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Artificial Intelligence
Handouts Lectures 12 - 14
1 Discussion on Problem Solving
In the previous chapter we studied problem solving in general and elaborated on various search strategies that help us solve problems through searching in problem trees. We kept the information about the tree traversal in memory (in the queues), thus we know the links that have to be followed to reach the goal. At times we don’t really need to remember the links that were followed. In many problems where the size of search space grows extremely large we often use techniques in which we don’t need to keep all the history in memory. Similarly, in problems where requirements are not clearly defined and the problem is ill-structured, that is, we don’t exactly know the initial state, goal state and operators etc, we might employ such techniques where our objective is to find the solution not how we got there.
Another thing we have noticed in the previous chapter is that we perfrom a sequential search through the search space. In order to speed up the techniques we can follow a parallel approach where we start from multiple locations (states) in the solution space and try to search the space in parallel.
2 Hill Climbing in Parallel
Suppose we were to climb up a hill. Our goal is to reach the top irrespective of how we get there. We apply different operators at a given position, and move in the direction that gives us improvement (more height). What if instead of starting from one position we start to climb the hill from different positions as indicated by the diagram below.
In other words, we start with different independent search instances that start from different locations to climb up the hill.
Further think that we can improve this using a collaborative approach where these instances interact and evolve by sharing information in order to solve the problem. You will soon find out that what we mean by interact and evolve.
However, it is possible to implement parallelism in the sense that the instances can interact and evolve to solve the solution. Such implementations and algorithms are
motivated from the biological concept of evolution of our genes, hence the name Genetic Algorithms , commonly terms as GA.
3 Comment on Evolution
Before we discuss Genetic Algorithms in detail with examples lets go though some basic terminology that we will use to explain the technique. The genetice algorithm technology comes from the concept of human evolution. The following paragraph gives a brief overview of evolution and introduces some terminologies to the extent that we will require for further discussion on GA. Individuals (animals or plants) produce a number of offspring (children) which are almost, but not entirely, like themselves. Variation may be due to mutation (random changes), or due to inheritance (offspring/children inherit some characteristics from each parent). Some of these offspring may survive to produce offspring of their own—some will not. The “better adapted” individuals are more likely to survive. Over time, generations become better and better adapted to survive.
4 Genetic Algorithm
Genetic Algorithms is a search method in which multiple search paths are followed in parallel. At each step, current states of different pairs of these paths are combined to form new paths. This way the search paths don't remain independent, instead they share information with each other and thus try to improve the overall performance of the complete search space.
Basic Genetic Algorithm
A very basic genetic algorithm can be stated as below.
Start with a population of randomly generated, (attempted) solutions to a problem
Repeatedly do the following: Evaluate each of the attempted solutions Keep the “best” solutions Produce next generation from these solutions (using “inheritance” and “mutation”)
Quit when you have a satisfactory solution (or you run out of time)
The two terms introduced here are inheritance and mutation. Inheritance has the same notion of having something or some attribute from a parent while mutation refers to a small random change. We will explain these two terms as we discuss the solution to a few problems through GA.
various individuals/ solutions for being better than others in the population. Notice that mutation can be as simple as just flipping a bit at random or any number of bits.
We go on repeating the algorithm until we either get our required word that is a 32- bit number with all ones, or we run out of time. If we run out of time, we either present the best possible solution (the one with most number of 1-bits) as the answer or we can say that the solution can’t be found. Hence GA is at times used to get optimal solution given some parameters.
5.2 Problem 2:
- Suppose you have a large number of data points (x, y), e.g., (1, 4), (3, 9), (5, 8), ...
- You would like to fit a polynomial (of up to degree 1) through these data points - That is, you want a formula y = mx + c that gives you a reasonably good fit to the actual data - Here’s the usual way to compute goodness of fit of the polynomial on the data points: - Compute the sum of (actual y – predicted y)^2 for all the data points - The lowest sum represents the best fit
- You can use a genetic algorithm to find a “pretty good” solution
By a pretty good solution we simply mean that you can get reasonably good polynomial that best fits the given data.
