Genetic Algorithms and Fuzzy Logic

Introduction

After scientists became disillusioned with classical and neo-classical attempts at modeling intelligence, they looked in other directions.

Two prominent fields arose, connectionism (neural networking, parallel processing) and evolutionary computing.

It is the latter that this essay deals with -genetic algorithms and genetic programming.

What is GA

A genetic algorithm (or GA) is a search technique used in computing to find true or approximate solutions to optimization and search problems.

Genetic algorithms are categorized as global search heuristics.

Genetic algorithms are a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover (also called recombination).

What is GA Genetic algorithms are implemented as a computer

simulation in which a population of abstract representations (called chromosomes or the genotype or the genome) of candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem evolves toward better solutions.

Traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible.

What is GA

The evolution usually starts from a population of randomly generated individuals and happens in generations.

In each generation, the fitness of every individual in the population is evaluated, multiple individuals are selected from the current population (based on their fitness), and modified (recombined and possibly mutated) to form a new population.

What is GA

The new population is then used in the next iteration of the algorithm.

Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population.

If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached.

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Simple Genetic Algorithmproduce an initial population of individuals

evaluate the fitness of all individuals

while termination condition not met do

select fitter individuals for reproduction

recombine between individuals

mutate individuals

evaluate the fitness of the modified individuals

generate a new population

End while

Introduction to Genetic Algorithms 8

The Evolutionary Cycle

selection

population evaluation

modification

discard

deleted members

parents

modifiedoffspring

evaluated offspring

initiate &

evaluate

Key terms Individual - Any possible solution Population - Group of all individuals Search Space - All possible solutions to the problem Chromosome - Blueprint for an individual Trait - Possible aspect (features) of an individual Allele - Possible settings of trait (black, blond, etc.) Locus - The position of a gene on the chromosome Genome - Collection of all chromosomes for an

individual

Chromosome, Genes andGenomes

Genotype and Phenotype

Genotype:– Particular set of genes in a genome

Phenotype:– Physical characteristic of the genotype (smart, beautiful, healthy, etc.)

Genotype and Phenotype

GA Requirements A typical genetic algorithm requires two things to be defined: a genetic representation of the solution domain, and a fitness function to evaluate the solution domain.

A standard representation of the solution is as an array of bits. Arrays of other types and structures can be used in essentially the same way.

The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, that facilitates simple crossover operation.

Variable length representations may also be used, but crossover implementation is more complex in this case.

Tree-like representations are explored in Genetic programming.

Representation

Chromosomes could be: Bit strings (0101 ... 1100) Real numbers (43.2 -33.1 ... 0.0 89.2) Permutations of element (E11 E3 E7 ... E1 E15) Lists of rules (R1 R2 R3 ... R22 R23) Program elements (genetic programming) ... any data structure ...

GA Requirements The fitness function is defined over the genetic representation

and measures the quality of the represented solution. The fitness function is always problem dependent. For instance, in the knapsack problem we want to maximize

the total value of objects that we can put in a knapsack of some fixed capacity.

A representation of a solution might be an array of bits, where each bit represents a different object, and the value of the bit (0 or 1) represents whether or not the object is in the knapsack.

Not every such representation is valid, as the size of objects may exceed the capacity of the knapsack.

The fitness of the solution is the sum of values of all objects in the knapsack if the representation is valid, or 0 otherwise. In some problems, it is hard or even impossible to define the fitness expression; in these cases, interactive genetic algorithms are used.

A fitness function

Basics of GA The most common type of genetic algorithm works like this: a population is created with a group of individuals created

randomly. The individuals in the population are then evaluated. The evaluation function is provided by the programmer and

gives the individuals a score based on how well they perform at the given task.

Two individuals are then selected based on their fitness, the higher the fitness, the higher the chance of being selected.

These individuals then "reproduce" to create one or more offspring, after which the offspring are mutated randomly.

This continues until a suitable solution has been found or a certain number of generations have passed, depending on the needs of the programmer.

General Algorithm for GA Initialization Initially many individual solutions are randomly

generated to form an initial population. The population size depends on the nature of the problem, but typically contains several hundreds or thousands of possible solutions.

