I am trying to solve this problem using genetic algorithm. But can't find any way to calculate fitness value. I am not writing a program instead I am just solving this on a paper.
Problem is I can't find the fitness value which caters the clashes. e.g. class1 and class2 can't be on same day.
Each professor can only teach one class at a time.
Days
Monday
Wednesday
Friday
The classes are:
Class 1 - Intro to Programming: meets from 8:00-9:00am.
Class 2 - Intro to Artificial Intelligence: meets from 8:30-9:30am
Class 3 - Natural Language Processing: meets from 9:00-10:00am
Class 4 - Computer Vision: meets from 9:00-10:00am
Class 5 - Machine Learning: meets from 9:30-10:30am
The professors are:
Professor A, who is available to teach Classes 3 and 4.
Professor B, who is available to teach Classes 2, 3, 4, and 5.
Professor C, who is available to teach Classes 1, 2, 3, 4, 5.
Any kind of help will be appreciated.
In your scenario, a fitness function needs to indicate how many clashes are created by a solution, so you would have something like negative values. E.g. -100 when classes clash, and -50 when profs clash. A fitness value of 0 would be then a solution that fulfills all required constraints.
Then you could also add some logic to honor solutions with good balancing of professor workload. A simple idea would be to take the difference between least and highest prof workload into account. The smaller the difference, the higher the fitness value should be. Either you subtract the difference again, like for the constraints, or you invert it, e.g. 100/diff.
Related
I understand why it is possible for zero-sum game, but I cannot understand whether it is possible for non-zero sum games with 2 players. Is it because both players do not necessarily have a conflict in their interests?
If the scores for two players are completely arbitrary, then you can't do any pruning. Suppose that the scores were {-10, 10} for P1 and P2 on one branch. P1 might be tempted to prune this branch because, perhaps, at the root P1 could get >= 5. But, perhaps the second branch is {100, 100} which would be a much better outcome. In general there is no guarantee that we won't find another branch with an arbitrarily large score for both players, so we can never prune.
If you make stronger assumptions, more can be done.
For instance, look at the M* algorithm which handles opponent models. There they handled the cases of bounded sum. An extended tech report is available online.
This is related to multi-player (meaning >=3) constant-sum games, which can use a different set of pruning techniques.
I have created a Gomoku(5 in a row) AI using Alpha-Beta Pruning. It makes moves on a not-so-stupid level. First, let me vaguely describe the grading function of the Alpha-Beta algorithm.
When it receives a board as an input, it first finds all repetitions of stones and gives it a score out of 4 possible values depending on its usefulness as an threat, which is decided by length. And it will return the summation of all the repetition scores.
But, the problem is that I explicitly decided the scores(4 in total), and they don't seem like the best choices. So I've decided to implement a genetic algorithm to generate these scores. Each of the genes will be one of 4 scores. So for example, the chromosome of the hard-coded scores would be: [5, 40000,10000000,50000]
However, because I'm using the genetic algorithm to create the scores of the grading function, I'm not sure how I should implement the genetic fitness function. So instead, I have thought of the following:
Instead of using a fitness function, I'll just merge the selection process together: If I have 2 chromosomes, A and B, and need to select one, I'll simulate a game using both A and B chromosomes in each AI, and select the chromosome which wins.
1.Is this a viable replacement to the Fitness function?
2.Because of the characteristics of the Alpha-Beta algorithm, I need to give the max score to the win condition, which in most cases is set to infinity. However, because I can't use Infinity, I just used an absurdly large number. Do I also need to add this score to the chromosome? Or because it's insignificant and doesn't change the values of the grading function, leave it as a constant?
3.When initially creating chromosomes, random generation, following standard distribution is said to be the most optimal. However, genes in my case have large deviation. Would it still be okay to generate chromosomes randomly?
Is this a viable replacement to the Fitness function?
Yes, it is. It's a fairly common way to define a fitness function for board games. Probably a single round is not enough (but you have to experiment).
A slight variant is something like:
double fitness(Agent_k)
fit = 0
repeat M times
randomly extract an individual Agent_i (i <> k)
switch (result of Agent_k vs Agent_i)
case Agent_k wins: fit = fit + 1
case Agent_i wins: fit = fit - 2
case draw: fit doesn't change
return fit
i.e. an agent plays against M randomly selected opponents from the population (with replacement but avoiding self match).
