f ( f ( x )) Solving Iterated Functions Using Genetic Programming - - PowerPoint PPT Presentation

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f ( f ( x )) Solving Iterated Functions Using Genetic Programming - - PowerPoint PPT Presentation

Oct 25, 2022 268 likes •411 views

f ( f ( x )) Solving Iterated Functions Using Genetic Programming Michael Schmidt Hod Lipson 2010 HUMIES Competition Iterated Functions Iterated Function: Answer: f ( f ( x )) = x f ( x ) = x f ( f ( x )) = x + 2 f ( x ) = x + 1 f ( f ( x ))

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Solving Iterated Functions Using Genetic Programming

Michael Schmidt Hod Lipson 2010 HUMIES Competition

f(f(x))

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SLIDE 2

Iterated Functions

f(f(x)) = x f(x) = x f(f(x)) = x + 2 f(x) = x + 1 f(f(x)) = x4 f(x) = x2 f(f(x)) = (x2 + 1)2 +1 f(x) = x2 + 1 f(f(x)) = x2 – 2 f(x) = ?

Iterated Function: Answer:

Why is this problem so hard for humans?
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Test of Intelligence: f(f(x)) = x2 – 2

  • B. A. Brown, A. R. Brown, and M. F. Shlesinger, "Solutions of Doubly
and Higher Order Iterated Equations," Journal of Statistical Physics, vol. 110, pp. 1087-1097, 2003. "Mathvn journal problems," in Mathvn. vol. 01/2009 mathvn.org, 2009. This problem has become famous in math and physics circles for requiring deep mathematical insight in order to solve. Appeared in mathematical competitions The rumored fastest solver Michael Fisher
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SLIDE 4

The known solution requires deep human insight to solve a special case

Assume f(f(x)) = g(a2g-1(x)): g(a2g-1(x)) = x2 – 2 Next assume a2 = 2 and let θ = g-1(x) : g(2θ) = x2 – 2, g(2θ) = g(θ)2 – 2, x2 – 2 = g(θ)2 – 2 x = g(θ) = 2 cos(θ), x = g(g-1(θ)) = 2 cos(g-1(θ)) By inspection:

              

2 cos 2 cos 2 ) (

1 x

x f

Double angle formula:

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But there are possibly many solutions

f(f(x)) = x f(x) = x f(x) = –x f(x) = 1/x

This a dark area of mathematics; Only a few special cases of functional problems have ever been solved.

Yet, Genetic Programming can find these solutions easily….
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SLIDE 6

f(f(x)) = x2 – 2

What is f(x)?

x f(f(x)) x2 – 2

Straightforward application of Symbolic Regression

Solutions iterated twice: Fitness of a candidate f(x) =

     

  n i i i x f f y n 1 2 1
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SLIDE 7

Solved in 81 seconds

10 10 1 10 2 10 3
  • 10
  • 8
  • 6
  • 4
  • 2
Time [seconds] Fitness [-error] 10 10 1 10 2 10 3 20 40 60 80 100 Time [seconds] Converged Runs [%] And Solved Reliably: 50 trials 50 trials
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SLIDE 8

) ) 10 16871 . 1 ( 2 4916 . 16 ( ) 10 16871 . 1 ( ) 10 16871 . 1 ( 2 4916 . 16 ) (

2 18 18 18

x x x x f            

 

2

2 2 lim ) ( ax b ax ax b x f

a

  

 

Nearly Perfect Fitness

The genetic program is trying to take a limit….

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SLIDE 9

       

 

2

) ( 2 ) ( ) ( 2 ) ( x f a b x f a x f a b x f f   

       

 

2

) ( 2 ) ( ) ( 2 lim ) ( lim x f a b x f a x f a b x f f

a a

  

   

 

2 ) ( lim

2 

 

x x f f

a

Exactly Correct Symbolicly

The solution is symbolicly correct
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New Solution Found with Genetic Programming

 

2

2 1 2 1 lim ) ( ax ax ax x f

a

  

 

f(f(x)) = x2 – 2

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Human Competitive:

  • Long-developed and infamous problem in

physics and mathematics

  • Has required deep human insight into

mathematics to solve special cases

  • No other general method exists
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Human Competitive:

  • Long-developed and infamous problem in

physics and mathematics

  • Has required deep human insight into

mathematics to solve special cases

  • No other general method exists

The Best Entry:

  • Entirely new solution found via GP
  • Fastest this problem has ever been solved
  • Potential impact in many fields, where such

problems have never been solved before

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Conclusions

Use GP to Solve Iterated Functions

f(f(x))