GENETIC ALGORITHMS AND GENETIC PROGRAMMING. John R. Koza Consulting Professor (Medical Informatics) Department of Medicine School of Medicine Consulting

GENETIC ALGORITHMS AND GENETIC PROGRAMMING

John R. KozaConsulting Professor (Medical Informatics)

Department of Medicine

School of Medicine

Consulting Professor

Department of Electrical Engineering

School of Engineering

Stanford University

Stanford, California 94305

[email protected]

http://www.smi.stanford.edu/people/koza/

DEFINITION OF THE GENETIC ALGORITHM (GA)

The genetic algorithm is a probabalistic search algorithm that iteratively transforms a set (called a population) of mathematical objects (typically fixed-length binary character strings), each with an associated fitness value, into a new population of offspring objects using the Darwinian principle of natural selection and using operations that are patterned after naturally occurring genetic operations, such as crossover (sexual recombination) and mutation.

GENETIC ALGORITHM (GA)

Generation 0 Generation 1

Individuals Fitness Offspring

011 $3 111

001 $1 010

110 $6 110

010 $2 010

HAMBURGER RESTAURANT PROBLEM

• Price1 = $ 0.50 price0 = $10.00 price

• Drink1 = Coca Cola0 = Wine

• Ambiance1 = Fast snappy service0 = Leisurely service with tuxedoed waiter

CHROMOSOME (GENOME) OF THE GLOBAL OPTIMUM

McDONALD's

1 1 1

THE SEARCH SPACE

• Alphabet size K=2, Length L=3

• Size of search space: KL=2L=23=8

1 000

2 001

3 010

4 011

5 100

6 101

7 110

8 111

IMPRACTICALITY OF RANDOM OR ENUMERATIVE SEARCH

• 81-bit problems are very small for GA

• However, even if L is as small as 81, 281 ~ 1027 = number of nanoseconds since the beginning of the universe 15 billion years ago

GA FLOWCHART

GENERATION 0

Generation 0

1 011 3

2 001 1

3 110 6

4 010 2

Total

Worst

Average

Best



PROBABILISTIC SELECTION BASED ON FITNESS

• Better individuals are preferred• Best is not always picked• Worst is not necessarily excluded• Nothing is guaranteed• Mixture of greedy exploitation and adventurous exploration• Similarities to simulated annealing (SA)

PROBABILISTIC SELECTION BASED ON FITNESS

0.5

0.08

0.250.17

DARWINIAN FITNESS PROPORTIONATE SELECTION

Generation 0 Mating pool

1 011 3 .25 011 3

2 001 1 .08 110 6

3 110 6 .50 110 6

4 010 2 .17 010 2

Total 12 17

Worst 1 2

Average 3.00 4.5

Best 6 6



MUTATION OPERATION

• Parent chosen probabilistically based on fitness

• Mutation point chosen at random

• One offspring

Parent

010

Parent

--0

Offspring

011

AFTER MUTATION OPERATION

Generation 0 Mating pool Generation 1

1 011 3 .25 011 3

2 001 1 .08 110 6

3 110 6 .50 110 6

4 010 2 .17 010 2 --- 011 3

Total 12 17

Worst 1 2

Average 3.00 4.5

Best 6 6

CROSSOVER OPERATION

• 2 parents chosen probabilistically based on fitness

Parent 1 Parent 2

011 110

CROSSOVER (CONTINUED)• Interstitial point picked at random

• 2 remainders

• 2 offspring produced by crossover

Remainder 1 Remainder 2

- - 1 - - 0

Offspring 1 Offspring 2

111 010

Fragment 1 Fragment 2

01- 11-

AFTER CROSSOVER OPERATION


1 011 3 .25 011 3 2 111 7

2 001 1 .08 110 6 2 010 2

3 110 6 .50 110 6

4 010 2 .17 010 2

Total 12 17

Worst 1 2

Average 3.00 4.5

Best 6 6

AFTER REPRODUCTION OPERATION


1 011 3 .25

2 001 1 .08

3 110 6 .50 110 6 --- 110 6

4 010 2 .17

Total 12 17

Worst 1 2

Average 3.00 4.5

Best 6 6



GENERATION 1


1 011 3 .25 011 3 2 111 7

2 001 1 .08 110 6 2 010 2

3 110 6 .50 110 6 --- 110 6

4 010 2 .17 010 2 --- 011 3

Total 12 17 18

Worst 1 2 2

Average 3.00 4.5 4.5

Best 6 6 7





PROBABILISTIC STEPS

• The initial population is typically random• Probabilistic selection based on fitness

- Best is not always picked- Worst is not necessarily excluded

• Random picking of mutation and crossover points

• Often, there is probabilistic scenario as part of the fitness measure

ANTENNA DESIGN

ANTENNA DESIGN

• The problem (Altshuler and Linden 1998) is to determine the x-y-z coordinates of the 3-dimensional position of the ends (X1, Y1, Z1, X2, Y2, Z2,… , X7, Y7, Z7) of 7 straight wires so that the resulting 7-wire antenna satisfies certain performance requirements

• The first wire starts at feed point (0, 0, 0) in the middle of the ground plane

• The antenna must fit inside the 0.5 cube

ANTENNA GENOME

• 105-bit chromosome (genome)• Each x-y-z coordinate is represented by 5 bits

(4-bit granularity for data plus a sign bit)• Total chromosome is 3 7 5 = 105 bits