Increasing M the noise decreases but longer simulation times are required (M=5 is a value used in some chess-related experiments).
2.Because of the characteristics of the Alpha-Beta algorithm...
Not sure of the question. A very large value is a standard approach for a static evaluation function signaling a winning condition.
The exact value isn't very important and shouldn't probably be subject to optimization.
3.When initially creating chromosomes, random generation, following standard distribution is said to be the most optimal. However, genes in my case have large deviation. Would it still be okay to generate chromosomes randomly?
This is somewhat related to the specific genetic algorithm "flavor" you are going to use.
A standard genetic algorithm could work better with not completely random initial values.
Other variants (e.g. Differential Evolution) could be less sensitive to this aspect.
Take also a look at this question / answer: Getting started with machine learning a zero sum game?
I'm using a genetic algorithm (GA) to optimise a traveling salesman problem (TSP). My problem is how I calculate the fitness of an individual.
Obviously solutions with shorter routes are fitter but how exactly do I assign a fitness value without knowing what the shortest possible route and longest possible route is to determine where my solution fits in that range?
Having fitness equals to path length is fine. Keep in mind that in genetic algorithms the fitness is only used for selecting individuals: consequently with usual selection procedures the scale does not matter, only the rank does.
Examples of implementation:
http://www.codeproject.com/Articles/1403/Genetic-Algorithms-and-the-Traveling-Salesman-Prob
http://khayyam.developpez.com/articles/algo/voyageur-de-commerce/genetique/ (use Google translate)
http://www.lalena.com/ai/tsp/
http://www.mathworks.com/matlabcentral/fileexchange/13680
More subtleties (2001 - Swarm Intelligence - Kennedy & Eberhart - page 249):
Pablo Moscato is a South American researcher who has pioneered the
study of memetic algorithms (e.g., Moscato, 1989). He and Michael
Norman, who is now in Scotland at the University of Edinburgh, began
working together in the 1980s at Caltech. In a recent paper they
describe the use of a memetic algorithm for optimization of a
traveling salesman problem (TSP) (Moscato and Norman, 1992). Recall
that the TSP requires finding the shortest path through a number of
cities, passing through each one only once. The problem has a rich
history in applied mathematics, as it is very hard to solve,
especially when the number of cities is large. TSP is an NP-hard
problem, which suggests that if a way is found to solve it, then a
large number of other problems will also have been solved. Moscato and
Norman use an algorithm with both cooperation and competition among
agents in the population, and implement a hybrid version of simulated
annealing for local search.
A population of individuals—these
researchers usually use a population size of 16—searches the problem
space, which is defined by permutations of the cities, called “tours.”
The population is conceptualized as a ring, where each individual is
attached to its two immediately adjacent neighbors, with whom it
competes in the search; individuals are also connected to others on
the far side of the ring, with whom they cooperate. Each individual in
the population comprises a tour of the cities. Competition is seen as
“challenge” and “battles” between pairs of individuals, where the tour
lengths of an individual and its neighbor are compared and a
probability threshold is set based on the difference. The difference
between the tours’ lengths affects the steepness of the sshaped curve;
when the difference is small or the temperature is cool, the
probability distribution becomes nearly uniform, and when the
difference in lengths between the two tours is great, the probability
is increased that tour 1 will be deleted and replaced with a copy of
tour 0.
Cooperation is used to let more successful individuals “mate”
with one another, rather than with less-fit members of the population.
The same rule that is used in deciding competitive interactions is
used to assess the desirability of partners for crossover, which is
implemented just as it is in GA. One individual “proposes” to another,
and if the proposition is accepted, that is, if the stochastic
decision favors their interaction, then the crossover operator is
implemented. Thus the next generation is created.
You could normalise all candidate solutions, such that the shortest path you've seen to date gets the fitness score 1.0 (or 10, or 42, or 3.14... whatever you like), and then scale all paths longer than this relatively. Same with the longest path - the longest path that you've observed is considered the worst possible score.
The trick comes with what you do when you find an even shorter path (given that you assigned some longer path the highest possible score, such as 1.0) - you have to then raise the ceiling on your normalisation function. Start assigning fitness 2.0, for example (or 1.1, or some other arbitrarily larger fitness score).
If your program is maximizing fitness values, you would want to maximize a fitness function
f = - Tour-Length
EDITED: I had added 1000000000000, an arbitrary number to the fitness, to make the fitness positive, On reading a few comments, I realize it is not necessary.
