The Game of Go

Game of Go

It is amazing how simple rules can result in breathtaking complexity. This post is about such another game :

Go

Chances are, you have already seen people play the game. I had seen it first in ‘Fist of Legend‘, Jet Li’s adaptation of the ‘Fist of Fury‘.

It is so old that it was mentioned in the ‘Analects of Confucius‘( 479 B.C ). In fact, the American Go Association cites this is as one of the top ten reasons to play Go.

When you play go, you are doing something that billions of people have done for thousands of years. Many centuries ago, people were doing the exact same thing, in the exact same way.

What an absurd reason to give a promising player to choose the game!!!

Crisply, the rules as specified here:

• Go-board(Goban):19 x 19 intersections.
• Back and White play alternatively.
• Black starts the game.
• Adjacent stones ( of the same colour ) are called a string.
• Liberties are the empty intersections next to the string.
• Stones do not move, they are only added and removed from the board.
• A string is removed if its number of liberties is 0.
• Score: Territories(number of occupied or surrounded intersections).

Plain and simple. There is also a large set of terms associated with the game which you will nedd to know if you are playing / coding a player. All of them, Japanese. I strongly agree with the subject of this post, so try to play a few games before you get coding. I would suggest GNUGo, with Quarry to make a good beginning.

However, Computer Science considers a human level Computer Go player an open problem. It belongs to the same class of problems as the ` P = NP ` and ` The Church Turing Hypothesis `.

A comparison with Chess, ought to bring more light on the subject.

• Computer Chess Programs have defeated Chess Grandmasters, and have become so good, that are now competing among themselves for Chess Supremacy.
• An average Go program, on the other hand, are routinely trounced by Asian school children.
• At any stage of the game, Chess, on average, has 37 possible moves, decreasing as the game progresses. ( Lesser coins, lesser moves )
• Go, on the other hand, averages 250 possible replies per move.
• There are terms in Go, look here, each of which needs a representation in the program.
• Chess has very little such terms, en passe, fortress, and the like, and provides lesser challenge to a programmer who has little knowledge of the game.

If you cannot truly assess the complexity of the problem, here’s one more trivia, Go, is associated with the number 10⁷⁰⁰. This number is almost definitely an overestimate considering that the estimated number of particles in the Universe, when I last checked, was 10⁹¹!! However, more reasonable estimates aren’t too far from this number either.

A brief Intro on how computers play ( board ) games:

Computers, play games by searching in a state space tree. Many use a theorem called the Minimax Theorem. Chess, Go and most other board games are said to be complete-information deterministic games in that both players know everything there is to be known from the beginning to the end of the game. There is no role for chance here. In 1928, Von Neumann proved that such games are always zero sum. In non formal terms, a zero-sum game is a game in which one player’s winnings equal the other player’s losses. It is pretty much like cake cutting. So a good move means that the opponent has got only worse moves, all of them in all variants leading to favourable positions which are therefore unfavourable for the opponent.

An example of the minimax algorithm should be very useful here :

Minimax Example

Consider a two-player game, where each player takes alternate turns ( which is the vast majority of the board games today ). Now as shown in the figure, you have three options, to each which your opponent has three options in reply. The tree’s root indicates that you’ll choose the move that lands you in the best position possible when you are to play next. The numbers in the circle indicates your position after that move. Level 2 is your opponent’s move. Initially, the tree is constructed with empty circles, and evaluation is done from the leaf to the root.

Some evaluation function, say f(x), evaluates the board position and returns the values at the leaf level. You take the minimum of the points your opponent’s move left you and put it in the corresponding higher level. Then, you take the maximum of all the points your move will leave you and put it in the higher level. What has happened? You are left with a guarantee that you’ll get at least +1 no matter what your opponent’s response is. IN simpler terms, you have maximised your minimum profit. And hence the term minimax.

** A minimax tree is generally more than 2 ply( 1 ply is a single player’s turn, turn itself being an ambiguous term, to be used across multiple games ) deep.

Here is more on zero-sum games. A wiki minimax example.

Here is an article that shows why MInimax Theorem is not always practically optimal.

Many optimizations have been made to this approach the prominent of them being, Alpha-Beta pruning.

