Early work in intelligent systems

Early work in intelligent systems

Alan Turing (1912 – 1954) Arthur Samuel (1901-1990)

Early work in intelligent systems

• Alan Turing (1912 – 1954)Father of computer science, mathematician, philosopher, codebreaker (WW II), homosexual

• The Turing Machine• The Turing Test (AI)

Early work in intelligent systems

Alan Turing (1950):We cannot expect to find a good child-machine at the first attempt. One must experiment with teaching one such machine and see how well it learns. One can then try another and see if it is better or worse. There is an obvious connection between this process and evolution, by the identifications:Structure of the child machine = Hereditary materialChanges of the child machine = MutationsNatural selection = Judgment of the experimenter.

Early work in intelligent systems

Arthur Samuel (1901-1990)• “How can computers learn to

solve problems without being explicitly programmed? In other words, how can computers be made to do what is needed to be done, without being told exactly how to do it?" (1959)

• “The aim is to get machines to exhibit behavior, which if done by humans, would be assumed to involve the use of intelligence.” (1983)

Genetic Programming

• Breed a population of computer programs to solve a given problem– An extension of genetic algorithms

– Selection, crossover, mutation

Preparatory Steps

John Koza: the human user supplies:(1)The set of terminals (e.g., the independent

variables of the problem, zero-argument functions, and random constants)

(2) The set of primitive functions for each branch of the program to be evolved

(3) The fitness measure (4) The parameters for controlling the run(5) The termination criteria

1. Terminal Set

• External inputs to the program

• Numerical constants (problem dependent?) , e, 0, 1, … , random numbers, …

2. Function Set

• Arithmetic functions

• Conditional branches (if statements)

• Problem specific functions (controllers, filters, integrators, differentiators, circuit elements, …)

3. Fitness Measure

• The GP measures the fitness of each individual (computer program)

• Fitness is usually averaged over a variety of different cases– Program inputs– Initial conditions– Different environments

4. Control Parameters

• Population size (thousands or millions)

• Selection method

• Crossover probability

• Mutation probability

• Maximum program size

• Elitism option

5. Termination Criterion

• Maximum number of generations / real time

• Convergence of highest / mean fitness

• …

GP Flowchart

Initialization

Initialization

Max((* x x) (+ x (* 3 y)))

Prefix notation (Lisp)

Max(x*x, x+3*y)

• Nodes (points, functions)

• Links, terminals

Program Tree

(+ 1 2 (IF (> TIME 10) 3 4))

If Time > 10 then x = 3

elsex = 4

Solution = 1 + 2 + x

Mutation

• Select one individual probabilistically• Pick one point in the individual• Delete the subtree at the chosen point• Grow a new subtree at the mutation point in

same way as for the initial random population• The result is a syntactically valid executable

program

Crossover

• Select two parents probabilistically based on fitness

• Randomly pick a node in the first parent (often internal nodes 90% of the time)

• Independently randomly pick a node in the second parent

• Swap subtrees at the chosen nodes

Reproduction

• Select an individual probabilistically based on fitness

• Copy it (unchanged) into the next generation of the population (cloning)

ExampleIndependent variable Dependent variable

-1.00 1.00

-0.80 0.84

-0.60 0.76

-0.40 0.76

-0.20 0.84

0.00 1.00

0.20 1.24

0.40 1.56

0.60 1.96

0.80 2.44

1.00 3.00

Generate a computer program with one input x whose output equals the given data y

( y = x2 + x + 1 )

Preparatory Steps

1 Terminal set: T = {x, Random Constants}

2 Function set: F = { +, -, *, % }

3 Fitness: The sum of the absolute value of the differences between the program’s output and the given data (low is good)

4 Parameters: Population size M = 4

5 Termination: An individual emerges whose sum of absolute errors is less than 0.1

Initialization

Fitness Evalution

x+1 x2+1 2 x

0.67 1.00 1.70 2.67

Reproduction

• Copy (a), the most fit individual• Mutate (c)

Crossover

Crossover

Interpreting a program tree

Interpreting a program tree

{ – [ + ( – 3 0 ) ( – x 1 ) ] [ / ( – 3 0 ) ( – x 2 ) ] }

What does this evaluate as?

What are the terminals, functions, and lists?

Interpreting a program tree

{ – [ + ( – 3 0 ) ( – x 1 ) ] [ / ( – 3 0 ) ( – x 2 ) ] }

[ (3 – 0) + (x – 1) ] – [ (3 – 0) / (x – 2) ]

• Terminals = { 3, 0, x, 1, 2}

• Functions = { –, +, / }

• Lists = ( – 3 0 ), [ + ( – 3 0 ) ( – x 1 ) ], …

Interpreting a program tree

recursion

\Re*cur"sion\ (-sh?n), n. [L. recursio.]