If your program is minimizing fitness values, you would want to minimize a fitness function
f = Tour-Length
I once wrote a Tetris AI that played Tetris quite well. The algorithm I used (described in this paper) is a two-step process.
In the first step, the programmer decides to track inputs that are "interesting" to the problem. In Tetris we might be interested in tracking how many gaps there are in a row because minimizing gaps could help place future pieces more easily. Another might be the average column height because it may be a bad idea to take risks if you're about to lose.
The second step is determining weights associated with each input. This is the part where I used a genetic algorithm. Any learning algorithm will do here, as long as the weights are adjusted over time based on the results. The idea is to let the computer decide how the input relates to the solution.
Using these inputs and their weights we can determine the value of taking any action. For example, if putting the straight line shape all the way in the right column will eliminate the gaps of 4 different rows, then this action could get a very high score if its weight is high. Likewise, laying it flat on top might actually cause gaps and so that action gets a low score.
I've always wondered if there's a way to apply a learning algorithm to the first step, where we find "interesting" potential inputs. It seems possible to write an algorithm where the computer first learns what inputs might be useful, then applies learning to weigh those inputs. Has anything been done like this before? Is it already being used in any AI applications?
In neural networks, you can select 'interesting' potential inputs by finding the ones that have the strongest correlation, positive or negative, with the classifications you're training for. I imagine you can do similarly in other contexts.
I think I might approach the problem you're describing by feeding more primitive data to a learning algorithm. For instance, a tetris game state may be described by the list of occupied cells. A string of bits describing this information would be a suitable input to that stage of the learning algorithm. actually training on that is still challenging; how do you know whether those are useful results. I suppose you could roll the whole algorithm into a single blob, where the algorithm is fed with the successive states of play and the output would just be the block placements, with higher scoring algorithms selected for future generations.
Another choice might be to use a large corpus of plays from other sources; such as recorded plays from human players or a hand-crafted ai, and select the algorithms who's outputs bear a strong correlation to some interesting fact or another from the future play, such as the score earned over the next 10 moves.
Yes, there is a way.
If you choose M selected features there are 2^M subsets, so there is a lot to look at.
I would to the following:
For each subset S
run your code to optimize the weights W
save S and the corresponding W
Then for each pair S-W, you can run G games for each pair and save the score L for each one. Now you have a table like this:
feature1 feature2 feature3 featureM subset_code game_number scoreL
1 0 1 1 S1 1 10500
1 0 1 1 S1 2 6230
...
0 1 1 0 S2 G + 1 30120
0 1 1 0 S2 G + 2 25900
Now you can run some component selection algorithm (PCA for example) and decide which features are worth to explain scoreL.
A tip: When running the code to optimize W, seed the random number generator, so that each different 'evolving brain' is tested against the same piece sequence.
I hope it helps in something!
I'm working with a couple of AI algorithms at school and I find people use the words Fuzzy Logic to explain any situation that they can solve with a couple of cases. When I go back to the books I just read about how instead of a state going from On to Off it's a diagonal line and something can be in both states but in different "levels".
I've read the wikipedia entry and a couple of tutorials and even programmed stuff that "uses fuzzy logic" (an edge detector and a 1-wheel self-controlled robot) and still I find it very confusing going from Theory to Code... for you, in the less complicated definition, what is fuzzy logic?
Fuzzy logic is logic where state membership is, essentially, a float with range 0..1 instead of an int 0 or 1. The mileage you get out of it is that things like, for example, the changes you make in a control system are somewhat naturally more fine-tuned than what you'd get with naive binary logic.
An example might be logic that throttles back system activity based on active TCP connections. Say you define "a little bit too many" TCP connections on your machine as 1000 and "a lot too many" as 2000. At any given time, your system has a "too many TCP connections" state from 0 (<= 1000) to 1 (>= 2000), which you can use as a coefficient in applying whatever throttling mechanisms you have available. This is much more forgiving and responsive to system behavior than naive binary logic that only knows how to determine "too many", and throttle completely, or "not too many", and not throttle at all.
I'd like to add to the answers (that have been modded up) that, a good way to visualize fuzzy logic is follows:
Traditionally, with binary logic you would have a graph whose membership function is true or false whereas in a fuzzy logic system, the membership function is not.