Naturally, it is important how deep your tree is. The evaluation of the game state is also just as crucial if not more, in the grand scheme of things. A good evaluation function can reduce how deep you need to look.

All of the most famous Chess engines of our time use this technique. For a game like Chess, with an average branching factor of 35~50, specialised hardware with pre-calculated open-game and end-game tables is enough to beat a Chess Grandmaster. In fact it even defeated the highest graded Grandmaster in history.

Such an approach becomes immediately inadequate in the game of Go. No hardware can search in such large a state space. Also, an accurate evaluation function is also very difficult to arrive at. Well, then how do today’s programs approach Go?

A look at the current Computer Go Scenario :

The most powerful computer Go programs in the world today are :

• GNUGo
• MoGo
• CrazyStone
• Go++
• NeuroGo

A rough analysis of these five should be of great interest. Updates on this post and further posts will be regarding this.

Request to readers

Please do leave a token on the page that you are interested. Perhaps a comment? Please do post comments, whatever they may be.

Sincerely,

Sanjeev

The Game of Life

This is not an article on philosophy.

This is about a zero player game called, Life.

Considering its a zero – player game, who would want to play it???

Gosper

But before that, let me just give you a brief intro on what it is and how it is played. It is what one would call in computer science terminology as a cellular automaton. Cellular automatons are studied primarily by biologists, mathematicians and Computer Scientists.

In the second year of our computer science course, we had a paper on Thoery of Computation. And “automata”

was the most uttered word in the class. The paper was complete theory and did not seem to give any scope for fun or enjoyment, unless you were a nerd, or a mathematician.

It was in Robert Kruse’s, Data Structures and Design, that I first came across Life. So here’s the deal.

The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead. Every cell interacts with its eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

1. Any live cell with fewer than two live neighbours dies, as if by loneliness.
2. Any live cell with more than three live neighbours dies, as if by overcrowding.
3. Any live cell with two or three live neighbours lives, unchanged, to the next generation.
4. Any dead cell with exactly three live neighbours comes to life.

The initial pattern constitutes the ’seed’ of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed — births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick. (In other words, each generation is a pure function of the one before.) The rules continue to be applied repeatedly to create further generations.

Simple. Very basic indeed, simulations of how Life may exist, overcrowding, loneliness, all only too familiar for us humans. Simplicity is always a defining characteristic of an automaton. So there would be starting population, and once the sequence starts, it continues, without any intervention.

I see no further point in explaining in theory about stuff that you really can enjoy once you see it for yourself. I’ll direct you here to a Java Applet at Math.comhere Somewhere in the page is a link to a Java applet. I had also seen an online Flash implementation, as an active essay, which I am unable to retrieve now. Anyway, Go ahead. Experiment. Try your own signature, and see what it results in. How about if you draw a smiley, what happens??? Chances are most likely that the population eventually dwindled into nothing or it stabilised, i.e.., split up and formed gangs, stopped interacting with other gangs, and eventually settled down.

When John Conway published his ideas in Scientific American in 1970, he had conjectured that there could not exist initial populations, that would neither die off nor stabilise. He offered $50 as prize money to anyone who could prove or disprove his conjecture. A team from MITlead by Bill Gosper won the prize. Gosper disproved Conway’s idea by developing an initial population, called the Glider Gun, which generated a glider after every 30 generations. Thats the picture at the top of the page. Eventually more and more initial populations were found that had traits similar to the Glider Gun. Many of them were more or less.., Guns. After every X generations the population would produce a unit that would fly away, and live an endless life. I have provided a link at the bottom, in case you are interested.

Glider

Gosper’s glider was a hit. So much that, this symbol called ‘Glider’, to the right was proposed to be adopted as the symbol of the hacker community. “The Hacker Community” I am referring to here is not the one that goes around busting the Internet. Any enthusiastic coder, who tweaks a software.., could be roughly called a programmer. I am obliged to give a reference here.