See recursion.

factorial ( n )

if n = = 0 then return 1

else return n * factorial (n – 1)

Interpreting a program treeRecursive function EVAL:

if EXPR is a list then // i.e., delimited by parenthesesPROC = EXPR(1)VAL = PROC [ EVAL(EXPR(2)), EVAL(EXPR(3)),

…]else // i.e., EXPR is a terminal

if EXPR is a variable or constant thenVAL = EXPR

else // i.e., EXPR is a function with no argumentsVAL = EXPR ( )

endend

Can computer programs create new inventions?

GP Inventions

• Two patents filed by Keane, Koza, and Streeter on July 12, 2002 1. Creation of Tuning Rules for PID

Controllers that Outperform the Ziegler-Nichols and Åström-Hägglund Tuning Rules

2. Creation of 3 Non-PID Controllers that Outperform a PID Controller that uses the Ziegler-Nichols or Åström-Hägglund Tuning Rules

GP for Antenna Design

• X band antenna – Jason Lohn, NASA Ames– Wide beamwidth for a circularly polarized wave

– Wide bandwidth

The evolution of genetic programmingHardware Years CPU Power Results

Texas Instruments LISP machine

1987–1994 1 (base) Example problems

64-node Transtech transputer

1994–1997 9 Human competitive results

64-node Parsytec parallel computer

1995-2000 22 Reproduction of 20th century patents

70-node Alpha parallel computer

1999-2001 7 Circuit synthesis

1,000-node Pentium II 2000-2002 9 Reproduction of 21st century patents

1,000-node Pentium II4 weeks of CPU time

2002 9 Two new patents

GP Computational Effort

• Human brain 1012 neurons1 msec 1015 operations per second 1 peta-op = 1 brain second (B-sec)• Keane, Koza, Streeter patents:

Pop. Gens. Hours Nodes MHz B-sec

Patent 1 100K 76 107 1K 350 135

Patent 2 100K 325 1409 1K 350 1775

When should you use GP?

• Problem involving many variables that are interrelated in highly nonlinear ways

• Relationships among variables is not well understood

• Discovery of the size and shape of the solution is a major part of the problem

• “Black art” problems (controller tuning)• Areas where you have no idea how to program

a solution, but you know what you want

When should you use GP?

• Problems where a good approximate solution is satisfactory– Design

– Control and estimation

– Bioinformatics

– Classification

– Data mining

– System identification

– Forecasting

When should you use GP?

Areas where large computerized databases are accumulating and computerized techniques are needed to analyze the data genome, protein, microarray data satellite image data astronomical data petroleum databases medical records marketing databases financial databases

Schema Theory for GP

• The # symbol represents “don’t care”

• Example: H = ( + ( – # y ) # ) instances are:( + ( – x y ) x ) → ( x – y ) + x

( + ( – x y ) y ) → ( x – y ) + y

( + ( – y y ) x ) → ( y – y ) + x

( + ( – y y ) y ) → ( y – y ) + y

Schema Theory for GP

Example: H = ( + ( – # y ) # )

• o(H) = number of defined symbolso(H) = ?

• Length N(H) = number of symbolsN(H) = ?

• Defining length L(H) = number of links joining defined symbolsL(H) = ?

Schema Theory for GP

All these schema sample the program ( + ( – 2 x ) y )What are the schema defining length L, order o, and length N?

+

– #

+ #

–

#

–#

2 x

#

# x

#

2 #

#

# #

Schema Theory for GP

L = 3 L = 2 L = 1 L = 0o = 4 o = 2 o = 2 o = 1N = 5 N = 5 N = 5 N = 5

+

– #

+ #

–

#

–#

2 x

#

# x

#

2 #

#

# #

Schema Theory for GP

How many schema match a tree of length N ?

For example, consider the program

( + ( – 2 x ) ( – 3 y ) )

+

–

2 x

–

3 y

Schema Theory for GP

Definitions:

• m(H, t) = number of schema H at generation # t

• G = structure of schema H

For example, if H = ( + ( – # y ) # )

then G = ( # ( # # # ) # )

Schema Theory for GP

• m(H, t) = number of schema H at gen. # t

• m(H, t+1/2) = number of schema selected for crossover / mutation

• m(H, t+1) = number of schema after crossover / mutation

• Fitness proportionate selection:m(H, t+1/2) = m(H, t) f(H, t) / fave

Schema Theory for GP

Crossover: two ways for destruction of schema H1. Program h H crosses with program g that has a

different structure than G Event D1

2. Program h H crosses with program g that has the same structure as G, but g H Event D2

Pr(crossover destruction) = Pr(D) = Pr(D1) + Pr( D2 )