X1 Y1 Z1 X2 Y2 Z2 …

+0010 -1110 +0001 +0011 -1011 +0011 …

ANTENNA FITNESS

• Antenna is for ground-to-satellite communications for cars and handsets

• We desire near-uniform gain pattern 10 above the horizon

• Fitness is measured based on the antenna's radiation pattern. The radiation pattern is simulated by National Electromagnetics Code (NEC)

ANTENNA FITNESS

• Fitness is sum of the squares of the difference between the average gain and the antenna's gain

• Sum is taken for angles between -90 and +90 and all azimuth angles from 0 to 180

• The smaller the value of fitness, the better

GRAPH OF ANTENNA FITNESS

U. S. PATENT 5,719,794

10-MEMBER TRUSS

10-MEMBER TRUSS

• Prespecified topological arrangement of the 10 members, the load, and the wall (Goldberg and Samtani 1986)

• Truss has 10 members (6 are length of 30 feet and 4 are length 30√2 = 41 feet)

• The problem is to determine the cross-sectional areas (A1, … , A10) of each of the 10 members so as to minimize weight of the material for a truss that supports the 2 loads

• The weight is based on volume (i.e., cross-sectional area length)

TRUSS GENOME

• 40-bit chromosome (genome)• 4-bit granularity for truss diameters• 0000 = smallest diameter• 1111 = largest diameter• Total chromosome is 4 10 = 40 bits

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

0010 1110 0001 0011 1011 0011 1111 0011 0011 1010

TRUSS FITNESS

• Two-part (multiobjective) fitness measure– First, fitness is computed by taking the sum,

over the 10 members, of the cross-sectional area of each member times the length of each member (30 feet or 30√2 = 41 feet).

– Second, a penalty (up to 10%) is imposed for violating the stress constraints. Stresses are computed using standard mechanical engineering techniques.

• The smaller the total fitness, the better

CELLULAR AUTOMATA

STATE TRANSITION TABLE

# WWW WW W X E EE EEE Rule

0 0 0 0 0 0 0 1 a0

1 0 0 0 0 0 1 0 a1

2 0 0 0 0 1 0 0 a2

3 0 0 0 0 1 1 0 a3

4 0 0 1 1 0 0 0 a4

… … … … … … … … …

127 1 1 1 1 1 1 1 a127

CELLULAR AUTOMATA

• 128-bit chromosome (genome)

A0 A1 A2 … A127

a0 a1 a2 … a127

PROBLEM-SPECIFIC GENOMES

N M GENOME

1 1 0 1 1 1 1

1 1 1 1 0 1 1

1 0 1 1 1 0 0

1 1 1 1 1 1 1

1 1 0 1 1 1 1

1 1 1 1 0 1 1

1 1 0 1 1 1 0

GENETIC ALGORITHM USING VARIABLE-LENGTH STRINGS

• 5-WIRE ANTENNA (5 15 = 75 bits)

• 4-WIRE ANTENNA (4 15 = 60 bits)

X1 Y1 Z1 … X5 Y5 Z5

+0010 -1110 +0001 … +0010 -1110 +0001

X1 Y1 Z1 … X4 Y4 Z4

+1010 -0110 +1101 … +1010 -0110 +1001

GENETIC PROGRAMMING

THE CHALLENGE

"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?"

Attributed to Arthur Samuel (1959)

CRITERION FOR SUCCESS

"The aim [is] ... to get machines to exhibit behavior, which if done by humans, would be assumed to involve the use of intelligence.“

Arthur Samuel (1983)

REPRESENTATIONS

• Decision trees• If-then production

rules• Horn clauses• Neural nets• Bayesian networks• Frames• Propositional logic

• Binary decision diagrams

• Formal grammars • Coefficients for

polynomials• Reinforcement

learning tables• Conceptual clusters • Classifier systems

A COMPUTER PROGRAM

GENETIC PROGRAMMING (GP)

• GP applies the approach of the genetic algorithm to the space of possible computer programs

• Computer programs are the lingua franca for expressing the solutions to a wide variety of problems

• A wide variety of seemingly different problems from many different fields can be reformulated as a search for a computer program to solve the problem.

GP MAIN POINTS

• Genetic programming now routinely delivers high-return human-competitive machine intelligence.

• Genetic programming is an automated invention machine.

• Genetic programming has delivered a progression of qualitatively more substantial results in synchrony with five approximately order-of-magnitude increases in the expenditure of computer time.

DEFINITION OF “HIGH-RETURN”

The AI ratio (the “artificial-to-intelligence” ratio) of a problem-solving method as the ratio of that which is delivered by the automated operation of the artificial method to the amount of intelligence that is supplied by the human applying the method to a particular problem

DEFINITION OF “ROUTINE”

A problem solving method is routine if it is general and relatively little human effort is required to get the method to successfully handle new problems within a particular domain and to successfully handle new problems from a different domain.

