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I'm teaching a kid programming, and am introducing some basic artificial intelligence concepts at the moment. To begin with we're going to implement a tic-tac-toe game that searches the entire game tree and as such plays perfectly. Once we finish that I want to apply the same concepts to a game that has too many positions to evaluate every single one, so that we need to implement a heuristic to evaluate intermediate positions.
The best thing I could think of was Dots and Boxes. It has the advantage that I can set the board size arbitrarily large to stop him from searching the entire tree, and I can make a very basic scoring function be the number of my boxes minus the number of opponent boxes. Unfortunately this means that for most of the beginning of the game every position will be evaluated equivalently with a score of 0, because it takes quite a few moves before players actually start making boxes.
Does anyone have any better ideas for games? (Or a better scoring function for dots and boxes)?
Another game choice could be Reversi aka Othello.
A naive heuristic would be to simply count the number of tiles gained by each valid move and choose the greatest. From there you can factor in board position and minimizing vulnerably to the opponent.
One game you may consider is Connect Four. Simple game with straightforward rules but more complicated that Tic-Tac-Toe.
Checkers will let you teach several methods. Simple lookahead, depth search of best-case-worst-case decisions, differences between short-term and long-term gains, and something they could continue to work on after learning what you want to teach them.
Personally I think that last bit is the most critical -- there are natural points in the AI development which are good to stop at, see if you can beat it, and then delve into deeper AI mechanisms. It keeps your student interested without being horribly frustrated, and gives them more to do on their own if they want to continue the project.
How about Reversi? It has a pretty nice space of heuristics based on number of pieces, number of edge pieces, and number of corner pieces.
How about Mancala? Only 6 possible moves each turn, and it's easy to calculate the resulting score for each, but it's important to consider the opponent's response, and the game tree gets big pretty fast.
Gomoku is a nice, simple game, and fun one to write AI for.
Rubik's Infinity's quite fun, it's a little bit like Connect Four but subtly different. Evauluating a position is pretty easy.
I knocked together a Perl script to play it a while back, and actually had to reduce the number of moves ahead it looked, or it beat me every time, usually with quite surprising tactics.
Four in a line Hard enough, but easy enough to come up with an easy working evaluation function, for example, (distance to four from my longest line - distance to four from my opponent's longest line)
I really like Connect Four. Very easy to program using a Minimax algorithm. A good evaluation function could be:
eval_score = 0
for all possible rows/lines/diagonals of length 4 on the board:
if (#player_pieces = 0) // possible to connect four here?
if (#computer_pieces = 4)
eval_score = 10000
break for loop
else
eval_score = eval_score + #computer_pieces
(less pieces to go -> higher score)
end if
else if (#player_pieces = 4)
eval_score = -10000
break for loop
end if
end for
To improve the program you can add:
If computer moves first, play in the middle column (this has been proven to be optimal)
Alpha-Beta Pruning
Move Ordering
Zobrist Hashes
How about starting your Dots and Boxes game with random lines already added. This can get you into the action quickly. Just need to make sure you don't start the game with any boxes.
Take a look at Go.
Simple enough for kid on very small boards.
Complexity scales infinitely.
Has a lot of available papers, algorithms and programs to use either as a scale or basis.
Update: reversi was mentioned, which is a simplified variant of Go. Might be a better choice.
In regards to a better heuristic for dots and boxes, I suggest looking at online strategy guides for the game. The first result on Google for "dots and boxes strategy" is quite helpful.
Knowing how to use the chain rule separates an OK player from a good one. Knowing when the chain rule will work against you is what separates the best players from the good ones.
Related
i put a small request on upwork where i am requesting help for a topic which is right now out of my skill zone.
The problem is a fitting problem of small rectangles in a big rectangle via a ANN.
Problem is the first freelancer baffled me a little bit with a comment.
So my thinking was, because the solution is easy verified and rewardable, that you can simply throw a ANN on this problem and with enough time it will perform better and better.
