtering

• Solve the original problem by stages.

At the first stage, easy constraints are obtained to narrow down the solution space to a smaller space .

At the second stage, hard constraints are handled only within .

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ExampleTraditional function optimization

Max f(x)Subject to: x=[0,1]

f is a continuously differentiable function. Method: obtain set by solving df(x)/dx=0 on xR

at the first stage and then solve Max f(x)

x ={0,1}

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ExampleIslanding operation for power systems:

Under local failures, to avoid collapse of the entire power system, it is separated into several small islands which can operate in safe conditions.

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ExampleIslanding operation for power systems [Zhao03a]

[Sun03]:

The balance of static power supply and load in each island is a necessary condition for each island to operate safely.

First stage: obtain solution set by search all separation operation keeping static power balance.

Second stage: search within true proper separation operation by simulation.

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A power system

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Surrogate

• Exploration

Learning by example:

Predict complex constraints/objective function with ANN

• Average

Noised observation by Mote Carlo simulation

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Example

• Q-learning

• Neural dynamic programming

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Goal soften

• Instead of asking best for sure, we ask good enough with high probability

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Example

• Ordinal Optimization

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F-matrix [Ho02]

x1

x2

x|X|

f1 f2 f|F|

y1

y1

y1

y1

y1

y2

y|Y|

y|Y|

y|Y|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y|

The number of all different problem instances is |Y||X|.

Note the sum for each row is the same.

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F-matrix

x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

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F-matrix

• Assumptions:

a) Finite world assumption: finite search space and finite set of performance values.

b) There is no constraint.

c) Only P (polynomial) solutions can be searched.

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Outline

• Optimization

• Modeling strategies

• General search strategies

• General design strategies

• Complexity in behavior of dynamic systems

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General search strategies

• Neighborhood search

• Random guess

• Parallel search

• Hybrid search

• Hill climbing

• Backtracking

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

Neighborhood search (one dimension)

Every element of designs in a neighborhood can be listed as nearby designs.

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

X X 2 1 2 1 1 1

Total computation effort consumed:

2+2+2+1+2+1+1+1

Number of successes: 6

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

Random guess

Not like neighborhood search, random guess jumps in the entire search space stochastically.

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

X 2 X 1 2 1 1 1

Total computation effort consumed:

2+2+2+1+2+1+1+1

Number of successes : 6

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

S0S1

S2

S3

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

X X 2 1 2 1 1 1X 2 X 1 2 1 1 1

S0S1

S2

S3

NS

RG

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n

k SP

k0

~;

n is the number of solutions.

Sk is the set of binary strings with exactly k 1s.

is 1 if the problem instance f has outcome 1 for at least one solution in the randomly picked P solutions {x’1,x’2,…,x’P}.

P~

;

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

Parallel search

Parallel search allow several search procedures work simultaneously.

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

P1 is a search procedure

P2 is another search procedure

P12: the iterative search process

For each search step of both procedures, results are reported to the controller.

P1 P2

P1 P2c

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1

X X X 1 X 1 1 1

x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

X 1 X X 1 1 X 1

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f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1

X X X 1 X 1 1 1

x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

X 1 X X 1 1 X 1

X 2 X 2 2 2 2 2

Total computation effort consumed:

2+2+2+2+2+2+2+2

Number of successes: 6

P12

P2

P1

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n is the number of solutions.

Sk is the set of binary strings with exactly k 1s.

is 1 if the problem instance f has outcome 1 for at least one design in P1 designs (decided by neighborhood search). is defined similarly.

},max{21 ;

0; P

n

k SP

k

},max{21 ;

0; P

n

k SP

k

n

k SPP

k0; 21

=

1;P

2;P

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

Hybrid

Simple search strategies can also be combined.

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n is the number of solutions.

Sk is the set of binary strings with exactly k 1s.

is 1 if the problem instance f has outcome 1 for at least one design in the randomly picked P1 designs.

is defined similarly.

},max{21

~;

0

~; P

n

k SP

k

},max{21

~;

0

~; P

n

k SP

k

n

k SPP

k0

~~; 21

=

1~

;P

2~

;P

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Hill climbing

The purpose of hill climbing is to find the maximum outcome of the given instance by search in an increasing direction. If it find a maximum, we say it makes a hit.

x1

x2

x|X|

f1 f2 f|F|

y1

y1

y1

y1

y1

y2

y|Y|

y|Y|

y|Y|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y-1|

y|Y|

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f

x1 0x2 1x3 2x4 3x5 2x6 4

…xn 0

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f1 f2 f3 f4 f5x1 0 1 0 1 2x2 1 0 1 2 0x3 0 1 2 0 1x4 1 2 0 1 0x5 2 0 1 0 1

3 2 1 X X

5 4 3 2 1HC

NS

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Back tracking

The purpose of back tracking is to return to a history point and pickup a different search direction so that the algorithm can traverse the whole solution space.