CRITERIA FOR “HUMAN-COMPETITIVENESS”

• Previously patented, an improvement over a patented invention, or patentable today

• Publishable in its own right as a new scientific result independent of the fact that the result was mechanically created

• Holds it own in regulated competition against humans (or programs)

• 5 other similar criteria that are “arms-length” from the fields of AI, ML, GP

PROGRESSION OF QUALITATIVELY MORE SUBSTANTIAL RESULTS PRODUCED BY GP

• Toy problems

• Human-competitive non-patent results

• 20th-century patented inventions

• 21st-century patented inventions

• Patentable new inventions

GP FLOWCHART

A COMPUTER PROGRAM IN C

int foo (int time){ int temp1, temp2; if (time > 10) temp1 = 3; else temp1 = 4; temp2 = temp1 + 1 + 2; return (temp2);}

OUTPUT OF C PROGRAMTime Output

0 6

1 6

2 6

3 6

4 6

5 6

6 6

7 6

8 6

9 6

10 6

11 7

12 7

PROGRAM TREE

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

CREATING RANDOM PROGRAMS

CREATING RANDOM PROGRAMS

• Available functions F = {+, -, *, %, IFLTE}

• Available terminals T = {X, Y, Random-Constants}

• The random programs are:– Of different sizes and shapes

– Syntactically valid

– Executable

GP GENETIC OPERATIONS

• Reproduction

• Mutation

• Crossover (sexual recombination)

• Architecture-altering operations

MUTATION OPERATION

MUTATION OPERATION

• Select 1 parent probabilistically based on fitness• Pick point from 1 to NUMBER-OF-POINTS• Delete subtree at the picked point• Grow new subtree at the mutation point in same

way as generated trees for initial random population (generation 0)

• The result is a syntactically valid executable program

• Put the offspring into the next generation of the population

CROSSOVER OPERATION

CROSSOVER OPERATION

• Select 2 parents probabilistically based on fitness

• Randomly pick a number from 1 to NUMBER-OF-POINTS for 1st parent

• Independently randomly pick a number for 2nd parent

• The result is a syntactically valid executable program

• Put the offspring into the next generation of the population

• Identify the subtrees rooted at the two picked points

REPRODUCTION OPERATION

• Select parent probabilistically based on fitness

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

FIVE MAJOR PREPARATORY STEPS FOR

GP

• Determining the set of terminals• Determining the set of functions• Determining the fitness measure • Determining the parameters for the run• Determining the method for designating a result and

the criterion for terminating a run

ILLUSTRATIVE GP RUN

SYMBOLIC REGRESSION

Independent variable X

Dependent variable Y

-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

PREPARATORY STEPS

Objective: Find a computer program with one input (independent variable X) whose output equals the given data

1 Terminal set: T = {X, Random-Constants}

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

3 Fitness: The sum of the absolute value of the differences between the candidate program’s output and the given data (computed over numerous values of the independent variable x from –1.0 to +1.0)

4 Parameters: Population size M = 4

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

SYMBOLIC REGRESSION

POPULATION OF 4 RANDOMLY CREATED INDIVIDUALS FOR GENERATION 0

SYMBOLIC REGRESSION x2 + x + 1

FITNESS OF THE 4 INDIVIDUALS IN GEN 0

x + 1 x2 + 1 2 x

0.67 1.00 1.70 2.67

SYMBOLIC REGRESSION x2 + x + 1

GENERATION 1

Copy of (a)

Mutant of (c) picking “2” as mutation point

First offspring of crossover of (a) and (b) picking “+” of parent (a) and left-most “x” of parent (b) as crossover points

Second offspring of crossover of (a) and (b) picking “+” of parent (a) and left-most “x” of parent (b) as crossover points

CLASSIFICATION

GP TABLEAU – INTERTWINED SPIRALS

Objective: Create a program to classify a given point in the x-y plane to the red or blue spiral

1 Terminal set: T = {X,Y,Random-Constants}

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

3 Fitness: The number of correctly classified points (0 – 194)

4 Parameters: M = 10,000. G = 51

5 Termination: An individual program scores 194

WALL-FOLLOWER

FITNESS

BEST OF GENERATION 57

BOX MOVER – BEST OF GEN 0

BOX MOVERGEN 45 – FITNESS CASE 1

TRUCK BACKER UPPER

TRUCK BACKER UPPER

• 4-Dimensional control problem– horizontal position, x– vertical position, y– angle between trailer and horizontal, t

– angle between trailer and cab, d

• One control variable (steering wheel turn angle)

• State transition equations map the 4 state variables into 1 output (the control variable)

• Simulation run over many initial conditions and over hundreds of time steps

GENETIC PROGRAMMING: ON THE PROGRAMMING OF COMPUTERS BY

MEANS OF NATURAL SELECTION

(Koza 1992)

2 MAIN POINTS FROM 1992 BOOK

• Virtually all problems in artificial intelligence, machine learning, adaptive systems, and automated learning can be recast as a search for a computer program.

• Genetic programming provides a way to successfully conduct the search for a computer program in the space of computer programs.

SOME RESULTS FROM 1992 BOOK

• Intertwined Spirals• Truck Backer Upper• Broom Balancer• Wall Follower• Box Mover• Artificial Ant• Differential Games• Inverse Kinematics• Central Place Foraging• Block Stacking

• Randomizer

• Cellular Automata

• Task Prioritization

• Image Compression

• Econometric Equation

• Optimization

• Boolean Function Learning

• Co-Evolution of Game-Playing Strategies


• Toy problems





COMPUTER PROGRAMS

• Subroutines provide one way to REUSE code possibly with different instantiations of the dummy variables (formal parameters)

