## Wednesday, December 30, 2009

### The answer from above

While we game AI programmers noodle with our gross simplfications and regrettable /understandably unavoidable but also fun and challenging real-time performance constraints, we occasionally look up for any new answers from above, normally we are too busy to stay up to date, but it's the holidays and I am bored, I have no ps3 devkit here, nor a PC with a dev. environment to do some brainless coding... so I had to be brainful and start reading and so I stumbled upon this: "the currently available theories do not explain or engender anything resembling human-level general intelligence" what is meant here is theories coming from Information-processing psychology e.g: Cognitive Science and Cognitive Neuroscience. (source: http://www.cs.umd.edu/~nau/cmsc722/)

I wonder when such theories will start to be discovered and what kind of processing power we will have at the time and if they will good enough to allow the ones who will be peeking there at the time and spotting low hanging fruits to become famous applying them to video games (and other applications) being again, at the right time and place.

## Sunday, November 29, 2009

### Now that's bad performance code! once and for all....

Translating an idea into a piece of code is an over-constrained problem, just like many other problems.

To decide how to code something, make a list of all points you think are important for it (maintainability, performance, easy to read by me, easy to reuse by me, flexible, many other pieces will depend on it, multi-platform, multi-compiler, link fast, compile fast, short names for faster typing, easy to read/understand/reuse for my colleagues, easy to read/understand/reuse for my clients, cryptic to prove I am 'old school' and can write assembly and you should be scared of discussing it with me, totally abstract to prove I don't care about performance and want to make a point that premature optimization is the source of all evil, totally lean and mean to prove that non-premature optimization is the road to a lame duck... you name it! I don't care what you put in there, the list can be very long and can include anything you like), score the points in your list based on their utility for the piece of code to be written with the very welcome possibility of zero utility for some of them (makes it less constrained).

You cannot compare apples to oranges? (e.g: maintainability vs. performance) ? yes you can (Yes son, you can compare apples to oranges... )! on top of that, you have no choice...

Finally, code/make compromises to maximize the total score, that's all there is to it and being an over-constrained problem for anything none-trivial it won't be completely obvious.

But the problem is clear, no need to call a programming style 'too old school' or another one 'too abstract' or 'too object oriented'. The higher the total score, the better ... that's it.

Now if u do not have the necessary coding skills, you might generate code that has a total score that is not the maximum possible ... but that is another topic.

.

To decide how to code something, make a list of all points you think are important for it (maintainability, performance, easy to read by me, easy to reuse by me, flexible, many other pieces will depend on it, multi-platform, multi-compiler, link fast, compile fast, short names for faster typing, easy to read/understand/reuse for my colleagues, easy to read/understand/reuse for my clients, cryptic to prove I am 'old school' and can write assembly and you should be scared of discussing it with me, totally abstract to prove I don't care about performance and want to make a point that premature optimization is the source of all evil, totally lean and mean to prove that non-premature optimization is the road to a lame duck... you name it! I don't care what you put in there, the list can be very long and can include anything you like), score the points in your list based on their utility for the piece of code to be written with the very welcome possibility of zero utility for some of them (makes it less constrained).

You cannot compare apples to oranges? (e.g: maintainability vs. performance) ? yes you can (Yes son, you can compare apples to oranges... )! on top of that, you have no choice...

Finally, code/make compromises to maximize the total score, that's all there is to it and being an over-constrained problem for anything none-trivial it won't be completely obvious.

But the problem is clear, no need to call a programming style 'too old school' or another one 'too abstract' or 'too object oriented'. The higher the total score, the better ... that's it.

Now if u do not have the necessary coding skills, you might generate code that has a total score that is not the maximum possible ... but that is another topic.

.

### AI room + blackboard = geek art

### PS3 game/FPS AI research

## Monday, October 19, 2009

### Tech-radio silence

It has been some time since I posted anything but I am still alive and still scratching my brain the whole time, the reason for the radio-silence is that since September 2009 I am an AI coder at Guerrilla Games, it is a great experience.

I was going to take some pics but I found this:

http://ps3life.nl/nieuws/4528-een-kijkje-rond-en-in-guerrilla-studios/ this is how it looks like in here currently.

I was also interviewed at AiGameDev (http://aigamedev.com/insider/event/event-career-journey/), if you want to laugh at how sleepy the tone of my voice makes you will be probably be able to see it when it's posted as a video capture sometime in the near future.

More to come...

I was going to take some pics but I found this:

http://ps3life.nl/nieuws/4528-een-kijkje-rond-en-in-guerrilla-studios/ this is how it looks like in here currently.

I was also interviewed at AiGameDev (http://aigamedev.com/insider/event/event-career-journey/), if you want to laugh at how sleepy the tone of my voice makes you will be probably be able to see it when it's posted as a video capture sometime in the near future.

More to come...

## Friday, August 21, 2009

### Irrationals on the border of existence and sqrt(2)

I have been reading a lot about abstract math, what numbers really are and are not, set theoretic number theory and related. The set theoretic approach even if I did not dig into the deepest depths of it, allowed me to be able to logically justify to myself the existence and nature of numbers.

I even bothered my wife (who could not care less) about the beauty I found in the irrational number: square root of 2, what I told her is the following:

I will prove to you how beautiful is math and that we should be grateful for all the people who contributed to it along the centuries of human thinking. I will give you a calculator and you can only use to to multiply, now with no other references, find me the exact square root of 2. Of course, one would proceed to multiply 1.1*1.1= 1.21 then 1.5*1.5=2.25, hence coming to the conclusion that 1.1 < style="font-weight: bold;">

-------------------------------------------------------------------------------------------------

The following reductio ad absurdum argument showing the irrationality of √2 is less well-known. It uses the additional information 2 > √2 > 1 so that 1 > √2 − 1 > 0.