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f

x1 0x2 1x3 2x4 3x5 2x6 0

…xn 4

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f1 f2 f3 f4 f5x1 0 1 0 1 2x2 1 0 1 2 0x3 0 1 2 0 1x4 1 2 0 1 0x5 2 0 1 0 1

3 2 1 4 5

5 4 3 2 1HC+BT

NS

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Types of search problems

• Easy

• Hard

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Number of difficult instances

Easy problem instances : the number of good outcomes k in the instance is large enough:

k/n>,

where is a threshold level such as 1%.

For easy instances, we can solve by random search. The total number of easy instances is sumk>n(|Sk|)

The number of difficulty instances is sumk<=n(|Sk|)

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Example

• Difficult problem: a problem including some difficult instances

Boolean satisfiability problem (SAT)Whether there is a x such that f(x)=1?

f1 f2 f3 f4 f5 f6 f7 f8x1 0 0 0 1 0 1 1 1x2 0 0 1 0 1 0 1 1x3 0 1 0 0 1 1 0 1

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Find a needle in a Haystack problem

Guess a N bit password. f1 f2 f8

x1 0 0 … 1x2 0 0 0x3 0 0 0x4 0 0 0x5 0 0 0x6 0 0 0x7 0 1 0x8 1 0 0

On average 2N-1 search steps are needed!

(N=3 example)

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Summary

• Neighborhood search, random guess, parallel search, hybrid, hill climbing plus back tracking are equivalent when no problem information are available.

• In other words, there is no universal search strategy-No Free Lunch Theorem [Wolpert97].

• Problem specific knowledge discovery should be honored.

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Outline

• Optimization

• Modeling strategies

• General search strategies

• General design strategies

• Complexity in behavior of dynamic systems

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Design Problem

• Obtain engineering systems with – High performance– Robustness– Safeness– High level of security– Low cost

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

Design problem

Find a specific row so that outcomes for all possible columns (called planned columns) are acceptable.

Possible columns

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General design strategies

• Modular design

• Hierarchical design

• Small world design

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Some facts from F-matrix

• The determination of planned columns is based on designer’s knowledge about the problem. It may be inaccurate.

• Catastrophes are not avoidable

• Designs for complex systems are robust yet fragile.

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x1

x2

x|X|

f1 f2 f|F|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

When an unplanned column happens, it may not give good outcomes.

Planned columns

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Example

• The design of airbags: although airbags can increase the safeness in general, but they may kill children which is not intended. This is an unexpected situation when implement the design.

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Gamma matrix [Ho04]

• All design when encoded as l-bit strings, can form a Gamma matrix

b1

b2

bl

x1 x2 x|X|

0

0

1

0

0

0 1

1

0 1

0

1 1

1

1

1

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Modular design

M1 M2

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Modularity in solution

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

f1x1 0x2 1x3 1x4 1

x5 1

x6 2x7 2x8 2

s1=1 for if b1=1

s23 =1 when b2 or b3 =1

f= s1 + s23

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x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

+

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f0 f1 f2x1 0 0 0x2 1 1 0x3 1 1 0x4 1 1 0

x5 0 1 1

x6 1 2 1x7 1 2 1x8 1 2 1

Case of module failure

Error in b1 Error in b2b3x1 x2 x3 x4 x5 x6 x7 x8

b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

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f1x1 0x2 2x3 2x4 2

x5 1

x6 1x7 1x8 1

s1=1 for x2,x3,x4,x5

s23 =1 for x2,x3,x4,x6,x7,x8

f= s1 + s23

When the design of two modules are coupled

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

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f0 f1 f2x1 0x2 1 2 0x3 1 2 0x4 1 2 0

x5 1

x6 1x7 1x8 1

Case of module failure for coupled design

Error in b1 Error in b2b3

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

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Hierarchical design

• The system is organized in a tree structure.

M2

M1 M3

The high level modules depends on the low level modules but the low modules does not depends on the high level modules. The system functions well only when the high level module works well.