The freelancer requested labeled data first before he can tackle the problem(thats the comment which confuses me).
I was thinking that unlabeled random Input data is enough for the start.
Do I think wrong?
here the link to the job post.
https://www.upwork.com/jobs/~01e040711c31ac0979
edit: directly the original job description
I want python code for training a ANN and using it in a productive enviroment.
The problem it needs to solve is a rectangle fitting problem.
Input are
1000 small Rectangles(groupid,width,heigth,Oriantion(free,restricted,hor or ver), value) --sRect
1 big Rectangles(width, heigth)--bRect
Layout(bool,bool,bool,xpos,ypos,Oriantaion(hor or ver))--Layout
Output
Layout
The bRect will be duplicated to 3 Rectangles where the sRects need to be fitted into.
The Worth of the solution is determined by the sum of the value of sRect inside the bRect.
Further is the value decreased if the sRect is placed in the second bRect or third bRect.
sum(sRect(value))*0.98^nth bRect
Not all sRect needs to be placed.
Layout is structered that the three bool at the start represent at which bRect the sRect is placed. If a sRect is placed at one of the bRect, then the Solution Layout muss stay for this sRect the same.
Restricted Ori means all of the sRect with the same group need to be Oriantated the same way. Hor means the sRect is not turned, ver the sRect is turned by 90degrees.
Other then that normal rules apply, like all sRect needs to be inside the bRect and not Overlapp between sRect.
Looking forward to replys and i am avaible for further explanations.
edit: example picture
important i dont want to optimise for maximum plate usage, because it can happen that a smaller sRect can have a higher value then a bigger sRect.
example fitting problem
Without expected output for each input you cannot use the most standard training methodology - supervised learning. If you only have a way to verify the solution (e.g. in a game of chess you can tell me if I won but you cant tell me how to win) then the most standard approach is reinforcement learning. That being said, it is much more complex problem, not something that say a newcomer to the field of ML will be capable of doing (while supervised learning is something that one can do essentially by following basic tutorials online)
I need a bit of counseling. I´m trying to reproduce one of M.C. Escher´s models in Actionscript, but I´m not entirely sure about where to begin. Ideally, I´d want to make something from his Circle Limit series look somewhat like this: http://vimeo.com/4154382
Could anyone provide any pointers as in what approach should I take? I am not an expert coder, so anything would help.
Thanks in advance,
Garfeel M.D.
The different copies of a hyperbolic transformation are related to one another via Möbius transformations which leave the circle fixed. You can represent them as transformations
(a+bi)z + (c+di)
z |-> ----------------
(c-di)z + (a-bi)
You might want to represent the switch from circle to half plane as a Möbius transformation as well, to avoid numeric issues with simple zooming.
I have tools available to make hyperbolic ornaments from Escher ornaments, and zoom into them in real time. But Escher isn't public domain yet, and in my experience the Escher foundation is less than enthusiastic in granting permission for derived works. So if you get ther OK, or decide on some other artist (possibly starting from a Euclidean ornament), feel free to contact me by e-mail to discuss this further.
I recently was a jury member foir an ornament competition where some submissions were hyperbolized from Euclidean drawings. Gaining permissions for those would likely be easier than from the Escher foundation.
I am working on a simple drawing application, and i need an algorithm to make flood fills.
The user workflow will look like this (similar to Flash CS, just more simpler):
the user draws straight lines on the workspace. These are treated as vectors, and can be selected and moved after they are drawn.
user selects the fill tool, and clicks on the drawing area. If the area is surrounded by lines in every direction a fill is applied to the area.
if the lines are moved after the fill is applied, the area of fill is changed accordingly.
Anyone has a nice idea, how to implement such algorithm? The main task is basically to determine the line segments surrounding a point. (and storing this information somehow, incase the lines are moved)
EDIT: an explanation image: (there can be other lines of course in the canvas, that do not matter for the fill algorithm)
EDIT2: a more difficult situation:
EDIT3: I have found a way to fill polygons with holes http://alienryderflex.com/polygon_fill/ , now the main question is, how do i find my polygons?