Traditionally, the population is generated randomly, covering the entire range of possible solutions (the search space).

Occasionally, the solutions may be "seeded" in areas where optimal solutions are likely to be found.

General Algorithm for GA Selection During each successive generation, a proportion of the existing

population is selected to breed a new generation. Individual solutions are selected through a fitness-based

process, where fitter solutions (as measured by a fitness function) are typically more likely to be selected.

Certain selection methods rate the fitness of each solution and preferentially select the best solutions. Other methods rate only a random sample of the population, as this process may be very time-consuming.

Most functions are stochastic and designed so that a small proportion of less fit solutions are selected. This helps keep the diversity of the population large, preventing premature convergence on poor solutions. Popular and well-studied selection methods include roulette wheel selection and tournament selection.

General Algorithm for GA

In roulette wheel selection, individuals are given a probability of being selected that is directly proportionate to their fitness.

Two individuals are then chosen randomly based on these probabilities and produce offspring.

General Algorithm for GARoulette Wheel’s Selection Pseudo Code:for all members of population

sum += fitness of this individualend for for all members of population

probability = sum of probabilities + (fitness / sum) sum of probabilities += probability

end for loop until new population is full

do this twice number = Random between 0 and 1 for all members of population

if number > probability but less than next probability then you have been selected

end for endcreate offspring

end loop

General Algorithm for GA Reproduction The next step is to generate a second generation population of

solutions from those selected through genetic operators: crossover (also called recombination), and/or mutation.

For each new solution to be produced, a pair of "parent" solutions is selected for breeding from the pool selected previously.

By producing a "child" solution using the above methods of crossover and mutation, a new solution is created which typically shares many of the characteristics of its "parents". New parents are selected for each child, and the process continues until a new population of solutions of appropriate size is generated.

General Algorithm for GA

These processes ultimately result in the next generation population of chromosomes that is different from the initial generation.

Generally the average fitness will have increased by this procedure for the population, since only the best organisms from the first generation are selected for breeding, along with a small proportion of less fit solutions, for reasons already mentioned above.

Crossover the most common type is single point crossover. In single

point crossover, you choose a locus at which you swap the remaining alleles from on parent to the other. This is complex and is best understood visually.

As you can see, the children take one section of the chromosome from each parent.

The point at which the chromosome is broken depends on the randomly selected crossover point.

This particular method is called single point crossover because only one crossover point exists. Sometimes only child 1 or child 2 is created, but oftentimes both offspring are created and put into the new population.

Crossover does not always occur, however. Sometimes, based on a set probability, no crossover occurs and the parents are copied directly to the new population. The probability of crossover occurring is usually 60% to 70%.

Crossover

Mutation

After selection and crossover, you now have a new population full of individuals.

Some are directly copied, and others are produced by crossover.

In order to ensure that the individuals are not all exactly the same, you allow for a small chance of mutation.

You loop through all the alleles of all the individuals, and if that allele is selected for mutation, you can either change it by a small amount or replace it with a new value. The probability of mutation is usually between 1 and 2 tenths of a percent.

Mutation is fairly simple. You just change the selected alleles based on what you feel is necessary and move on. Mutation is, however, vital to ensuring genetic diversity within the population.

Mutation

General Algorithm for GA

Termination This generational process is repeated until a

termination condition has been reached. Common terminating conditions are:

A solution is found that satisfies minimum criteria Fixed number of generations reached Allocated budget (computation time/money) reached The highest ranking solution's fitness is reaching or has

reached a plateau such that successive iterations no longer produce better results

Manual inspection Any Combinations of the above

GA Pseudo-code Choose initial populationEvaluate the fitness of each individual in the population Repeat

Select best-ranking individuals to reproduce

Breed new generation through crossover and mutation (genetic operations) and give birth to offspring

Evaluate the individual fitnesses of the offspring

Replace worst ranked part of population with offspring

Until <terminating condition>

Symbolic AI VS. Genetic Algorithms

Most symbolic AI systems are very static. Most of them can usually only solve one given

specific problem, since their architecture was designed for whatever that specific problem was in the first place.

Thus, if the given problem were somehow to be changed, these systems could have a hard time adapting to them, since the algorithm that would originally arrive to the solution may be either incorrect or less efficient.