• Loops (and iterations) provide a 2nd way to REUSE code• Recursion provide a 3rd way to REUSE code• Memory provides a 4th way to REUSE the results of

executing code

SYMBOLIC REGRESSION

Fitness case L0 W0 H0 L1 W1 H1 Dependent variable D

1 3 4 7 2 5 3 54

2 7 10 9 10 3 1 600

3 10 9 4 8 1 6 312

4 3 9 5 1 6 4 111

5 4 3 2 7 6 1 -18

6 3 3 1 9 5 4 -171

7 5 9 9 1 7 6 363

8 1 2 9 3 9 2 -36

9 2 6 8 2 6 10 -24

10 8 1 10 7 5 1 45

EVOLVED SOLUTION

(- (* (* W0 L0) H0) (* (* W1 L1) H1))

DIFFERENCE IN VOLUMES

D = L0W0H0 – L1W1H1

AUTOMATICALLY DEFINED FUNCTION volume

AUTOMATICALLY DEFINED FUNCTION volume

(progn

(defun volume (arg0 arg1 arg2)

(values

(* arg0 (* arg1 arg2))))

(values

(- (volume L0 W0 H0)

(volume L1 W1 H1))))

AUTOMATICALLY DEFINED FUNCTIONS

• ADFs provide a way to REUSE code• Code is typically reused with different

instantiations of the dummy variables (formal parameters)

ADDITION OF V0 AND V1

Fitness case L0 W0 H0 L1 W1 H1 V0 V1 D

1 3 4 7 2 5 3 84 30 54

2 7 10 9 10 3 1 630 30 600

3 10 9 4 8 1 6 360 48 312

4 3 9 5 1 6 4 135 24 111

5 4 3 2 7 6 1 24 42 -18

6 3 3 1 9 5 4 9 180 -171

7 5 9 9 1 7 6 405 42 363

8 1 2 9 3 9 2 18 54 -36

9 2 6 8 2 6 10 96 120 -24

10 8 1 10 7 5 1 80 35 45

DIVIDE AND CONQUER

DIVIDE AND CONQUER

• Decompose a problem into sub-problems

• Solve the sub-problems

• Assemble the solutions of the sub-problems into a solution for the overall problem

CHANGE OF REPRESENTATION

CHANGE OF REPRESENTATION

• Identify regularities

• Change the representation

• Solve the overall problem

ADF IMPLEMENTATION

• Each overall program in population includes – a main result-producing branch (RPB) and– function-defining branch (i.e., automatically defined

function, ADF)

• In generation 0, create random programs with different ingredients for the RPB and the ADF– Terminal set for ADF typically contains dummy

arguments (formal parameters), such as ARG0, ARG1, …

– Function set of the RPB contains ADF0– ADFs are private and associated with a particular

individual program in the population

ADF MUTATION

• Select parent probabilistically on the basis of fitness

• Pick a mutation point from either RPB or an ADF

• Delete sub-tree rooted at the picked point • Grow a new sub-tree at the picked point

composed of the allowable ingredients appropriate for the picked point

• The offspring is a syntactically valid executable program

ADF CROSSOVER

• Select parent probabilistically on the basis of fitness

• Pick a crossover point from either RPB or an ADF of the FIRST patent

• The choice of crossover point in the SECOND parent is RESTRICTED to the picked RPB or to the picked ADF

• The sub-trees are swapped• The offspring are syntactically valid executable

programs

GENETIC PROGRAMMING II: AUTOMATIC DISCOVERY OF REUSABLE PROGRAMS

(Koza 1994)

MAIN POINTS OF 1994 BOOK

• Scalability is essential for solving non-trivial problems in artificial intelligence, machine learning, adaptive systems, and automated learning

• Scalability can be achieved by reuse• Genetic programming provides a way to

automatically discover and reuse subprograms in the course of automatically creating computer programs to solve problems

COMPUTER PROGRAMS



executing code

MEMORY

Settable (named)

variables

Indexed vector

memory

Matrix

memory Relational memory

LANGDON'S DATA STRUCTURES

• Stacks

• Queues

• Lists

• Rings

COMPUTER PROGRAMS



executing code

AUTOMATICALLY DEFINED ITERATION (ADI)

• The overall program includes an iteration-performing branch (IPB) in addition to a result-producing branch (RPB) and function-defining branches (ADF)

• There are no infinite loops because the iteration is performed over a known, fixed set – protein or DNA sequence (of varying length)– time-series data– two-dimensional array of pixels

• Memory is usually involved and is used to communicate between IPB, RPB, and ADF

TRANSMEMBRANE SEGMENT IDENTIFICATION PROBLEM

Goal is to classify a given protein segment as being a transmembrane domain or

non-transmembrane area of the protein


(progn (defun ADF0 ()(ORN (ORN (ORN (I?) (H?)) (ORN (P?) (G?))) (ORN (ORN (ORN (Y?)

(N?)) (ORN (T?) (Q?))) (ORN (A?) (H?)))))) (defun ADF1 ()(values (ORN (ORN (ORN (A?) (I?)) (ORN (L?) (W?))) (ORN (ORN

(T?) (L?)) (ORN (T?) (W?)))))) (defun ADF2 ()(values (ORN (ORN (ORN (ORN (ORN (D?) (E?)) (ORN (ORN (ORN

(D?) (E?)) (ORN (ORN (T?) (W?)) (ORN (Q?) (D?)))) (ORN (K?) (P?)))) (ORN (K?) (P?))) (ORN (T?) (W?))) (ORN (ORN (E?) (A?)) (ORN (N?) (R?))))))