One could almost argue such numbers do not really exist, in the end, they are not called crazy/irrational (and have been fought) for no reason! The way I see it is that they don't at least as written out numbers, they do exist if we set a desired precision, this is why I am liking what I call a 'computationally theoretic number theory', no idea if it exists but you get my point. By setting a precision we can work with those things. One could argue that the number exists and that it's representation is sqrt(2), but this is not a number, the way I see it is that this is a rational (existing) number combined with an algorithm (or call it a function) that can transform this into another in this case irrational number, so either we imprecisely write down a number that approximates it to a given precision or we represent it as an algortihm (sqrt) and data (2) that expand to this 'inexisting' number, this is all layman terms and layman talk, and mathematicians will laugh, but I am recording these thoughts because, since I am a bit satisfied with what I know about this now, I will stop digging and go back to the actual reason I started to look into math again, and that is to solidify my math needed for a self designed auto-didactic machine learning 'course' in my free time. Another sqrt(2) existence thought occured to me in the car last weekend, imagine you have a piece of rubber of length 1, now you take it and stretch it to length 2, did you pass by sqrt(2)? you must have, so it exists? can one measure it? again only to some precision ... (even on the atomic/quantum level). almost mind boggling, this infinity of numbers, but it also makes sense, we allowed for it the moment we allowed ourselves to have a coma and numbers after it, after that recursively you have infinities of infinities of infinities ... but all still allow me not to explode when I look at the set theoretic number theory, basically it is ordering and number of things (in layman, mathematician laugh terms) ... between 1 and 2 there is an infinity of numbers, same as between 1 and 1.1 and 1 and 1.0001 ... in any case ... back to less mind boggling and much more practical stuff, in the spirit of the way ppl have been using numbers since ages for practical matter and not even really understanding what they are. and let me reiterate, please excuse the layman :( he's just trying to make sense of it within a very limited amount of time.

Addendum:

* bourbaki, one of my favorite persons for math discussions, does not think this post is utter non-sense and he just pointed me to:

http://en.wikipedia.org/wiki/List_of_paradoxes, http://en.wikipedia.org/wiki/Continuum_hypothesis

-------------------------------------------------------------------------------------------------

Set theory is the branch of mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....

that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....

and Dedekind in the 1870s. After the discovery of paradoxes

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics....

in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms

Zermelo?Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundations of mathematics....

, with the axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...

, are the best-known.

Set theory, formalized using first-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....

, is the most common foundational system for mathematics.

----------------------------------------------------------------------

Some References:

* http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html

* http://www.absoluteastronomy.com/topics/Naive_set_theory

* http://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality

* The essence of discrete mathematics book

* ...

.

I even bothered my wife (who could not care less) about the beauty I found in the irrational number: square root of 2, what I told her is the following:

I will prove to you how beautiful is math and that we should be grateful for all the people who contributed to it along the centuries of human thinking. I will give you a calculator and you can only use to to multiply, now with no other references, find me the exact square root of 2. Of course, one would proceed to multiply 1.1*1.1= 1.21 then 1.5*1.5=2.25, hence coming to the conclusion that 1.1 < style="font-weight: bold;">

-------------------------------------------------------------------------------------------------

The following reductio ad absurdum argument showing the irrationality of √2 is less well-known. It uses the additional information 2 > √2 > 1 so that 1 > √2 − 1 > 0.

- Assume that √2 is a rational number. This would mean that there exist positive integers
*m*and*n*with*n*≠ 0 such that*m*/*n*= √2. Then*m*=*n*√2 and*m*√2 = 2*n*. - We may assume that
*n*is the smallest integer so that*n*√2 is an integer. That is, that the fraction*m*/*n*is in lowest terms. - Then
- Since 1 > √2 − 1 > 0, it follows that
*n*>*n*(√2 − 1) =*m*−*n*> 0. - So the fraction
*m*/*n*for √2, which according to (2) is already in lowest terms, is represented by (3) in strictly lower terms. This is a contradiction, so the assumption that √2 is rational must be false.

One could almost argue such numbers do not really exist, in the end, they are not called crazy/irrational (and have been fought) for no reason! The way I see it is that they don't at least as written out numbers, they do exist if we set a desired precision, this is why I am liking what I call a 'computationally theoretic number theory', no idea if it exists but you get my point. By setting a precision we can work with those things. One could argue that the number exists and that it's representation is sqrt(2), but this is not a number, the way I see it is that this is a rational (existing) number combined with an algorithm (or call it a function) that can transform this into another in this case irrational number, so either we imprecisely write down a number that approximates it to a given precision or we represent it as an algortihm (sqrt) and data (2) that expand to this 'inexisting' number, this is all layman terms and layman talk, and mathematicians will laugh, but I am recording these thoughts because, since I am a bit satisfied with what I know about this now, I will stop digging and go back to the actual reason I started to look into math again, and that is to solidify my math needed for a self designed auto-didactic machine learning 'course' in my free time. Another sqrt(2) existence thought occured to me in the car last weekend, imagine you have a piece of rubber of length 1, now you take it and stretch it to length 2, did you pass by sqrt(2)? you must have, so it exists? can one measure it? again only to some precision ... (even on the atomic/quantum level). almost mind boggling, this infinity of numbers, but it also makes sense, we allowed for it the moment we allowed ourselves to have a coma and numbers after it, after that recursively you have infinities of infinities of infinities ... but all still allow me not to explode when I look at the set theoretic number theory, basically it is ordering and number of things (in layman, mathematician laugh terms) ... between 1 and 2 there is an infinity of numbers, same as between 1 and 1.1 and 1 and 1.0001 ... in any case ... back to less mind boggling and much more practical stuff, in the spirit of the way ppl have been using numbers since ages for practical matter and not even really understanding what they are. and let me reiterate, please excuse the layman :( he's just trying to make sense of it within a very limited amount of time.

