G5AI AI Introduction to AI
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Transcript of G5AI AI Introduction to AI
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Graham Kendall
www.cs.nott.ac.uk/~gxk
+44 (0) 115 846 6514
G5AIAIIntroduction to AI
Graham KendallCombinatorial Explosion
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G5G5AIAIAIAI History of AI History of AI
The Travelling Salesman Problem
• A salesperson has to visit a number of cities
• (S)He can start at any city and must finish at that same city
• The salesperson must visit each city only once
• The number of possible routes is (n!)/2 (where n is the number of cities)
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Combinatorial Explosion
Travelling Salesman Problem
0
500000
1000000
1500000
2000000
1 2 3 4 5 6 7 8 9 10
Cities
Ro
ute
s
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Combinatorial Explosion
Cities Routes
1 12 13 34 125 606 3607 25208 201609 18144010 181440011 19958400
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Combinatorial ExplosionA 10 city TSP has 181,000 possible solutions
A 20 city TSP has 10,000,000,000,000,000 possible solutions
A 50 City TSP has 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 possible solutions
There are 1,000,000,000,000,000,000,000 litres of water on the planet
Mchalewicz, Z, Evolutionary Algorithms for Constrained Optimization Problems, CEC 2000 (Tutorial)
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
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Combinatorial Explosion - Towers of Hanoi
• How many moves does it take to move four rings?
• You might like to try writing a towers of hanoi program (and you may well have to in one of your courses!)
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G5G5AIAIAIAI History of AI History of AI
Combinatorial Explosion - Towers of Hanoi
• If you are interested in an algorithm here is a very simple one
• Assume the pegs are arranged in a circle
• 1. Do the following until 1.2 cannot be done– 1.1 Move the smallest ring to the peg residing next to
it, in clockwise order
– 1.2 Make the only legal move that does not involve the smallest ring
• 2. Stop
• P. Buneman and L.Levy (1980). The Towers of Hanoi Problem, Information Processing Letters, 10, 243-4
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Combinatorial Explosion - Towers of Hanoi
• A time analysis of the problem shows that the lower bound for the number of moves is
2N-1
• Since N appears as the exponent we have an exponential function
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Combinatorial Explosion - Towers of Hanoi
Pegs 2N-1
3 74 155 326 63… …10 1023
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Combinatorial Explosion - Towers of Hanoi
• The original problem was stated that a group of tibetan monks had to move 64 gold rings which were placed on diamond pegs.
• When they finished this task the world would end.
• Assume they could move one ring every second (or more realistically every five seconds).
• How long till the end of the world?
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Combinatorial Explosion - Towers of Hanoi
• > 500,000 years!!!!! Or 3 Trillion years
• Using a computer we could do many more moves than one a second so go and try implementing the 64 rings towers of hanoi problem.
• If you are still alive at the end, try 1,000 rings!!!!
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Combinatorial Explosion - Optimization
• Optimize f(x1, x2,…, x100)
• where f is complex and xi is 0 or 1
• The size of the search space is 2100 1030
• An exhaustive search is not an option– At 1000 evaluations per second– Start the algorithm at the time the universe was
created– As of now we would have considered 1% of all
possible solutions
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Combinatorial Explosion
11E+141E+281E+421E+561E+701E+841E+981E+1121E+1261E+1401E+1541E+1681E+1821E+1961E+2101E+2241E+2381E+2521E+2661E+280
2 4 8 16 32 64 128 256 512 1024 2048
5N
N^3
N^5
N^10
1.2^N
2^N
N^N
Microseconds in a Day
Microseconds since Big Bang
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Combinatorial Explosion
10 20 50 100 200
N2
N5
1/10,000 second
1/2500 second
1/400 second
1/100 second
1/25 second
1/10 second
3.2 seconds
5.2 minutes
2.8 hours
3.7 days
2N
NN
1/1000 second
1 second
35.7 years
> 400 trillion
centuries
45 digit no. of centuries
2.8 hours
3.3 trillion years
70 digit no. of
centuries
185 digit no. of
centuries
445 digit no. of
centuries
Running on a computer capable of 1 million instructions/second
Ref : Harel, D. 2000. Computer Ltd. : What they really can’t do, Oxford University Press
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G5AIAIIntroduction to AI
Graham KendallEnd Combinatorial Explosion