Crossover Destruction – Type 1

+

yx

+

–

2 x

–

3 y

( + ( – 2 x ) ( – 3 y ) )

( + x y )

Crossover results in

( + y ( – 3 y ) )

( + x ( – 2 x ) )

Both schema are destroyed

+

y

x

+

–

2 x

–

3 y

Crossover Destruction – Type 2

If h = (+ x y) H = ( # x y) and g = (g1 y x) Hthen crossover between the + and x gives:

( + y x ) and (g1 x y ) H, schema preserved

But if h = (+ x y) H = ( + x #) and g = ( g1 y x) H then crossover between the + and x gives:

( + y x ) and (g1 x y ) H, schema destroyed

(unless g1 = “+”)

Crossover Destruction – Type 1

Program h H crosses with program g that has a different structure than G Event D1

M = population size

Pr(D1) = Pr(D | g G) Pr(g G)

Pr(g G) = [M – m(G, t+1/2)] / M

Pr(D | g G) = Pdiff

Crossover Destruction – Type 2

Program h H crosses with program g that has the same structure as G but g H Event D2

Pr(D2) = Pr(D | g G) Pr(g G)

Pr(g G) = m(G, t+1/2) / M

Pr(D | g G) = Pr(D | g H) Pr(g H | g G)

Pr(g H | g G) =

[ m(G, t+1/2) – m(H, t+1/2) ] / m(G, t+1/2)

Crossover Destruction – Type 2

Pr(D | g H) ≤ L(H) / [ N(H) – 1 ]

Therefore,

Pr(D2) ≤ { L(H) / [ N(H) – 1 ] }

[ m(G, t+1/2) – m(H, t+1/2) ] / M

Crossover Destruction

Pr(D) = Pr (D1) + Pr ( D2 ) ≤

{ [M – m(G, t+1/2)] / M } Pdiff +

{ L(H) / [ N(H) – 1 ] }

[ m(G, t+1/2) – m(H, t+1/2) ] / M

Crossover Destruction

Crossover occurs with probability pc

m(H, t+1) = (1 – pc ) m(H, t+1/2) +

pc m(H, t+1/2) [1 – Pr(D)]

m(H, t+1) = m(H, t+1/2) [1 – pc Pr(D)]

Mutation Destruction

Pr(Mutation Destruction) =

1 – (1 – pm)o(H) pm o(H)

m(H, t+1) =

m(H, t+1/2) [1 – pc Pr(D)] [1 – pm o(H) ]

Combine previous results to obtain a lower bound for m(H, t+1)

Schema Theory for GP

m(H, t+1)

[ m(H, t) f(H, t) / fave ] [ 1 - pm o(H) ] ×

[1 – pc { ( 1 – m(G,t) f(G,t) / M fave) Pdiff +

( L(H) / [ N(H) – 1 ] )

[ m(G,t) f(G,t) – m(H,t) f(H,t) ] / M fave) } ]

Slightly more complex than GA schema theorem

Schema Theory for GP

Simplification: early in the GP run (highdiversity) we have:

• Pr(D | g G) = Pdiff 1

• m(G,t) f(G,t) / M fave << 1

Schema Theory for GP

m(H, t+1)

[ m(H, t) f(H, t) / fave ] (1 - pm o(H)) ×

[1 – pc { ( 1 – m(G,t) f(G,t) / M fave) Pdiff +

( L(H) / [ N(H) – 1 ] )

(m(G,t) f(G,t) – m(H,t) f(H,t)) / M fave) } ]

[ m(H, t) f(H, t) / fave ] (1 - pm o(H)) ×

[1 – pc { 1 + ( L(H) / [ N(H) – 1 ] )

(– m(H,t) f(H,t)) / M fave) } ]

Schema Theory for GPs

For short schema, L(H) / [ N(H) – 1 ] << 1 and

m(H, t) f(H, t) / fave (1 - pm o(H)) ×

[1 – pc { 1 + ( L(H) / [ N(H) – 1 ] )

(– m(H,t) f(H,t)) / M fave) } ]

[ m(H, t) f(H, t) / fave ] (1 - pm o(H)) (1 – pc )

Schema Theory for GPs

m(H, t+1)

[ m(H, t) f(H, t) / fave ] (1 - pm o(H)) (1 – pc )

Nearly the same as the GA schema theorem:

m(H, t+1)

[ m(H, t) f(H, t) / fave ] (1 - pm o(H)) ×

(1 – pc L(H) / [ N(H) – 1 ] )

GP references

• www.genetic-programming.org

• www.genetic-programming.com

• cswww.essex.ac.uk/staff/poli