You're looking for a point location algorithm. It's not overly complex, but it's not simple enough to explain here. There's a good chapter on it in this book: http://www.cs.uu.nl/geobook/
When I get home I'll get my copy of the book and see if I can try anyway. There's just a lot of details you need to know about. It all boils down to building a DCEL of the input and maintain a datastructure as lines are added or removed. Any query with a mouse coord will simply return an inner halfedge of the component, and those in particular contain pointers to all of the inner components, which is exactly what you're asking for.
One thing though, is that you need to know the intersections in the input (because you cannot build the trapezoidal map if you have intersecting lines) , and if you can get away with it (i.e. input is few enough segments) I strongly suggest that you just use the naive O(n²) algorithm (simple, codeable and testable in less than 1 hour). The O(n log n) algorithm takes a few days to code and use a clever and very non-trivial data structure for the status. It is however also mentioned in the book, so if you feel up to the task you have 2 reasons to buy it. It is a really good book on geometric problems in general, so for that reason alone any programmer with interest in algorithms and datastructures should have a copy.
Try this:
http://keith-hair.net/blog/2008/08/04/find-intersection-point-of-two-lines-in-as3/
The function returns the intersection (if any) between two lines in ActionScript. You'll need to loop through all your lines against each other to get all of them.
Of course the order of the points will be significant if you're planning on filling them - that could be harder!
With ActionScript you can use beginFill and endFill, e.g.
pen_mc.beginFill(0x000000,100);
pen_mc.lineTo(400,100);
pen_mc.lineTo(400,200);
pen_mc.lineTo(300,200);
pen_mc.lineTo(300,100);
pen_mc.endFill();
http://www.actionscript.org/resources/articles/212/1/Dynamic-Drawing-Using-ActionScript/Page1.html
Flash CS4 also introduces support for paths:
http://www.flashandmath.com/basic/drawpathCS4/index.html
If you want to get crazy and code your own flood fill then Wikipedia has a decent primer, but I think that would be reinventing the atom for these purposes.
while playing to this game I wondered how an AI controlling either the detectives either the criminal could work.
For lazy people the aim of the game is simple:
the board game is an undirected graphs that has 4 kinds of edges (that can also overlap for same pair or vertices), each kind is a type of transport that requires a specific kind of ticket
detectives have a bunch of tickets to move around this graph, one move per turn (which means from a node to another node). The criminal can do the same set of moves (plus 3 exclusive paths) but with no limits on tickes
the criminal is usually hidden to detectives but it has to show up himself in 5 specific turns (and then hide again)
if detectives are able to catch him (one of them must occupy the same cell of the criminal) before 24 moves then they win, otherwise the criminal wins
the criminal has to show which ticket he uses each turn but he also has 1 black ticket per detective (let's assume 5) that can be used to vanify this thing
the criminal also has two 2x tickets that allow him to use two tickets (and so two movements) in the same turn
I can think effectively about an AI for the criminal that it would be just a minmax tree that tries to choose movements that maximize the number of moves needed by detectives to reach him (it seems to be a good metric) but I cannot think anything enough cool for detectives which should cooperate and try to guess where the criminal can be by looking at tickets it uses.
It's just for fun but do you now any cool ideas to work out something quite clever?
You've asked how to model this, not how to solve this efficiently:
It can be easily modeled as a partially observable markov decision process (wiki link). This works both for the detectives and the criminal. POMDPs are a very generic model.
I love this game, and I think for the detectives you want to model the probability that the criminal is at each location. Every once in a while you know the exact position of the criminal, and then you can take into account the following moves he makes to determine which spots he could possibly be at.
Once you have this, I'm not quite sure how to optimize the detectives moves. You can move the detectives to reduce the set of possibilities, effectively corraling the criminal. But I'm sure there is also some higher level strategy needed surrounding the tickets and not running out of them.