(progn (loop-over-residues (SETM0 (+ (- (ADF1) (ADF2)) (SETM3 M0))))

(values (% (% M3 M0) (% (% (% (- L -0.53) (* M0 M0)) (+ (% (% M3 M0) (% (+ M0 M3) (% M1 M2))) M2)) (% M3 M0))))))


• in-sample correlation of 0.976

• out-of-sample correlation of 0.968

• out-of-sample error rate 1.6%

AUTOMATICALLY DEFINED LOOP (ADL)

• loop initialization branch, LIB• loop condition branch, LCB• loop body branch, LBB• loop update branch, LUB

ADL

COMPUTER PROGRAMS



executing code

AUTOMATICALLY DEFINED RECURSION (ADR)

• recursion condition branch, RCB• recursion body branch, RBB• recursion update branch, RUB• recursion ground branch, RGB

ADR

HUMAN-COMPETITIVE RESULTS(NOT RELATED TO PATENTS)

Transmembrane segment identification problem for proteins

Motifs for D–E–A–D box family and manganese superoxide dismutase family of proteins

Cellular automata rule for Gacs-Kurdyumov-Levin (GKL) problem

Quantum algorithm for the Deutsch-Jozsa “early promise” problem

Quantum algorithm for Grover’s database search problem

Quantum algorithm for the depth-two AND/OR query problem

Quantum algorithm for the depth-one OR query problem

Protocol for communicating information through a quantum gate

Quantum dense coding

Soccer-playing program that won its first two games in the 1997 Robo Cup competition

Soccer-playing program that ranked in the middle of field in 1998 Robo Cup competition

Antenna designed by NASA for use on spacecraft

Sallen-Key filter


• Toy problems





GENETIC PROGRAMMING III: DARWINIAN INVENTION AND PROBLEM SOLVING

(Koza, Bennett, Andre, Keane 1999)

SUBROUTINE DUPLICATION

SUBROUTINE CREATION

SUBROUTINE DELETION

ARGUMENT DUPLICATION

ARGUMENT DELETION

16 ATTRIBUTES OF A SYSTEM FOR AUTOMATICALLY CREATING COMPUTER

PROGRAMS

• Starts with "What needs to be done"

• Tells us "How to do it"• Produces a computer

program• Automatic determination of

program size• Code reuse• Parameterized reuse• Internal storage• Iterations, loops, and

recursions

• Self-organization of hierarchies

• Automatic determination of program architecture

• Wide range of programming constructs

• Well-defined• Problem-independent• Wide applicability• Scalable• Competitive with human-

produced results

GENETIC PROGRAMMING PROBLEM SOLVER (GPPS)

AUTOMATIC SYNTHESIS OF BOTH THE TOPOLOGY AND

SIZING OF ANALOG ELECTRICAL CIRCUITS BY MEANS OF

DEVELOPMENTAL GENETIC PROGRAMMING

AUTOMATED CIRCUIT SYNTHESIS

• The topology of a circuit includes specifying the gross number of components in the circuit, the type of each component (e.g., a capacitor), and a netlist specifying where each lead of each component is to be connected.

• Sizing involves specifying the values (typically numerical) of each of the circuit's components.

COMPONENT-CREATING FUNCTIONS

• Resistor R function

• Capacitor C function

• Inductor L function

• Diode D function

• Transistor Q function (3-leaded)

COMPONENT-CREATING FUNCTIONS

TOPOLOGY-MODIFYING FUNCTIONS

• SERIES division• PARALLEL division• VIA• FLIP

TOPOLOGY-MODIFYING FUNCTIONS

DEVELOPMENT-CONTROLLING FUNCTIONS

• END function• NOP (No Operation) function• SAFE_CUT function

THE INITIAL CIRCUIT

DEVELOPMENTAL GP

(LIST (C (– 0.963 (– (– -0.875 -0.113) 0.880)) (series (flip end) (series (flip end) (L -0.277 end) end) (L (– -0.640 0.749) (L -0.123 end)))) (flip (nop (L -0.657 end)))))

CAPACITOR-CREATING FUNCTION

(LIST (C (– 0.963 (– (– -0.875

-0.113) 0.880)) (series (flip

end) (series (flip end) (L –

0.277 end) end) (L (– -0.640

0.749) (L -0.123 end)))) (flip

(nop (L -0.657 end)))))

CAPACITOR-CREATING FUNCTION

SERIES DIVISION FUNCTION

(LIST (C (– 0.963 (– (– -0.875

-0.113) 0.880)) (series (flip

end) (series (flip end) (L –

0.277 end) end) (L (– -0.640

0.749) (L -0.123 end)))) (flip

(nop (L -0.657 end)))))

SERIES DIVISION

DEVELOPMENTAL GP

EVALUATION OF FITNESS

Program Tree

+ IN OUT z0

Embryonic Circuit

Fully Designed Circuit (NetGraph)

Circuit Netlist (ascii)

Circuit Simulator (SPICE)

Circuit Behavior (Output)