Genetic algorithms (or GA) were created to combat these problems; they are basically algorithms based on natural biological evolution.

Symbolic AI VS. Genetic Algorithms

The architecture of systems that implement genetic algorithms (or GA) are more able to adapt to a wide range of problems.

A GA functions by generating a large set of possible solutions to a given problem.

It then evaluates each of those solutions, and decides on a "fitness level" (you may recall the phrase: "survival of the fittest") for each solution set.

These solutions then breed new solutions. The parent solutions that were more "fit" are more likely to

reproduce, while those that were less "fit" are more unlikely to do so.

In essence, solutions are evolved over time. This way you evolve your search space scope to a point where you can find the solution.

Genetic algorithms can be incredibly efficient if programmed correctly.

Genetic Programming

In programming languages such as LISP, the mathematical notation is not written in standard notation, but in prefix notation. Some examples of this:

+ 2 1 : 2 + 1 * + 2 1 2 : 2 * (2+1) * + - 2 1 4 9 : 9 * ((2 - 1) + 4) Notice the difference between the left-hand side to the right?

Apart from the order being different, no parenthesis! The prefix method makes it a lot easier for programmers and compilers alike, because order precedence is not an issue.

You can build expression trees out of these strings that then can be easily evaluated, for example, here are the trees for the above three expressions.

Genetic Programming

Genetic Programming You can see how expression evaluation is thus a lot

easier. What this have to do with GAs? If for example you

have numerical data and 'answers', but no expression to conjoin the data with the answers.

A genetic algorithm can be used to 'evolve' an expression tree to create a very close fit to the data.

By 'splicing' and 'grafting' the trees and evaluating the resulting expression with the data and testing it to the answers, the fitness function can return how close the expression is.

Genetic Programming

The limitations of genetic programming lie in the huge search space the GAs have to search for - an infinite number of equations.

Therefore, normally before running a GA to search for an equation, the user tells the program which operators and numerical ranges to search under.

Uses of genetic programming can lie in stock market prediction, advanced mathematics and military applications .

Evolving Neural Networks

Evolving the architecture of neural network is slightly more complicated, and there have been several ways of doing it. For small nets, a simple matrix represents which neuron connects which, and then this matrix is, in turn, converted into the necessary 'genes', and various combinations of these are evolved.

Evolving Neural Networks Many would think that a learning function could be

evolved via genetic programming. Unfortunately, genetic programming combined with neural networks could be incredibly slow, thus impractical.

As with many problems, you have to constrain what you are attempting to create.

For example, in 1990, David Chalmers attempted to evolve a function as good as the delta rule.

He did this by creating a general equation based upon the delta rule with 8 unknowns, which the genetic algorithm then evolved.

Other Areas

Genetic Algorithms can be applied to virtually any problem that has a large search space.

Al Biles uses genetic algorithms to filter out 'good' and 'bad' riffs for jazz improvisation.

The military uses GAs to evolve equations to differentiate between different radar returns.

Stock companies use GA-powered programs to predict the stock market.

Checkboard example We are given an n by n checkboard in which every field

can have a different colour from a set of four colors. Goal is to achieve a checkboard in a way that there are no

neighbours with the same color (not diagonal)

1 2 3 4 5 6 7 8 9 10

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Checkboard example Cont’d Chromosomes represent the way the checkboard is colored. Chromosomes are not represented by bitstrings but by

bitmatrices The bits in the bitmatrix can have one of the four values 0,

1, 2 or 3, depending on the color. Crossing-over involves matrix manipulation instead of

point wise operating. Crossing-over can be combining the parential matrices in a

horizontal, vertical, triangular or square way. Mutation remains bitwise changing bits in either one of the

other numbers.

Checkboard example Cont’d

• This problem can be seen as a graph with n nodes and (n-1) edges, so the fitness f(x) is defined as:

f(x) = 2 · (n-1) ·n

Example

f(x) = {MAX(x2): 0 <= x <= 32 } Encode Solution: Just use 5 bits (1 or 0). Generate initial population.