- Your formula is y = mx + c
- Your unknowns are m and c; where m and c are integers
- Your representation is the array [m, c]
- Your evaluation function for one array is:
- For every actual data point (x, y)
- Compute ý = mx + c
- Find the sum of (y – ý) 2 over all x
- The sum is your measure of “badness” (larger numbers are worse)
- Example: For [5, 7] and the data points (1, 10) and (2, 13):
- ý = 5x + 7 = 12 when x is 1
- ý = 5x + 7 = 17 when x is 2
- (10 - 12) 2 + (13 – 17) 2 = 2^2 + 4^2 = 20
- If these are the only two data points, the “badness” of [5, 7] is 20
- Your algorithm might be as follows:
- Create two-element arrays of random numbers
- Repeat 50 times (or any other number):
- For each of the arrays, compute its badness (using all data points)
- Keep the best arrays (with low badness)
- From the arrays you keep, generate new arrays as follows:
- Convert the numbers in the array to binary, toggle one of the bits at random
- Quit if the badness of any of the solution is zero
- After all 50 trials, pick the best array as your final answer
Let us solve this problem in detail. Consider that the given points are as follows.
We start will the following initial population which are the arrays representing the solutions (m and c).
Compute badness for [2 7]
- ý = 2x + 7 = 9 when x is 1
- ý = 2x + 7 = 13 when x is 3
- (5 – 9) 2 + (9 – 13)^2 = 4^2 + 4^2 = 32
- ý = 1x + 3 = 4 when x is 1
- ý = 1x + 3 = 6 when x is 3
- (5 – 4) 2 + (9 – 6)^2 = 1^2 + 3^2 = 10
- Lets keep the one with low “badness” [1 3]
- Representation [001 011]
- Apply mutation to generate new arrays [011 011]
- Now we have [1 3] [3 3] as the new population considering that we keep the two best individuals
Second iteration
- (x, y) : {(1,5) (3, 9)}
- [1 3][3 3]
- ý = 1x + 3 = 4 when x is 1
- ý = 1x + 3 = 6 when x is 3
- (5 – 4) 2 + (9 – 6)^2 = 1^2 + 3^2 = 10
- ý = 3x + 3 = 6 when x is 1
- ý = 3x + 3 = 12 when x is 3
- (5 – 6) 2 + (9 – 12)^2 = 1 + 9 = 10
- Lets keep the [3 3]
- Representation [011 011]
- Apply mutation to generate new arrays [010 011]
- Now we have [3 3] [2 3] as the new population
In the 32-bit word problem, the (two-parent, no mutation) approach, if it succeeds, is likely to succeed much faster because up to half of the bits change each time, not just one bit. However, with no mutation, it may not succeed at all. By pure bad luck, maybe none of the first (randomly generated) words have (say) bit 17 set to 1. Then there is no way a 1 could ever occur in this position. Another problem is lack of genetic diversity. Maybe some of the first generation did have bit 17 set to 1, but none of them were selected for the second generation. The best technique in general turns out to be a combination of both, i.e., crossover with mutation.
6 Eight Queens Problem
Let us now solve a famous problem which will be discussed under GA in many famous books in AI. Its called the Eight Queens Problem.
The problem is to place 8 queens on a chess board so that none of them can attack the other. A chess board can be considered a plain board with eight columns and eight rows as shown below.
The possible cells that the Queen can move to when placed in a particular square are shown (in black shading)
We now have to come up with a representation of an individual/ candidate solution representing the board configuration which can be used as individuals in the GA.
We will use the representation as shown in the figure below.
Where the 8 digits for eight columns specify the index of the row where the queen is placed. For example, the sequence 2 6 8 3 4 5 3 1 tells us that in first column the queen is placed in the second row, in the second column the queen is in the 6 th^ row so on till in the 8th^ column the queen is in the 1st^ row.
Now we need a fitness function, a function by which we can tell which board position is nearer to our goal. Since we are going to select best individuals at every step, we need to define a method to rate these board positions or individuals. One fitness function can be to count the number of pairs of Queens that are not attacking each other. An example of how to compute the fitness of a board configuration is given in the diagram on the next page.
Similarly, the mutation step can be done as under.
That is, we represent the individual in binary and we flip at random a certain number of bits. You might as well decide to flip 1, 2, 3 or k number of bits, at random position. Hence GA is totally a random technique.
This process is repeated until an individual with required fitness level is found. If no such individual is found, then the process is repeated till the overall fitness of the population or any of its individuals gets very close to the required fitness level. An upper limit on the number of iterations is usually used to end the process in finite time.
One of the solutions to the problem is shown as under whose fitness value is 8.
The following flow chart summarizes the Genetic Algorithm.
You are encouraged to explore the internet and other books to find more applications of GA in various fields like:
- Genetic Programming
- Evolvable Systems
- Composing Music
- Gaming
- Market Strategies
- Robotics
- Industrial Optimization
and many more.
No Yes
Start
Initialize Population
Evaluate Fitness of Population
Solution Found? End individuals inMate population
Apply mutation