Fitness

DESIRED BEHAVIOR OF A LOWPASS FILTER

EVOLVED CAMPBELL FILTER

U. S. patent 1,227,113George Campbell

American Telephone and Telegraph1917

EVOLVED ZOBEL FILTER

U. S. patent 1,538,964Otto Zobel

American Telephone and Telegraph Company1925

EVOLVED SALLEN-KEY FILTER

EVOLVED DARLINGTON EMITTER-FOLLOWER SECTION

U. S. patent 2,663,806

Sidney Darlington

Bell Telephone Laboratories

1953

NEGATIVE FEEDBACK

HAROLD BLACK’S RIDE ON THE LACKAWANNA FERRY

Courtesy of Lucent Technologies

20th-CENTURY PATENTS

Campbell ladder topology for filters

Zobel “M-derived half section” and “constant K” filter sections

Crossover filter

Negative feedback

Cauer (elliptic) topology for filters

PID and PID-D2 controllers

Darlington emitter-follower section and voltage gain stage

Sorting network for seven items using only 16 steps

60 and 96 decibel amplifiers

Analog computational circuits

Real-time analog circuit for time-optimal robot control

Electronic thermometer

Voltage reference circuit

Philbrick circuit

NAND circuit

Simultaneous synthesis of topology, sizing, placement, and routing


• Toy problems





SIX POST-2000 PATENTED INVENTIONS

EVOLVED HIGH CURRENT LOAD CIRCUIT

REGISTER-CONTROLLED CAPACITOR CIRCUIT

LOW-VOLTAGE CUBIC CIRCUIT

VOLTAGE-CURRENT-CONVERSION CIRCUIT

LOW-VOLTAGE BALUN CIRCUIT

TUNABLE INTEGRATED ACTIVE FILTER

21st-CENTURY PATENTED INVENTIONS

Low-voltage balun circuit

Mixed analog-digital variable capacitor circuit

High-current load circuit

Voltage-current conversion circuit

Cubic function generator

Tunable integrated active filter


• Toy problems





NOVELTY-DRIVEN EVOLUTION

• Two factors in fitness measure– Circuit’s behavior in the frequency domain

– Largest number of nodes and edges (circuit components) of a subgraph of the given circuit that is isomorphic to a subgraph of a template representing the prior art. Graph isomorphism algorithm with the cost function being based on the number of shared nodes and edges (instead of just the number of nodes).

NOVELTY-DRIVEN EVOLUTION

• For circuits not scoring the maximum number of hits (101), the fitness of a circuit is the product of the two factors.

• For circuits scoring 101 hits (100%-compliant individuals), fitness is the number of shared nodes and edges divided by 10,000.

PRIOR ART TEMPLATE

NON-INFRINGING SOLUTION NO. 1

NON-INFRINGING SOLUTION NO. 5

GP AS AN INVENTION MACHINE

CIRCUIT-CONSTRUCTING PROGRAM TREE WITH ADFs

LOWPASS FILTER WITH ADFs

ADF0

AUTOMATIC SYNTHESIS OF CIRCUIT LAYOUT

INCLUDING THE PLACEMENT OF COMPONENTS AND ROUTING OF

WIRES

ALONG WITH THE TOPOLOGY AND SIZING

CIRCUIT LAYOUT

• Circuit placement involves the assignment of each of the circuit's components to a particular physical location on a printed circuit board or silicon wafer.

• Routing involves the assignment of a particular physical location to the wires between the leads of the circuit's components.

LAYOUT

LAYOUT — GENERATION 0

100%-COMPLIANT LOWPASS FILTER

GENERATION 25 WITH 5 CAPACITORS AND 11 INDUCTORS AREA OF 1775.2


GENERATION 30 WITH 10 INDUCTORS AND 5 CAPACITORS AREA OF 950.3


BEST-OF-RUN CIRCUIT OF GENERATION 138 WITH 4 INDUCTORS AND 4 CAPACITORS AREA OF 359.4

LAYOUT 60 DB AMPLIFIER

AUTOMATIC SYNTHESIS OF BOTH THE TOPOLOGY

AND TUNING OF CONTROLLERS

PROGRAM TREE FOR A CONTROLLER

CONTROLLER BLOCKS

• gain

• integrator

• differentiator

• adder

• subtractor

• multiplier

• differential input integrators

• inverter

• lead

• lag

• two-parameter lag

• absolute value

• limiter

• divider

• delay

• conditional operators (switches)

FUNCTION SET FOR CONTROLLER SYNTHESIS

F = {GAIN,INVERTER,LEAD,LAG,LAG2, DIFFERENTIAL_INPUT_INTEGRATOR, DIFFERENTIATOR, ADD_SIGNAL, SUB_SIGNAL,ADD_3_SIGNAL,ADF0, ADF1,ADF2,ADF3,ADF4}

TERMINAL SET FOR CONTROLLER SYNTHESIS

T = { REFERENCE_SIGNAL, CONTROLLER_OUTPUT, PLANT_OUTPUT}

CONSTRAINED SYNTACTIC STRUCTURE

• A grammar that specifies what functions and terminals are allowed as arguments to particular functions

• For example, the first argument of the GAIN function must be a value-setting subtree whereas the second can be from the general pool of functions

• Also known as strong typing

TWO-LAG PLANT

8 FITNESS CASES

• 8 elements of the fitness measure represent 2 2 2 choices:– 2 different values of the plant's internal gain,

K (1.0 and 2.0), – 2 different values of the plant's time constant

(0.5 and 1.0), – 2 different values for the height of the

reference signal (rising from 0 to 1 volts or from 0 to 1 microvolts at t = 100 milliseconds

FITNESS MEASURE

• For each of these 8 fitness cases, a transient analysis (time domain) is performed using the SPICE simulator.

• The contribution to fitness for the 8 elements is

– Integral of time-weighted absolute error (ITAE)

– e(t) is difference between plant output and reference signal.