Addendum:

* bourbaki, one of my favorite persons for math discussions, does not think this post is utter non-sense and he just pointed me to:

http://en.wikipedia.org/wiki/List_of_paradoxes, http://en.wikipedia.org/wiki/Continuum_hypothesis

-------------------------------------------------------------------------------------------------

Set theory is the branch of mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....

that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Cantor

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....

and Dedekind in the 1870s. After the discovery of paradoxes

Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics....

in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms

Zermelo–Fraenkel set theory

Zermelo?Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundations of mathematics....

, with the axiom of choice

Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...

, are the best-known.

Set theory, formalized using first-order logic

First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....

, is the most common foundational system for mathematics.

----------------------------------------------------------------------

Some References:

* http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html

* http://www.absoluteastronomy.com/topics/Naive_set_theory

* http://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality

* The essence of discrete mathematics book

* ...

.

## Saturday, July 25, 2009

### At the end of the day, he would still be staring at the same blank sheet of paper.

Did it ever happen to you that you would sit to try to solve a new problem, and the more you would think about it the less it would make sense to you? if you would do that at your desk, would you then be considered non-productive? if you were a game developer be it technical, artistic or manager, sitting there and not typing for hours without making any progress would that be bad? well, Bertrand Russell, one of the most famous logicians of all times did exactly that, so you are ok :)

S = {x : x is a set and x !∈ x}.

In other words, S is the set of all sets that do not contain themselves.

In more 'naive' words:

* In Seville, there’s a barber who shaves all those people who do not shave themselves. Does the

barber shave himself or not? This is known as the “Barber of Seville problem”.

* Imagine a card. On one side is written, “The statement on the other side of this card is true.” and

on the other side is written, “The statement on the other side of this card is false.”

Bertrand Russell, one of the most famous logicians ever, struggled with this problem for a long time. In his autobiography, he describes just how hard he found the problem. Every morning, he said, he would sit down at his desk with a blank piece of paper in front of him. At the end of the day, he would still be staring at the same blank sheet of paper.

Russell’s final resolution to the problem is described in his “Principia Mathematica”, written with Alfred North Whitehead, in which he introduced a “Theory of Types” to get around his paradox. The basic idea was this: sets cannot contain themselves....

http://www.geometer.org/mathcircles/nothing.pdf

.

S = {x : x is a set and x !∈ x}.

In other words, S is the set of all sets that do not contain themselves.

In more 'naive' words:

* In Seville, there’s a barber who shaves all those people who do not shave themselves. Does the

barber shave himself or not? This is known as the “Barber of Seville problem”.

* Imagine a card. On one side is written, “The statement on the other side of this card is true.” and

on the other side is written, “The statement on the other side of this card is false.”

Bertrand Russell, one of the most famous logicians ever, struggled with this problem for a long time. In his autobiography, he describes just how hard he found the problem. Every morning, he said, he would sit down at his desk with a blank piece of paper in front of him. At the end of the day, he would still be staring at the same blank sheet of paper.

Russell’s final resolution to the problem is described in his “Principia Mathematica”, written with Alfred North Whitehead, in which he introduced a “Theory of Types” to get around his paradox. The basic idea was this: sets cannot contain themselves....

http://www.geometer.org/mathcircles/nothing.pdf

.

## Tuesday, July 21, 2009

### My steam gamer card, join to talk AI while blasting baddies :P

### Yes son, you can compare apples to oranges...

One of the things that bothered me while tweaking and tuning the Keltis AI heuristics, was that ultimately things sometimes boiled down to the need to compare apples to oranges, unfortunately I do not remember the exact details and I am too lazy to dig them up, but I know that I had to compare values that I was not able to reduce to a common unit to measure by (like risk per example), it was really a matter of preference, this is not a new problem, and with my head in the details, I failed to notice the obvious, this is an old topic called utility that economists have been using for decades, of course, as usual, I said 'aha' just after getting my head out of the details and shipping.

It was no big deal though, I ended up using utility without knowing it.

Utility is 'the' way to compare apples to oranges, but what brings me to today's rant is that I remembered this while reading in the context of my ongoing current research in applying Reinforcement Learning to Animation planning.

The question in question is about a very valid question [ :) :D :P ] about the 'essence' of Reinforcement Learning (similar to http://rlai.cs.ualberta.ca/RLAI/rewardhypothesis.html) :

source: http://www.eecs.umich.edu/~baveja/RLMasses/node5.html#SECTION00032000000000000000Is it sensible to treat all preferences as numeric rewards on a single scale?Theoretically, yes. There is a theorem (North [4]) that if you believe four fairly simple axioms about preferences, then you can derive the existence of a real-valued utility function. (The only mildly controversial axiom is substitutability: that if you preferAtoB, then you must prefer a coin flip betweenAandCto a coin flip betweenBandC.) Practically, it depends. Users often find it hard to articulate their preferences as numbers. (Example: you have to design the controller for a nuclear power plant. How many dollars is a human life worth?)