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x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

s1=1 for b1=1

s2 =1 for b2=1

s3 =1 for b3=1

f= s2 (s1 + s3)

f1x1 0x2 0x3 0x4 1

x5 0

x6 0x7 1x8 2

The system requires the high level module M2 and at least one low level module (M1 or M3) works.

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f0 f1 f2x1 0x2 0x3 0x4 1

x5 0

x6 0x7 1x8 1 2 0

Case of module failure for hierarchical design

Error in b1 or b3 Error in b2

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

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Small world design

• The system is organized in asymmetry flat pattern. Some modules (head modules) makes more contribution than other modules.

M3M1

M2

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x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

f1x1 0x2 1x3 2x4 3

x5 1

x6 2x7 3x8 4

s1=1 for b1=1

s2 =1 for b2=1

s3 =1 for b3=1

f= s1 + 2s2 + s3

A small world design

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A small world designf0 f1 f2

x1 0x2 1x3 2x4 3

x5 1

x6 2x7 3x8 3 4 2

Error in b1 or b3 Error in b2

x1 x2 x3 x4 x5 x6 x7 x8b1 0 0 0 0 1 1 1 1b2 0 0 1 1 0 0 1 1b3 0 1 0 1 0 1 0 1

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Example

• This is consistent to what happens in the small world network.

For example, when a famous website like Google is attacked by DoS, the average number of hops one want to find a web page on the internet will increase a lot. But if DoS happens only for a hospital website, generally we will not feel much change in search on the web.

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Summary

• The benefits of distributed design are:

Helps us to achieve robustness in design such that a design can degenerate performance gradually when random failure happens.

It helps to implement the system when central control is too expensive to design or implement.

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Finding good design for problems with modularity

• Hill climbing or neighborhood search can be used to improve the quality of solution incrementally. How to divide the system into proper set of modules and which module is to design first is a search problem which needs try and error.

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Outline

• Optimization

• Modeling strategies

• General search strategies

• General design strategies

• Complexity in behavior of dynamic systems

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Complexity in behavior of dynamic systems

• Reachability problem

• Abstraction points

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Some facts

• Halting problem of Turing machine is undecidable.

• Reachability of given state in a discrete event simulation model is NP-hard [Jacobson99].

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Some facts

Attraction point problem of Dynamic Boolean Networks (DBN) is NP-hard [Zhao03].

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References[Ho92] HO, Y.C., SREENIVAS, R.S., and VAKILI, P., Ordinal

Optimization of Discrete Event Dynamic Systems, Journal of Discrete Event Dynamic Systems, Vol. 2, pp. 61-68, 1992.

[Ho02] Ho and Pepyne, "Simple Explanation of the No-Free-Lunch Theorem and Its Implications," JOTA, Vol. 115, No. 3, 2002.

[Ho03] YC Ho, QC Zhao and DL Pepyne, The No Free Lunch Theorem, Complexity, and Computer Security, IEEE Trans. Automat. Contr., 48 (5): 783-793, 2003.

[Ho04] YC Ho and DL Pepyne, Conceptual Framework for Optimization and Distributed Intelligence, Submitted to CDC04.

[Jacobson99] SH Jacobson, On the complexity of verifying structural properties of discrete event simulation models, Operations Research, 47(3), 476-481,1999.

[Sun03] K Sun, DZ Zheng and Q Lu, Splitting Strategies for Islanding Operation of Large-Scale Power Systems Using OBDD-Based Methods, IEEE Transactions on Power Systems, 18(2), 912-923, 2003.

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References

[Watts98] DJ Watts and SH Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393, 1998, pp. 440-442.

[Wolpert97] Wolpert, D.H. and W.G. Macready, No Free Lunch Theorems for Optimization, IEEE TEC, Vol. 1, No. 1, April 1997.

[Zhao03a] QC Zhao, K Sun, DZ Zheng, J Ma and Q Lu, A Study of System Splitting Strategies for Island Operation of Power System: A Two-phase Method Based on OBDDs, IEEE Transactions on Power Systems, 18(4), 1556-1565, 2003.

[Zhao03b] QC Zhao, Inseparablity of min-max systems is co-NP hard, Chinese Control Conference, 454-458, 2003.

[Zhao04] QC Zhao, YC Ho and QS Jia, Vector Ordinal Optimization, Journal of Optimization Theory and Applications, to be published.

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Thanks!