Evaluate each solution against objective.Sol. String Fitness % of Total

A 01101 169 14.4

B 11000 576 49.2

C 01000 64 5.5

D 10011 361 30.9

A 0 1 1 0 1

B 1 1 0 0 0

C 0 1 0 0 0

D 1 0 0 1 1

Example Cont’d

Create next generation of solutions Probability of “being a parent” depends on the fitness.

Ways for parents to create next generation Reproduction

Use a string again unmodified.

Crossover Cut and paste portions of one string to another.

Mutation Randomly flip a bit.

COMBINATION of all of the above.

Checkboard example We are given an n by n checkboard in which every field

can have a different colour from a set of four colors. Goal is to achieve a checkboard in a way that there are no

neighbours with the same color (not diagonal)

1 2 3 4 5 6 7 8 9 10

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Checkboard example Cont’d Chromosomes represent the way the checkboard is colored. Chromosomes are not represented by bitstrings but by

bitmatrices The bits in the bitmatrix can have one of the four values 0,

1, 2 or 3, depending on the color. Crossing-over involves matrix manipulation instead of

point wise operating. Crossing-over can be combining the parential matrices in a

horizontal, vertical, triangular or square way. Mutation remains bitwise changing bits in either one of the

other numbers.

Checkboard example Cont’d

• This problem can be seen as a graph with n nodes and (n-1) edges, so the fitness f(x) is defined as:

f(x) = 2 · (n-1) ·n

Checkboard example Cont’d• Fitnesscurves for different cross-over rules:

0 100 200 300 400 500130

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Verical Cutting Crossing Over

Fuzzy LogicWHERE DID FUZZY LOGIC COME FROM?

The concept of Fuzzy Logic (FL) was conceived by Lotfi Zadeh, aprofessor at the University of California at Berkley, and presented notas a control methodology, but as a way of processing data by allowingpartial set membership rather than crisp set membership or non-membership. This approach to set theory was not applied to controlsystems until the 70's due to insufficient small-computer capabilityprior to that time. Professor Zadeh reasoned that people do not requireprecise, numerical information input, and yet they are capable ofhighly adaptive control. If feedback controllers could be programmedto accept noisy, imprecise input, they would be much more effectiveand perhaps easier to implement. Unfortunately, U.S. manufacturershave not been so quick to embrace this technology while theEuropeans and Japanese have been aggressively building realproducts around it.

Fuzzy LogicWHAT IS FUZZY LOGIC?

In this context, FL is a problem-solving control system methodologythat lends itself to implementation in systems ranging from simple,small, embedded micro-controllers to large, networked, multi-channelPC or workstation-based data acquisition and control systems. It canbe implemented in hardware, software, or a combination of both. FLprovides a simple way to arrive at a definite conclusion based uponvague, ambiguous, imprecise, noisy, or missing input information.FL's approach to control problems mimics how a person would makedecisions, only much faster..

HOW IS FL DIFFERENT FROM CONVENTIONAL CONTROL METHODS?

FL incorporates a simple, rule-based IF X AND Y THEN Z approachto a solving control problem rather than attempting to model a systemmathematically. The FL model is empirically-based, relying on anoperator's experience rather than their technical understanding of thesystem. For example, rather than dealing with temperature control interms such as "SP =500F", "T <1000F", or "210C <TEMP <220C",terms like "IF (process is too cool) AND (process is getting colder)THEN (add heat to the process)" or "IF (process is too hot) AND(process is heating rapidly) THEN (cool the process quickly)" areused. These terms are imprecise and yet very descriptive of what mustactually happen. Consider what you do in the shower if thetemperature is too cold: you will make the water comfortable veryquickly with little trouble. FL is capable of mimicking this type ofbehavior but at very high rate.

HOW DOES FL WORK?

FL requires some numerical parameters in order to operate such aswhat is considered significant error and significant rate-of-change-of-error, but exact values of these numbers are usually not critical unlessvery responsive performance is required in which case empiricaltuning would determine them. For example, a simple temperaturecontrol system could use a single temperature feedback sensor whosedata is subtracted from the command signal to compute "error" andthen time-differentiated to yield the error slope or rate-of-change-of-error, hereafter called "error-dot". Error might have units of degs Fand a small error considered to be 2F while a large error is 5F. The"error-dot" might then have units of degs/min with a small error-dotbeing 5F/min and a large one being 15F/min. These values don't haveto be symmetrical and can be "tweaked" once the system is operatingin order to optimize performance. Generally, FL is so forgiving thatthe system will probably work the first time without any tweaking.