I could not find the original in free electronic format: "D. W. North. A tutorial introduction to decision theory.

*IEEE Transactions on Systems Man and Cybernetics*, SSC-4(3), Sept. 1968. "

If anyone can provide it I would be grateful, it is always very insightful to read about the essence of these things, this usually involves reading very old papers, and from my experience it is always worth it, it gives lots of confidence when applying things later and when doubts appear, because much thought and critical thinking went into each and every 'fact' we take today for granted, and for too naive tomorrow.

.

## Saturday, July 11, 2009

### Jad the Naive Mathematician, the absurdity of logic

Here I present my brain, it has been learning and evolving for some time, and recently, it noticed that, logically, the math it thought makes sense, actually doesn't.

The source of 'Math'

This goes some time back into the past, when I suddenly felt the urge to see where Math starts, because logically, and this is something I remember was the base of proving stuff, you need to base yourself on something that is true to prove something else. Anybody who knows a little bit about this knows that this directly leads to Axioms, Occam's razor, Goedel and co...

Useless education

Funny we have been thinking we know our very basic math, but we really do not even know that.

Even the pythagorean theorem seems not logical looking at it this way. Looking at proofs, the proofs themselves either use geometric manipulation of squares, triangles making assertions about areas, and some of those proofs came from periods were an area was something intuitive and not really formalized, come to think of it the concept of area itself is pretty much elusive, and looking for the rigorous math definition leads you to Reimann and others, and that's pretty recent in history. What's more annoying, I made it through school and a Bachelor in Engineering and I never once heard of them. What is even more annoying, I felt I knew what an 'area' is although, if I had thought critically and logically, I would have came ot the conclusion that there is something elusive about it, just like I did recently.

All of this post comes after lots of going back to trying to understand the roots of math, using wikipedia and google, some of this are listed in the bottom of the post.

Proof of a proof

One nice idea from this quest is Goedel and his Incompleteness theorem, naively for me right now it means you need to start from something to make any proofs, and that something you started from cannot be proved. I will not go back and read the details, but while taking a shower just now, I became curious as to how Goedel proved this, did he use an axiom as a base, if this axiom was removed, not even this could be proven? This got me to think about what logic is, and about the 'axioms' of logic. Logic seems to be something the brain can very easily accept and use as a base. Again going back to Engineering, much of what is left is the logic. But why? and what is logic, isn't it absurd by itself? what is the logic that logic is based on? Why does the brain readily accept it? (without 'proof').

A group of 'things', excluding 'Neo, the source'

This got me to realize that there is a certain group of things, that all fall into some category for which I don't have a name: Logic (needing logic to make sense), time (continuous/discrete), infinity, zero (1 over infinity!), space and it's size both endless and not being absurd (same for time). All these things feel like one and the same, or belonging to one category. We end up accepting them and even using them, but few of us really grasp them.

Think versus. Grasp

I also remembered vaguely something that I think Einstein said about things a human brain will never grasp, comparing to a table with eyes looking down never being able to see what is above it (I am not sure about the exactness of any of this). But what I recently found interesting, is the fact that we are able to think about these things, even though we might not be able to understand them (by construction?), why this separation? why can't we only think about things we can understand? Does this boundary mean something? and what?

Dump and live on

I wrote this post mainly for one reason: get it off my brain to free it for thinking about more practical stuff.

Feel free to express your opinion about this at

http://forums.aigamedev.com/showthread.php?p=15004#post15004

Some of the references

http://www.mathacademy.com/pr/prime/articles/fta/index.asp?LEV=&TBM=&TAL=&TAN=&TBI=&TCA=&TCS=&TDI=&TEC=&TFO=&TGE=&TGR=&THI=&TNT=&TPH=&TST=&TTO=&TTR=&TAD=

http://www.mathacademy.com/pr/prime/articles/irr2/index.asp

http://www.google.de/search?q=proof+square+root+of+2+is+irrational&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a

http://en.wikipedia.org/wiki/Well-order

http://en.wikipedia.org/wiki/Infinite_descent

http://en.wikipedia.org/wiki/Square_root_of_2

http://en.wikipedia.org/wiki/Rational_number

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470211520.html

http://en.wikipedia.org/wiki/Commensurability_(mathematics)

http://www.boost.org/doc/libs/1_37_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html

http://en.wikipedia.org/wiki/Elliptic_integral

http://sci.tech-archive.net/Archive/sci.math/2006-09/msg04719.html

http://books.google.de/books?id=RM1D3mFw2u0C&pg=PA7&lpg=PA7&dq=%22rigorous+definition+of+area%22&source=bl&ots=jiarfVKaP5&sig=OAi9X-H7Hnp92BdfIuiIA911KSc&hl=en&ei=jJdXSomENIed_AahldSdCQ&sa=X&oi=book_result&ct=result&resnum=7

http://www.amazon.co.uk/gp/offer-listing/0133459438/ref=dp_olp_1?ie=UTF8&qid=1247256551&sr=8-1

http://www.amazon.com/gp/product/images/0486439461/ref=dp_image_0?ie=UTF8&n=283155&s=books

http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=Discrete+Mathematics&x=0&y=0

http://www.mathkb.com/Uwe/Forum.aspx/math/16463/Concept-of-measure-in-undergraduate-mathematics

http://www.google.de/search?hl=en&safe=off&client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&hs=iW1&num=100&q=%22rigorous+definition+of+area%22&btnG=Search

http://www.youtube.com/results?search_query=The+Fundamental+Theorem+of+Calculus&search_type=&aq=f

http://www.youtube.com/watch?v=MOnnMlMM70Q&feature=PlayList&p=D4E266DF4E3352B1&index=18

The source of 'Math'

This goes some time back into the past, when I suddenly felt the urge to see where Math starts, because logically, and this is something I remember was the base of proving stuff, you need to base yourself on something that is true to prove something else. Anybody who knows a little bit about this knows that this directly leads to Axioms, Occam's razor, Goedel and co...