1|
| /\
| / \
| / \
0|/ \
------------
a b c d
Assume for a second that the function is "likes peanuts"
a. kinda likes peanuts
b. really likes peanuts
c. kinda likes peanuts
d. doesn't like peanuts
The function itself doesn't have to be triangular and often isn't (it's just easier with ascii art).
A fuzzy system will likely have many of these, some even overlapping (even opposites) like so:
1| A B
| /\ /\ A = Likes Peanuts
| / \/ \ B = Doesn't Like Peanuts
| / /\ \
0|/ / \ \
------------
a b c d
so now c is "kind likes peanuts, kinda doesn't like peanuts" and d is "really doesn't like peanuts"
And you can program accordingly based on that info.
Hope this helps for the visual learners out there.
The best definition of fuzzy logic is given by its inventor Lotfi Zadeh:
“Fuzzy logic means of representing problems to computers in a way akin to the way human solve them and the essence of fuzzy logic is that everything is a matter of degree.”
The meaning of solving problems with computers akin to the way human solve can easily be explained with a simple example from a basketball game; if a player wants to guard another player firstly he should consider how tall he is and how his playing skills are. Simply if the player that he wants to guard is tall and plays very slow relative to him then he will use his instinct to determine to consider if he should guard that player as there is an uncertainty for him. In this example the important point is the properties are relative to the player and there is a degree for the height and playing skill for the rival player. Fuzzy logic provides a deterministic way for this uncertain situation.
There are some steps to process the fuzzy logic (Figure-1). These steps are; firstly fuzzification where crisp inputs get converted to fuzzy inputs secondly these inputs get processed with fuzzy rules to create fuzzy output and lastly defuzzification which results with degree of result as in fuzzy logic there can be more than one result with different degrees.
Figure 1 – Fuzzy Process Steps (David M. Bourg P.192)
To exemplify the fuzzy process steps, the previous basketball game situation could be used. As mentioned in the example the rival player is tall with 1.87 meters which is quite tall relative to our player and can dribble with 3 m/s which is slow relative to our player. Addition to these data some rules are needed to consider which are called fuzzy rules such as;
if player is short but not fast then guard,
if player is fast but not short then don’t guard
If player is tall then don’t guard
If player is average tall and average fast guard
Figure 2 – how tall
Figure 3- how fast
According to the rules and the input data an output will be created by fuzzy system such as; the degree for guard is 0.7, degree for sometimes guard is 0.4 and never guard is 0.2.
Figure 4-output fuzzy sets
On the last step, defuzzication, is using for creating a crisp output which is a number which may determine the energy that we should use to guard the player during game. The centre of mass is a common method to create the output. On this phase the weights to calculate the mean point is totally depends on the implementation. On this application it is considered to give high weight to guard or not guard but low weight given to sometimes guard. (David M. Bourg, 2004)
Figure 5- fuzzy output (David M. Bourg P.204)
Output = [0.7 * (-10) + 0.4 * 1 + 0.2 * 10] / (0.7 + 0.4 + 0.2) ≈ -3.5
As a result fuzzy logic is using under uncertainty to make a decision and to find out the degree of decision. The problem of fuzzy logic is as the number of inputs increase the number of rules increase exponential.
For more information and its possible application in a game I wrote a little article check this out
To build off of chaos' answer, a formal logic is nothing but an inductively defined set that maps sentences to a valuation. At least, that's how a model theorist thinks of logic. In the case of a sentential boolean logic:
(basis clause) For all A, v(A) in {0,1}
(iterative) For the following connectives,
v(!A) = 1 - v(A)
v(A & B) = min{v(A), v(B)}
v(A | B) = max{v(A), v(B)}
(closure) All sentences in a boolean sentential logic are evaluated per above.
A fuzzy logic changes would be inductively defined:
(basis clause) For all A, v(A) between [0,1]
(iterative) For the following connectives,
v(!A) = 1 - v(A)
v(A & B) = min{v(A), v(B)}
v(A | B) = max{v(A), v(B)}
(closure) All sentences in a fuzzy sentential logic are evaluated per above.
Notice the only difference in the underlying logic is the permission to evaluate a sentence as having the "truth value" of 0.5. An important question for a fuzzy logic model is the threshold that counts for truth satisfaction. This is to ask: for a valuation v(A), for what value D it is the case the v(A) > D means that A is satisfied.