Fuzzy LogicFL was conceived as a better method for sorting and handling data buthas proven to be a excellent choice for many control systemapplications since it mimics human control logic. It can be built intoanything from small, hand-held products to large computerizedprocess control systems. It uses an imprecise but very descriptivelanguage to deal with input data more like a human operator. It is veryrobust and forgiving of operator and data input and often works whenfirst implemented with little or no tuning.

WHY USE FL?FL offers several unique features that make it a particularly good choice for many control problems.1) It is inherently robust since it does not require precise, noise-free inputs and can be programmed to fail safely if a feedback sensor quits or is destroyed. The output control is a smooth control function despite a wide range of input variations.2) Since the FL controller processes user-defined rules governing the target control system, it can be modified and tweaked easily to improve or drastically alter system performance. New sensors can easily be incorporated into the system simply by generating appropriate governing rules.3) FL is not limited to a few feedback inputs and one or two control outputs, nor is it necessary to measure or compute rate-of-change parameters in order for it to be implemented. Any sensor data that provides some indication of a system's actions and reactions is sufficient. This allows the sensors to be inexpensive and imprecise thus keeping the overall system cost and complexity low.

WHY USE FL?

FL offers several unique features that make it a particularly goodchoice for many control problems.4) Because of the rule-based operation, any reasonable number ofinputs can be processed (1-8 or more) and numerous outputs (1-4 ormore) generated, although defining the rulebase quickly becomescomplex if too many inputs and outputs are chosen for a singleimplementation since rules defining their interrelations must also bedefined. It would be better to break the control system into smallerchunks and use several smaller FL controllers distributed on thesystem, each with more limited responsibilities.5) FL can control nonlinear systems that would be difficult orimpossible to model mathematically. This opens doors for controlsystems that would normally be deemed unfeasible for automation.

HOW IS FL USED?1) Define the control objectives and criteria: What am I trying to control? What do I have to do to control the system? What kind of response do I need? What are the possible (probable) system failure modes?2) Determine the input and output relationships and choose a minimum number of variables for input to the FL engine (typically error and rate-of-change-of-error).3) Using the rule-based structure of FL, break the control problem down into a series of IF X AND Y THEN Z rules that define the desired system output response for given system input conditions. The number and complexity of rules depends on the number of input parameters that are to be processed and the number fuzzy variables associated with each parameter. If possible, use at least one variable and its time derivative. Although it is possible to use a single, instantaneous error parameter without knowing its rate of change, this cripples the system's ability to minimize overshoot for a step inputs.

HOW IS FL USED?

4) Create FL membership functions that define the meaning (values) of Input/Output terms used in the rules.5) Create the necessary pre- and post-processing FL routines if implementing in S/W, otherwise program the rules into the FL H/W engine.6) Test the system, evaluate the results, tune the rules and membership functions, and retest until satisfactory results are obtained.

Fuzzy LogicLINGUISTIC VARIABLES

In 1973, Professor Lotfi Zadeh proposed the concept of linguistic or"fuzzy" variables. Think of them as linguistic objects or words, ratherthan numbers. The sensor input is a noun, e.g. "temperature","displacement", "velocity", "flow", "pressure", etc. Since error is justthe difference, it can be thought of the same way. The fuzzy variablesthemselves are adjectives that modify the variable (e.g. "largepositive" error, "small positive" error ,"zero" error, "small negative"error, and "large negative" error). As a minimum, one could simplyhave "positive", "zero", and "negative" variables for each of theparameters. Additional ranges such as "very large" and "very small"could also be added to extend the responsiveness to exceptional orvery nonlinear conditions, but aren't necessary in a basic system.