[ A similar, but not related concept is the Collatz Sequence. A simple rule indicating what one particular number would mutate into next. The sequence of numbers end, when you reach one. The conjecture is that from any number, on applying the rules, it always ends in the same pattern => 16-8-4-2-1 ]

It only gets more interesting. This, “Game of Life is said to be Turing Complete“. A Turing machine is essentially an infinite tape with a pointer attached to it. But, amusingly, you can do EVERYTHING using a Turing machine that you can do using a modern day computer. It may sound , a little insane perhaps, when I say that a tape can simulate a PC. I restate, this Game can simulate your PC. Look at it this way. You would believe it if I say that AND, OR and NOT gates could simulate a PC. Naturally, that’s what PCs are made of right? Maybe it would sound a little more sane or credible to say that the game can simulate all of these gates.

Hash Life

Consider implementing the Game of Life using say, Java. Most normal approach is to keep two copies of a 2D Boolean array and juggle between them as the current population. Bill Gosper also developed an algorithm for a much faster implementation of Life called Hash Life. There are various places where one could tweak the original approach, but Hash Life remains the one of the fastest algorithms known. I hear that there are only 200 or so implementations of the algorithm all over the world. Maybe one day, I could implement one on my own…

In retrospect, John Conway’s implementation was using using stones and pebbles.

A crazy Proposition

I have been recently getting more interested in Evolution, physiology and Genetic Algorithms. And I started wondering if one could evolve an initial population that would keep multiplying. It seems utterly non-sense in one sense in that the “World” is utterly chaos. How do you judge the fitness of populations? How would one implement a cross-over?? Anyway, just a crazy idea. I just wanted to put it down, in case, somebody stumbles upon it and gives me a hint.

Further Reading:

• Lots of cellular Automata
• Link from Wonders of Math describing very interesting formations.

A new kind of a Game

It doesn’t look visually enticing, but I am pretty sure its darn good. No, I have not played the game yet.

NERO

Now, I wasn’t expecting to write a post on a game, any time soon. Not even on DotA. But this is just too good a game for me to resist. I have been quite intrigued by the concept of Evolutionary Computation. One lazy Sunday, I started looking for more stuff on Evolutionary Computation. I found a lot. Particularly, a post-doctoral researcher’s blog. Julian Togelius. He’s published a paper on solving Sudoku using Genetic Algorithms. Serious stuff that! One of his other papers have been judged as one of the top ten AI game innovations of the year. This dude was one of the organisers of a Simulated Car Racing Championship atWCCI 2008. It seems there are not a lot of games available that actually use the latest Learning Techniques.

It was very funny in the beginning when I looked at the games’ graphics( WCCI ). Not a ‘Need for Speed’ then. On further reading, I found more on the rules of the game. From a programmer’s stand point, I thought, not very easy to program a controller. All the various cases. How could it adapt?? Perhaps learn?? Neural Networks then?? Neural Networks are to put it very very simple terms, like a brain. ( Please note that this is a gross over-estimation, it resembles the brain only in that it can learn new stuff. It is not even close to the functioning of a brain ). Perhaps neural networks would have been the answer if the same question had been put forth a few years ago. Today, ANNs alone will not suffice!!

Genetic Algorithms are now used to train Neural Networks. OK. Imagine you are given the job of teaching something to someone. If you couple GA with ANN, it’d something like, you just have to tell the person, how well he has learnt the subject, and he’ll keep improving all by himself. ( Of course, this is another gross over-estimation. If this is very non-technical please refer to one of the links given in the side )

It just got more interesting after that. I learnt that one of the Car controllers was evolved using rtNEAT( real time NeuroEvolving Augmenting Technologies ) developed by Kenneth O Stanley at University of Texas at Austin. NEAT not only evolves the best neural network of a given structure, it also evolves the best structure for the purpose. Fantastic stuff. As I kept reading, link after link. I get here. The page gives a detail overview of how NERO is played and how it is being developed. Please do read it. Its an interesting read. You’ll start wondering what kind of games will be the next big trend.

I have added on my links lists some of the more interesting blogs I have found during my research. Anyone interested in AI, please do have a look at these pages. I definitely plan on writing more posts on my own discoveries on the Web about AI, more particularly, Evolutionary Computing. And next time, I hope I try to convey more technically what I actually wish to convey. Cheers.

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