Useless education

Funny we have been thinking we know our very basic math, but we really do not even know that.

Even the pythagorean theorem seems not logical looking at it this way. Looking at proofs, the proofs themselves either use geometric manipulation of squares, triangles making assertions about areas, and some of those proofs came from periods were an area was something intuitive and not really formalized, come to think of it the concept of area itself is pretty much elusive, and looking for the rigorous math definition leads you to Reimann and others, and that's pretty recent in history. What's more annoying, I made it through school and a Bachelor in Engineering and I never once heard of them. What is even more annoying, I felt I knew what an 'area' is although, if I had thought critically and logically, I would have came ot the conclusion that there is something elusive about it, just like I did recently.

All of this post comes after lots of going back to trying to understand the roots of math, using wikipedia and google, some of this are listed in the bottom of the post.

Proof of a proof

One nice idea from this quest is Goedel and his Incompleteness theorem, naively for me right now it means you need to start from something to make any proofs, and that something you started from cannot be proved. I will not go back and read the details, but while taking a shower just now, I became curious as to how Goedel proved this, did he use an axiom as a base, if this axiom was removed, not even this could be proven? This got me to think about what logic is, and about the 'axioms' of logic. Logic seems to be something the brain can very easily accept and use as a base. Again going back to Engineering, much of what is left is the logic. But why? and what is logic, isn't it absurd by itself? what is the logic that logic is based on? Why does the brain readily accept it? (without 'proof').

A group of 'things', excluding 'Neo, the source'

This got me to realize that there is a certain group of things, that all fall into some category for which I don't have a name: Logic (needing logic to make sense), time (continuous/discrete), infinity, zero (1 over infinity!), space and it's size both endless and not being absurd (same for time). All these things feel like one and the same, or belonging to one category. We end up accepting them and even using them, but few of us really grasp them.

Think versus. Grasp

I also remembered vaguely something that I think Einstein said about things a human brain will never grasp, comparing to a table with eyes looking down never being able to see what is above it (I am not sure about the exactness of any of this). But what I recently found interesting, is the fact that we are able to think about these things, even though we might not be able to understand them (by construction?), why this separation? why can't we only think about things we can understand? Does this boundary mean something? and what?

Dump and live on

I wrote this post mainly for one reason: get it off my brain to free it for thinking about more practical stuff.

Feel free to express your opinion about this at

http://forums.aigamedev.com/showthread.php?p=15004#post15004

Some of the references

http://www.mathacademy.com/pr/prime/articles/fta/index.asp?LEV=&TBM=&TAL=&TAN=&TBI=&TCA=&TCS=&TDI=&TEC=&TFO=&TGE=&TGR=&THI=&TNT=&TPH=&TST=&TTO=&TTR=&TAD=

http://www.mathacademy.com/pr/prime/articles/irr2/index.asp

http://www.google.de/search?q=proof+square+root+of+2+is+irrational&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a

http://en.wikipedia.org/wiki/Well-order

http://en.wikipedia.org/wiki/Infinite_descent

http://en.wikipedia.org/wiki/Square_root_of_2

http://en.wikipedia.org/wiki/Rational_number

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470211520.html

http://en.wikipedia.org/wiki/Commensurability_(mathematics)

http://www.boost.org/doc/libs/1_37_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html

http://en.wikipedia.org/wiki/Elliptic_integral

http://sci.tech-archive.net/Archive/sci.math/2006-09/msg04719.html

http://books.google.de/books?id=RM1D3mFw2u0C&pg=PA7&lpg=PA7&dq=%22rigorous+definition+of+area%22&source=bl&ots=jiarfVKaP5&sig=OAi9X-H7Hnp92BdfIuiIA911KSc&hl=en&ei=jJdXSomENIed_AahldSdCQ&sa=X&oi=book_result&ct=result&resnum=7

http://www.amazon.co.uk/gp/offer-listing/0133459438/ref=dp_olp_1?ie=UTF8&qid=1247256551&sr=8-1

http://www.amazon.com/gp/product/images/0486439461/ref=dp_image_0?ie=UTF8&n=283155&s=books

http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=Discrete+Mathematics&x=0&y=0

http://www.mathkb.com/Uwe/Forum.aspx/math/16463/Concept-of-measure-in-undergraduate-mathematics

http://www.google.de/search?hl=en&safe=off&client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&hs=iW1&num=100&q=%22rigorous+definition+of+area%22&btnG=Search

http://www.youtube.com/results?search_query=The+Fundamental+Theorem+of+Calculus&search_type=&aq=f

http://www.youtube.com/watch?v=MOnnMlMM70Q&feature=PlayList&p=D4E266DF4E3352B1&index=18

## Friday, June 26, 2009

### A* / HPA* links and scribblings shared

There just was a question on the AiGameDev forums asking about A* (Astar), I remembered I had my own old links and scribblings somewhere so I shared them, so here they are if anybody needs them:

A* / HPA* links, references, implementation considerations:

http://docs.google.com/View?id=dcm3hb4r_30mjc2cj4j

A* basic theory scribblings:

http://jadnohra.net/release/AStar_Basic_Theory.pdf

If you have been following the blog you will see that I recently got into Reinforcement Learning and Dynamic Programming (see previous post), this gave me a much better overview about the 'essence' A* is and how it came to be, really just a case of applying dynamic programming.