– Multiplication by B (106 or 1) makes both reference signals equally influential.

– Additional weighting function, A, heavily penalizes non-compliant amounts of overshoot. A weights all variations up to 2% above the reference signal by 1.0, but bigger variations by 10.0.

EVOLVED CONTROLLER FOR TWO-LAG PLANT

LESS ITAE AND OVERSHOOT

BETTER DISTURBANCE REJECTION

REVERSE ENGINEERING OF METABOLIC PATHWAYS

EVOLVED PATHWAY

ANTENNA SYNTHESIS USING GP

(PROGN3 (TURN-RIGHT 0.125) (LANDMARK (REPEAT 2 (PROGN2 (DRAW 1.0 HALF-MM-WIRE) (DRAW 0.5 NO-WIRE))) (TRANSLATE-RIGHT 0.125 0.75))

USING A TURTLE TO DRAW TWO-DIMENSIONAL ANTENNA

BEST-OF-RUN ANTENNA FROM GENERATION 90

3-DIMENSIONAL ANTENNA

NASA EVOLVED ANTENNA

To be on satellite to be launched in 2004

OTHER STRUCTURES

GENETIC NETWORK FOR lac operon

EVOLVED NETWORK

(IF (< LACTOSE_LEVEL 9.139 ) (IF (<REPRESSOR_LEVEL 6.270 ) (IF (> GLUCOSE_LEVEL5.491 ) 2.02 (IF (< CAP_LEVEL 0.639 ) 2.033 (IF(< CAP_LEVEL 4.858 ) (IF (> LACTOSE_LEVEL 2.511 )(IF (> CAP_LEVEL 7.807 ) 5.586 (IF (>LACTOSE_LEVEL 2.114 ) 1.978 2.137 ) ) 0.0 ) (IF(> REPRESSOR_LEVEL 4.015 ) 0.036 (IF (<GLUCOSE_LEVEL 5.128 ) 10.0 (IF (< REPRESSOR_LEVEL4.268 ) 2.022 9.122 ) ) ) ) ) ) (IF (> CAP_LEVEL0.842 ) 0.0 5.97 ) ) (IF (< CAP_LEVEL 1.769 )2.022 (IF (< GLUCOSE_LEVEL 2.382 ) (IF (>LACTOSE_LEVEL 1.256 ) (IF (> LACTOSE_LEVEL 1.933) (IF (> GLUCOSE_LEVEL 2.022 ) (IF (<GLUCOSE_LEVEL 5.183 ) 6.323 (IF (> CAP_LEVEL1.208 ) 9.713 0.842 ) ) 10.0 ) (IF (>GLUCOSE_LEVEL 6.270 ) 2.109 ) 1.965 ) ) 0.665 )

1.982 ) ) )

IN C-STYLE PSEUDO CODE

if(LACTOSE_LEVEL < 9.139) { if(REPRESSOR_LEVEL <

6.270) { LAC_mRNA_LEVEL =

2.022; } else { LAC_mRNA_LEVEL =

0.0; }}

else{ if(CAP_LEVEL < 1.769) { LAC_mRNA_LEVEL =

2.022; } else { if(GLUCOSE_LEVEL <

2.382) { LAC_mRNA_LEVEL

= 10.0; } else { LAC_mRNA_LEVEL

= 1.982; } } }

PARAMETERIZED TOPOLOGIES

• One of the most important characteristics of computer programs is that they ordinarily contain inputs (free variables) and conditional operations

PARAMETERIZED TOPOLOGY FOR LOWPASS FILTER

PARAMETERIZED TOPOLOGY FOR HIGHPASS FILTER

PARAMETERIZED TOPOLOGY FOR GENERAL-PURPOSE CONTROLLER

EVOLVED EQUATIONS FOR GENERAL-PURPOSE CONTROLLER

EVOLVED EQUATIONS FOR GENERAL-PURPOSE CONTROLLER

PATENTABLE NEW INVENTIONS

PID tuning rules that outperform the Ziegler-Nichols and Åström-Hägglund tuning rules

General-purpose controllers outperforming Ziegler-Nichols and Åström-Hägglund rules


• Toy problems





PARALLELIZATION WITH SEMI-ISOLATED SUBPOPULATIONS

GP PARALLELIZATION

• Like Hormel, Get Everything Out of the Pig, Including the Oink

• Keep on Trucking

• It Takes a Licking and Keeps on Ticking

• The Whole is Greater than the Sum of the Parts

PETA-OPS

• Human brain operates at 1012 neurons operating at 103 per second = 1015 ops per second

• 1015 ops = 1 peta-op = 1 bs (brain second)

EVOLVABLE HARDWARECORNER OF XILINX XC6216

FUNCTION UNIT FOR CELL OF XILINIX XC6216

SORTING NETWORK

H G F E D

C B A

EVOLVED SORTING NETWORK

GP 1987–2002

System Dates Speed-up over first system

Human-competitive results

Problem Category

Serial LISP 1987–1994 1 (base) 0 toy problems 64 transputers 1994–1997 9 2 human-competitive

results not related to patented inventions

64 PowerPC’s 1995–2000 204 12 20th-century patented inventions

70 Alpha’s 1999–2001 1,481 2 20th-century patented inventions

1,000 Pentium II’s 2000–2002 13,900 12 21st-century patented inventions

4-week runs on 1,000 Pentium II’s

2002-2003 130,000 2 patentable new inventions

PROMISING GP APPLICATION AREAS

• Problem areas involving many variables that are interrelated in highly non-linear ways