Fuzzy Logic

THE RULE MATRIXIn the last article the concept of linguistic variables was presented. The fuzzy parameters of error (command-feedback) and error-dot (rate-of-change-of-error) were modified by the adjectives "negative", "zero", and "positive". To picture this, imagine the simplest practical implementation, a 3-by-3 matrix. The columns represent "negative error", "zero error", and "positive error" inputs from left to right. The rows represent "negative", "zero", and "positive" "error-dot" input from top to bottom. This planar construct is called a rule matrix. It has two input conditions, "error" and "error-dot", and one output response conclusion (at the intersection of each row and column). In this case there are nine possible logical product (AND) output response conclusions.

Fuzzy LogicTHE RULE MATRIXAlthough not absolutely necessary, rule matrices usually have an odd number of rows and columns to accommodate a "zero" center row and column region. This may not be needed as long as the functions on either side of the center overlap somewhat and continuous dithering of the output is acceptable since the "zero" regions correspond to "no change" output responses the lack of this region will cause the system to continually hunt for "zero". It is also possible to have a different number of rows than columns. This occurs when numerous degrees of inputs are needed. The maximum number of possible rules is simply the product of the number of rows and columns, but definition of all of these rules may not be necessary since some input conditions may never occur in practical operation. The primary objective of this construct is to map out the universe of possible inputs while keeping the system sufficiently under control.

Fuzzy Logic

The first step in implementing FL is to decide exactly what is to becontrolled and how. For example, suppose we want to design a simpleproportional temperature controller with an electric heating elementand a variable-speed cooling fan. A positive signal output calls for 0-100 percent heat while a negative signal output calls for 0-100 percentcooling. Control is achieved through proper balance and control ofthese two active devices.

Fuzzy Logic..

Fuzzy LogicFigure 1 - A simple block diagram of the control system.It is necessary to establish a meaningful system for representing the linguistic variables in the matrix.

"N" = "negative" error or error-dot input level"Z" = "zero" error or error-dot input level"P" = "positive" error or error-dot input level"H" = "Heat" output response"-" = "No Change" to current output"C" = "Cool" output response

Define the minimum number of possible input product combinations and corresponding output response conclusions using these terms. For a three-by-three matrix with heating and cooling output responses, all nine rules will need to be defined. The conclusions to the rules with the linguistic variables associated with the output response for each rule are transferred to the matrix.

Fuzzy Logic..

Fuzzy LogicFigure 2 - Typical control system response Figure 2 shows what command and error look like in a typical control system relative to the command setpoint as the system hunts for stability. Definitions are also shown for this example.

DEFINITIONS:INPUT#1: ("Error", positive (P), zero (Z), negative (N))INPUT#2: ("Error-dot", positive (P), zero (Z), negative (N))CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C))INPUT#1 System StatusError = Command-FeedbackP=Too cold, Z=Just right, N=Too hotINPUT#2 System StatusError-dot = d(Error)/dtP=Getting hotter Z=Not changing N=Getting colderOUTPUT Conclusion & System ResponseOutput H = Call for heating - = Don't change anything C = Call for cooling

SYSTEM OPERATING RULESLinguistic rules describing the control system consist of two parts;an antecedent block (between the IF and THEN) and a consequentblock (following THEN). Depending on the system, it may not benecessary to evaluate every possible input combination (for 5-by-5& up matrices) since some may rarely or never occur. By makingthis type of evaluation, usually done by an experienced operator,fewer rules can be evaluated, thus simplifying the processing logicand perhaps even improving the FL system performance.

Fuzzy Logic

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Figures 3 & 4 - The rule structure.After transferring the conclusions from the nine rules to thematrix there is a noticeable symmetry to the matrix. This suggests(but doesn't guarantee) a reasonably well-behaved (linear)system. This implementation may prove to be too simplistic forsome control problems, however it does illustrate the process.Additional degrees of error and error-dot may be included if thedesired system response calls for this. This will increase therulebase size and complexity but may also increase the quality ofthe control. Figure 4 shows the rule matrix derived from theprevious rules.

Fuzzy Logic

Fuzzy Logic

SUMMARYLinguistic variables are used to represent an FL system's operating parameters. The rule matrix is a simple graphical tool for mapping the FL control system rules. It accommodates two input variables and expresses their logical product (AND) as one output response variable. To use, define the system using plain-English rules based upon the inputs, decide appropriate output response conclusions, and load these into the rule matrix.

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