The essence:

.

A* / HPA* links, references, implementation considerations:

http://docs.google.com/View?id=dcm3hb4r_30mjc2cj4j

A* basic theory scribblings:

http://jadnohra.net/release/AStar_Basic_Theory.pdf

If you have been following the blog you will see that I recently got into Reinforcement Learning and Dynamic Programming (see previous post), this gave me a much better overview about the 'essence' A* is and how it came to be, really just a case of applying dynamic programming.

The essence:

*"Principle of Optimality: An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision."*(written in 1957 by Bellman, a genius).

## Sunday, June 21, 2009

### Pure math versus game animation technology

Today was yet another enlightening day, the game developer part of me likes to call those days "Level-up" days, if you look at my AiGameDev forum posts, my twits on twitter, the discussions we have on #gameai IRC channel, my emails with Mathematicians I have never met, you could see the conext of today's enlightment, but here is the full story:

Questionable game animation technology research

Working on the AIGameDev animation system with Alex Champandard, we reached a rather experimental (for now) stage of wanting to use Reinforcement learning to learn the heuristic of an A* planner used for planning locomotion based on a given, non-annotated, step based, automatically generated motion graph.

The motivation of doing this might be questionable, but the whole thing developed step by step.

In summary we had the motion graph builders, written by Alex, he also had the brilliant idea of making them step based, using very sensible points in time to make animation transitions, we had the generic A* code I have written, we wanted to combine the two to have something moving on the screen, showing the usefulness of this foot-skate free motion graph approach.

This gave birth to the A* motion planner, A planner that is not very suitable for multiple characters in real time, a planner that only needs a motion graph, and no extra code in form of hand written controllers, that gets you from A to B in a fast, high transition quality way.

It was not very good at reaching the destination with an orientation constraint, it was possible, but it made the heuristic too complicated, almost like writing a manual controller, and much

slower. In theory, this should be very easy, or at least easily learnable.

Reinforcement Learning

There are papers that use RL for motion planning, pretty recent ones (starting 2007), like "Real-Time Planning for Parametrized Human Motion" or "Near Optimal Character Animation with Continuous Control", I mention those two because of I have them printed on my running-out-of-space desk, along with countless "Gleicher"/"Kovar" papers about animation and motion graphs. It was Alex who drove me into all of this, so any straying is both his fault, and also his merit, I am grateful, I have long been searching for something interesting, some of my previous jobs failed big there.

The idea was to learn the heuristic, making the planner suitable for real time, this might be questionable for some people, the industry mostly drives it's animations, no need for planning at all, simple, effective, KISS, and full of foot-skate.

At the AiGameDev conference in Paris this month, there was one panel where we discussed the uncanny valley, the need or no-need to cross it, I also discussed this afterwards with Chrisitiaan Moleman and Markus Mohr, Remco Straatman and others, I mention those because I was extremely pleasantly surprised about how passionate and nice those people and everybody else who was at the conference are, I am not getting tired of repeating this because it was remarkable.

So to go back, the motivation, is by all means questionable, except if we decide that we want to see how a next step in animation would look like, I am not saying we should be planning the whole time, there will be times for driving the animations and times for planning, but I'm keeping this under wraps for the moment, although it's not very difficult to imagine what I mean.

Applying Reinforcement Learning

So all of this got me into Reinforcement Learning, a classical reference is the Sutton/Barto book "Reinforcement Learning, an Introduction" available online for free. I also bought a couple of related books. I read the parts of online book I needed the 1st time and designed an RL approach to learning what we needed to learn to help the A* heuristic, the result was that it basically 'worked' but was a naive first take that needed refinement to be really useful, the details will be available to look at in the AiGamDev sandbox at some point. At this stage I had read parts of the online book some in more detailed than other, and I would say I had revisited some parts 3 or 4 times. I was at the stage where I understood what we need, make a first attempt, understood the problem much better because of it, and was ready to design the second take, which needed a technique that is a bit more evolved that the 1st technique I had used.

Reading in detail and understanding the Blues

I decided, that I do not feel like coming back to the basics anymore, and to read the full book in maximum detail, all while writing a take-away and never look back. This alone was very enlightening, there were many subtle issues that at the level of detail that I had read until then, blurred together and looked like one simple thing, I like to compare this with music (being an ex-quarter-musician). When you listen to Progressive Metal, Jazz, Blues the first time, and you listen to one second, then the second, you feel that they are the same thing ... it's all the same, and this is true, they are, and then fun is in the differences and nuances, which are more detailed and intricate than in other styles, in blues you have a small selection of rhythmic and chord patterns that it looks extremely limiting at first sight, it turns out, there is such a huge richness to express thanks to this limitation, it sets the context and allows you to play with the rest.