If you really want to found out more about non-classical logics like fuzzy logic, I would recommend either An Introduction to Non-Classical Logic: From If to Is or Possibilities and Paradox
Putting my coder hat back on, I would be careful with the use of fuzzy logic in real world programming, because of the tendency for a fuzzy logic to be undecidable. Maybe it's too much complexity for little gain. For instance a supervaluational logic may do just fine to help a program model vagueness. Or maybe probability would be good enough. In short, I need to be convinced that the domain model dovetails with a fuzzy logic.
Maybe an example clears up what the benefits can be:
Let's say you want to make a thermostat and you want it to be 24 degrees.
This is how you'd implement it using boolean logic:
Rule1: heat up at full power when
it's colder than 21 degrees.
Rule2:
cool down at full power when it's
warmer than 27 degrees.
Such a system will only once and a while be 24 degrees, and it will be very inefficient.
Now, using fuzzy logic, it would be like something like this:
Rule1: For each degree that it's colder than 24 degrees, turn up the heater one notch (0 at 24).
Rule2: For each degree that it's warmer than 24 degress, turn up the cooler one notch (0 at 24).
This system will always be somewhere around 24 degrees, and it only once and will only once and a while make a tiny adjustment. It will also be more energy-efficient.
Well, you could read the works of Bart Kosko, one of the 'founding fathers'. 'Fuzzy Thinking: The New Science of Fuzzy Logic' from 1994 is readable (and available quite cheaply secondhand via Amazon). Apparently, he has a newer book 'Noise' from 2006 which is also quite approachable.
Basically though (in my paraphrase - not having read the first of those books for several years now), fuzzy logic is about how to deal with the world where something is perhaps 10% cool, 50% warm, and 10% hot, where different decisions may be made on the degree to which the different states are true (and no, it wasn't entirely an accident that those percentages don't add up to 100% - though I'd accept correction if needed).
A very good explanation, with a help of Fuzzy Logic Washing Machines.
I know what you mean about it being difficult to go from concept to code. I'm writing a scoring system that looks at the values of sysinfo and /proc on Linux systems and comes up with a number between 0 and 10, 10 being the absolute worst. A simple example:
You have 3 load averages (1, 5, 15 minute) with (at least) three possible states, good, getting bad, bad. Expanding that, you could have six possible states per average, adding 'about to' to the three that I just noted. Yet, the result of all 18 possibilities can only deduct 1 from the score. Repeat that with swap consumed, actual VM allocated (committed) memory and other stuff .. and you have one big bowl of conditional spaghetti :)
Its as much a definition as it is an art, how you implement the decision making process is always more interesting than the paradigm itself .. whereas in a boolean world, its rather cut and dry.
It would be very easy for me to say if load1 < 2 deduct 1, but not very accurate at all.
If you can teach a program to do what you would do when evaluating some set of circumstances and keep the code readable, you have implemented a good example of fuzzy logic.
Fuzzy Logic is a problem-solving methodology that lends itself to implementation in systems ranging from simple, small, embedded micro-controllers to large, networked, multi-channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. Fuzzy Logic provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. Fuzzy Logic approach to control problems mimics how a person would make decisions, only much faster.
Fuzzy logic has proved to be particularly useful in expert system and other artificial intelligence applications. It is also used in some spell checkers to suggest a list of probable words to replace a misspelled one.
To learn more, just check out: http://en.wikipedia.org/wiki/Fuzzy_logic.
The following is sort of an empirical answer.
A simple (possibly simplistic answer) is that "fuzzy logic" is any logic that returns values other than straight true / false, or 1 / 0. There are a lot of variations on this and they tend to be highly domain specific.
For example, in my previous life I did search engines that used "content similarity searching" as opposed to then common "boolean search". Our similarity system used the Cosine Coefficient of weighted-attribute vectors representing the query and the documents and produced values in the range 0..1. Users would supply "relevance feedback" which was used to shift the query vector in the direction of desirable documents. This is somewhat related to the training done in certain AI systems where the logic gets "rewarded" or "punished" for results of trial runs.
Right now Netflix is running a competition to find a better suggestion algorithm for their company. See http://www.netflixprize.com/. Effectively all of the algorithms could be characterized as "fuzzy logic"
Fuzzy logic is calculating algorithm based on human like way of thinking. It is particularly useful when there is a large number of input variables. One online fuzzy logic calculator for two variables input is given:
http://www.cirvirlab.com/simulation/fuzzy_logic_calculator.php