• Inter-relationship of variables is not well understood

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

• "Black art" problems

PROMISING GP APPLICATION AREAS (CONTINUED)

• Areas where you simply have no idea how to program a solution, but where you know what you want


• Problem areas where a good approximate solution is satisfactory

design

control

bioinformatics

classification

data mining

system identification

forecasting


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

financial databases

medical records

marketing databases


Areas for which humans find it very difficult to write good programs

parallel computers cellular automata multi-agent strategies field-programmable game arrays digital signal processors swarm intelligence

CHARACTERISTICS SUGGESTING USE OF GP

(1) discovering the size and shape of the solution, (2) reusing substructures, (3) discovering the number of substructures, (4) discovering the nature of the hierarchical references among

substructures, (5) passing parameters to a substructure,(6) discovering the type of substructures (e.g., subroutines, iterations,

loops, recursions, or storage),(7) discovering the number of arguments possessed by a substructure, (8) maintaining syntactic validity and locality by means of a

developmental process, or(9) discovering a general solution in the form of a parameterized

topology containing free variables

DESIGNING A GIRAFFE

• Long neck

• Long tongue

• Vegetable-digesting enzymes in stomach

• 4 legs

• Long legs

• Brown coloration

THE DESIGN OF A GOOD GIRAFE

Neck length

Tongue length

Carnivorous? Number of legs

Leg length

Coloration

15.11 feet

14 inches No 4 9.96 feet Brown

Floating point

Floating point

Boolean Integer Floating point

Categorical

NON-LINEARITY — GIRAFE

• Taken one-by-one, some gene values found in a giraffe, such as the long neck contribute (alone) negatively to fitness– requires considerable material to construct

– requires considerable energy to maintain

– prone to injury (thereby hurting rate of survival and reproduction)

• Thus, maximizing any one variable will not lead to the global optimum solution

NON-LINEARITY (CONTINUED)

• When the variables are taken in pairs (there are 15 possible pairs), many combinations of pairs (e.g., Long neck and long tongue) are doubly detrimental

NON-LINEARITY (CONTINUED)

• But, certain combinations of traits, when taken together, are "co-adapted sets of alleles" that yield a very fit animal for eating high acacia leaves in the jungle environment, having good camouflage, having high escape velocity when faced with predators, and exploiting a niche (and avoiding competition) with other animals feeding on low-hanging vegetation

SEARCH METHODS IN GENERAL

• Initial structure(s)

• Fitness measure

• Operations for creating new structures

• Parameters

• Termination criterion and method of designating the result

SPACE WITH MANY LOCAL OPTIMA

SEARCH METHODS

• Blind random search does not use acquired information in deciding on the future direction of the search

• Hill combing and gradient descent use acquired information; however, they are prone to becoming trapped on local optima

• The previous point is especially true for non-trivial search spaces

7 DIFFERENCES BETWEEN GP AND ARTIFICIAL INTELLIGENCE

AND MACHINE LEARNING APPROACHES

REPRESENTATION

Genetic programming overtly conducts it

search for a solution to the given problem

in program space

ROLE OF POINT-TO-POINT TRANSFORMATIONS IN THE

SEARCH

Genetic programming does not conduct its

search by transforming a single point in the

search space into another single point, but

instead transforms a set of points into

another set of points

ROLE OF HILL CLIMBING IN THE SEARCH

Genetic programming does not rely

exclusively on greedy hill climbing to

conduct its search, but instead allocates a

certain number of trials, in a principled

way, to choices that are known to be

inferior

DETERMINISM IN THE SEARCH

Genetic programming conducts its search

probabilistically

ROLE OF AN EXPLICIT KNOWLEDGE BASE

Genetic programming does NOT make use

of a knowledge base

ROLE OF FORMAL LOGIC IN THE SEARCH

Genetic programming does not utilize

formal logic in it’s search strategy. Contradictory alternatives are created and actively maintained.

UNDERPINNINGS OF THE TECHNIQUE

Biologically inspired

TURING (1948)

Turing made the connection between searches and the challenge of getting a computer to solve a problem without explicitly programming it in his 1948 essay “Intelligent Machines”

"Further research into intelligence of machinery will probably be very greatly concerned with 'searches' ... “

TURING’S 3 APPROACHES TO MACHINE INTELLIGENCE (1948)

LOGIC-BASED SEARCH

One approach that Turing identified is asearch through the space of integers representing candidate computer programs.

TURING’S 3 APPROACHES (CONTINUED)

CULTURAL SEARCH

A second approach is the "cultural search“which relies on knowledge and expertiseacquired over a period of years fromothers (akin to present-day knowledge-based systems).

TURING’S 3 APPROACHES (CONTINUED)

GENETICAL OR EVOLUTIONARY SEARCH

"There is the genetical or evolutionarysearch by which a combination of genes islooked for, the criterion being the survivalvalue.“

TURING (1950)

From Turing’s 1950 paper "ComputingMachinery and Intelligence" …

“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”

TURING (1950) (CONTINUED)

“Structure of the child machine =

Hereditary material

“Changes of the child machine =

Mutations

“Natural selection = Judgment of the experimenter”

Documents

GENETIC ALGORITHMS AND GENETIC PROGRAMMING. John R. Koza Consulting Professor (Medical Informatics) Department of Medicine School of Medicine Consulting