In a way I also find this related to trying to write solutions that are extremely general, I like generalizations, I like unified theories of everything (I hope the theoretical physicists will find a unifying theory that will allow us to do now cool practical stuff before I seize to exist, as an alternative, I hope they scientists find a way to escape immortality, another alternative is vampires really existing and after reading my wish here visiting me), I like being 'lazy' and writing code that will be reused by me and others in many different contexts without needing adapt it, but it turns out, in practice KISS is the way to go, just like Blues setting the limits, it is very important to set your own limits when choosing your problem, per example when designing your next AI, physics, graphics or gameplay technology. It is not easy, because what ends up limiting you are your computational resources and maybe the talent of your team, both are difficult if not impossible to measure for a designer.

So, I decided I wanted to give the book the full detail treatment, all with a take away that I am sharing online at: http://docs.google.com/View?id=dcm3hb4r_19gkzkdbdr.

So I started reading and thinking iteratively, taking breaks to think away from my desk to let things solidify. All looked good until I reached http://www.cs.ualberta.ca/~sutton/book/ebook/node34.html, Equation 3.10 line 3 to 4. Intuitively and logically, this made complete sense, but I felt the way this was written was trying to tell me that there is rigorous math derivation at work here, I asked 2 people by email about this, those were people I had met by accident on twitter, one of them very knowledgeable about Math, the other about AI, I also wrote at the AiGameDev forums, that was yesterday, I received a few replies, but I was not satisfied, eventually Alex took the time to discuss it with me and we agreed that there was no rigouros math involved, it was an mathematical expression of the backup diagram.

One additional help was the way this was presented in "http://paginas.fe.up.pt/~eol/schaefer/diplom/ReinforcementLearning.htm" saying:

This was our conclusion although I was expecting I would find a step by step expansion based on Mathematical rules and RL definitions, maybe using the rules of Linearity in Expectation and Iterated Expectation, applied to the RL definitions of environment models.

Rigorousness, a waste of time

In general I am very skeptical in accepting things without asking lots of 'why's.

I also have been getting a bit into rigorous math, because I see this as one of the current barrier standing between mean and world domination (ooops now everybody knows). I am trying to put a bit of time into improving and updating my math skills because it is a fact, that some academic papers or writing, no matter how useful like to use cryptic (for me currently) equations which could be expressed in a much nicer way in English. This is a long topic by itself, some months ago I was wondering that in order to prove something rigorously, you need to base it on another thing, but obviously, I thought, it must start somewhere, I set myself to search for that something in math, I found something probably obvious to all mathematicians, you need axioms to start with, you don't prove axioms. Now I know this from school, but I never thought of it this way, this journey took me many places mostly on wikipedia ending at the "There Ain't No Such Thing As A Free Lunch" theorem and it's sibling 'Full employment' which I thought was quite amazing and increased my new found love to going back to Math, one can actually logicall prove that computer scienetists will never run out of jobs, well done Math!

Enlightenment, I am not alone

Alex was of the opinion that I am taking it too far again and that this is not useful, but I was able to partially convince of the usefulness because it would make many more papers and academic writing accessible to me. Anyway, I kept digging, this time into the base of the equations I wanted to detail, the Bellman Equations, which led me to this document: http://www.wu.ac.at/usr/h99c/h9951826/bellman_dynprog.pdf, this is the main topic of this post, why? because this genius named Richard Bellman, who writes "At Stanford I had a chance to do analytic number theory, which I had wanted to do since I was sixteen." !! touched on many of the topics I am worried about and that I described here and constantly trying to understand better, it was extremely enlightening, Level up, here are some quotes and their relation to this post:

A lot of work and time goes into what becomes a one line 'obvious' equation, it was not always obvious! not even to the person that came up with it, treating it as obvious is almost a crime. However I also understand that there is the danger of tying one's legs to the chair if one wants to do this for every tiny bit of theory, not getting any new progress done is a crime as well, balance is key, as usual. This exact dilemma is what started me on today's enlightenment Journey, and again Bellman touches on that :), using a writing style 100x superior to mine of course, but the idea is there ...

Conclusion

Level up.

.

Questionable game animation technology research

Working on the AIGameDev animation system with Alex Champandard, we reached a rather experimental (for now) stage of wanting to use Reinforcement learning to learn the heuristic of an A* planner used for planning locomotion based on a given, non-annotated, step based, automatically generated motion graph.

The motivation of doing this might be questionable, but the whole thing developed step by step.

In summary we had the motion graph builders, written by Alex, he also had the brilliant idea of making them step based, using very sensible points in time to make animation transitions, we had the generic A* code I have written, we wanted to combine the two to have something moving on the screen, showing the usefulness of this foot-skate free motion graph approach.

This gave birth to the A* motion planner, A planner that is not very suitable for multiple characters in real time, a planner that only needs a motion graph, and no extra code in form of hand written controllers, that gets you from A to B in a fast, high transition quality way.

It was not very good at reaching the destination with an orientation constraint, it was possible, but it made the heuristic too complicated, almost like writing a manual controller, and much

slower. In theory, this should be very easy, or at least easily learnable.

Reinforcement Learning

There are papers that use RL for motion planning, pretty recent ones (starting 2007), like "Real-Time Planning for Parametrized Human Motion" or "Near Optimal Character Animation with Continuous Control", I mention those two because of I have them printed on my running-out-of-space desk, along with countless "Gleicher"/"Kovar" papers about animation and motion graphs. It was Alex who drove me into all of this, so any straying is both his fault, and also his merit, I am grateful, I have long been searching for something interesting, some of my previous jobs failed big there.

The idea was to learn the heuristic, making the planner suitable for real time, this might be questionable for some people, the industry mostly drives it's animations, no need for planning at all, simple, effective, KISS, and full of foot-skate.