I'd imagine some kind of a monte carlo implementation would be an excellent candidate for this, ie. simulating thousands of combinations and choosing the one that ends with the best result most of the time. Since the criminal has to be visible for 5 turns, the branching factor should stay well under control, although MC has also been shown to be a very good technique in games of high branching factor, ie. Go.
In order to get teamwork going between the detectives you need to model them as a team rather than as individuals. Minimax is still a good way to go but (sadly) your branching factor is going to soar.
Instead of stepping through all the detectives making what appears to be the best for each instead for your team of detectives you work out each permutation of moves they could make. If teamwork helps in this game then the minimax will favour the permutations in which the detectives are working together.
I'm not sure if it will be practical, 5 detectives for 24 ply might be too much work but it'd be fun to try and that's the point right?
I have a simple question regarding the Minimax algorithm: for example for the tic-tac-toe game, how do I determine the utility function's for each player plays? It doesn't do that automatically, does it? I must hard-code the values in the game, it can't learn them by itself, does it?
No, a MiniMax does not learn. It is a smarter version of a brute-force tree search.
Typically you would implement the utility function directly. In this case the algorithm would not learn how to play the game, it would use the information that you had explicitly hard-coded in the implementation.
However, it would be possible to use genetic programming (GP) or some equivalent technique to automatically derive a utility function. In this case you would not have to encode any explicit strategy. Instead the evolution would discover its own way of playing the game well.
You could either combine your minimax code and the GP code into a single (probably very slow) adaptive program, or you could run the GP first, find a good utility function and then add this function to your minimax code just as you would any hand-coded function.
Tic-Tac-Toe is small enough to run the game to the end and assign 1 for win, 0 for draw and -1 for lose.
Otherwise you have to provide a function which determines the value of a position heuristically. In chess for example a big factor is the value of the material, but also who controls the center or how easily the pieces can move.
As for learning, you can add weight factors to different aspects of the position and try to optimize those by repeatedly playing games.
How do determine the utility function for each play?
Carefully ;-) This article shows how a slightly flawed evaluation function (one for ex. which either doesn't go "deep" enough in looking ahead in the tree of possible plys, or one which fails to capture the relative strengh of some board positions) results in an overall weak algorithm (one that looses more often).
it can't learn them by itself, does it?
No, it doesn't. There are ways, however, to make the computer learn the relative strength of board positions. For example by looking into Donald Mitchie and his MENACE program you'll see how a stochastic process can be used to learn the board without any a priori knowledge but the rules of the game. The funny part is that while this can be implemented in computers, a few hundred colored beads and match boxes are all that is required, thanks to the relatively small size of the game space, and also thanks to various symmetries.
After learning such a cool way of teaching the computer how to play, we may not be so interested in going back to MinMax as applied to Tic-Tac-Toe. After all MinMax is a relatively simple way of pruning a decision tree, which is hardly needed with tic-tac-toe's small game space. But, if we must ;-) [go back to MinMax]...
We can look into the "matchbox" associated with the next play (i.e. not going deep at all), and use the percentage of beads associated with each square, as an additional factor. We can then evaluate a traditional tree, but only going, say 2 or 3 moves deep (a shallow look-ahead depth which would typically end in usually in losses or draws) and rate each next move on the basis of the simple -1 (loss), 0 (draw/unknown), +1 (win) rating. By then combining the beads percentage and the simple rating (by say addition, certainly not by multiplication), we are able to effectively use MinMax in a fashion that is more akin to the way it is used in cases when it is not possible to evaluate the game tree to its end.
Bottom line: In the case of Tic-Tac-Toe, MinMax only becomes more interesting (for example in helping us explore the effectiveness of a particular utility function) when we remove the deterministic nature of the game, associated with the easy evaluation the full tree. Another way of making the game [mathematically] interesting is to play with a opponent which makes mistakes...