At the AiGameDev conference in Paris this month, there was one panel where we discussed the uncanny valley, the need or no-need to cross it, I also discussed this afterwards with Chrisitiaan Moleman and Markus Mohr, Remco Straatman and others, I mention those because I was extremely pleasantly surprised about how passionate and nice those people and everybody else who was at the conference are, I am not getting tired of repeating this because it was remarkable.

So to go back, the motivation, is by all means questionable, except if we decide that we want to see how a next step in animation would look like, I am not saying we should be planning the whole time, there will be times for driving the animations and times for planning, but I'm keeping this under wraps for the moment, although it's not very difficult to imagine what I mean.

Applying Reinforcement Learning

So all of this got me into Reinforcement Learning, a classical reference is the Sutton/Barto book "Reinforcement Learning, an Introduction" available online for free. I also bought a couple of related books. I read the parts of online book I needed the 1st time and designed an RL approach to learning what we needed to learn to help the A* heuristic, the result was that it basically 'worked' but was a naive first take that needed refinement to be really useful, the details will be available to look at in the AiGamDev sandbox at some point. At this stage I had read parts of the online book some in more detailed than other, and I would say I had revisited some parts 3 or 4 times. I was at the stage where I understood what we need, make a first attempt, understood the problem much better because of it, and was ready to design the second take, which needed a technique that is a bit more evolved that the 1st technique I had used.

Reading in detail and understanding the Blues

I decided, that I do not feel like coming back to the basics anymore, and to read the full book in maximum detail, all while writing a take-away and never look back. This alone was very enlightening, there were many subtle issues that at the level of detail that I had read until then, blurred together and looked like one simple thing, I like to compare this with music (being an ex-quarter-musician). When you listen to Progressive Metal, Jazz, Blues the first time, and you listen to one second, then the second, you feel that they are the same thing ... it's all the same, and this is true, they are, and then fun is in the differences and nuances, which are more detailed and intricate than in other styles, in blues you have a small selection of rhythmic and chord patterns that it looks extremely limiting at first sight, it turns out, there is such a huge richness to express thanks to this limitation, it sets the context and allows you to play with the rest.

In a way I also find this related to trying to write solutions that are extremely general, I like generalizations, I like unified theories of everything (I hope the theoretical physicists will find a unifying theory that will allow us to do now cool practical stuff before I seize to exist, as an alternative, I hope they scientists find a way to escape immortality, another alternative is vampires really existing and after reading my wish here visiting me), I like being 'lazy' and writing code that will be reused by me and others in many different contexts without needing adapt it, but it turns out, in practice KISS is the way to go, just like Blues setting the limits, it is very important to set your own limits when choosing your problem, per example when designing your next AI, physics, graphics or gameplay technology. It is not easy, because what ends up limiting you are your computational resources and maybe the talent of your team, both are difficult if not impossible to measure for a designer.

So, I decided I wanted to give the book the full detail treatment, all with a take away that I am sharing online at: http://docs.google.com/View?id=dcm3hb4r_19gkzkdbdr.

So I started reading and thinking iteratively, taking breaks to think away from my desk to let things solidify. All looked good until I reached http://www.cs.ualberta.ca/~sutton/book/ebook/node34.html, Equation 3.10 line 3 to 4. Intuitively and logically, this made complete sense, but I felt the way this was written was trying to tell me that there is rigorous math derivation at work here, I asked 2 people by email about this, those were people I had met by accident on twitter, one of them very knowledgeable about Math, the other about AI, I also wrote at the AiGameDev forums, that was yesterday, I received a few replies, but I was not satisfied, eventually Alex took the time to discuss it with me and we agreed that there was no rigouros math involved, it was an mathematical expression of the backup diagram.

One additional help was the way this was presented in "http://paginas.fe.up.pt/~eol/schaefer/diplom/ReinforcementLearning.htm" saying:

The diagram shows that when initially in state s action a is selected, the successor state is s1 and reward r1 is expected but also r2 is expected passing to s2. If in state s action b is chosen, reward r3 is expected and leads to s1 but also reward r4 is expected and leads to state s2.

This diagram can be described by the following Bellman-Equation: ...

This was our conclusion although I was expecting I would find a step by step expansion based on Mathematical rules and RL definitions, maybe using the rules of Linearity in Expectation and Iterated Expectation, applied to the RL definitions of environment models.

Rigorousness, a waste of time

In general I am very skeptical in accepting things without asking lots of 'why's.

I also have been getting a bit into rigorous math, because I see this as one of the current barrier standing between mean and world domination (ooops now everybody knows). I am trying to put a bit of time into improving and updating my math skills because it is a fact, that some academic papers or writing, no matter how useful like to use cryptic (for me currently) equations which could be expressed in a much nicer way in English. This is a long topic by itself, some months ago I was wondering that in order to prove something rigorously, you need to base it on another thing, but obviously, I thought, it must start somewhere, I set myself to search for that something in math, I found something probably obvious to all mathematicians, you need axioms to start with, you don't prove axioms. Now I know this from school, but I never thought of it this way, this journey took me many places mostly on wikipedia ending at the "There Ain't No Such Thing As A Free Lunch" theorem and it's sibling 'Full employment' which I thought was quite amazing and increased my new found love to going back to Math, one can actually logicall prove that computer scienetists will never run out of jobs, well done Math!

Enlightenment, I am not alone

Alex was of the opinion that I am taking it too far again and that this is not useful, but I was able to partially convince of the usefulness because it would make many more papers and academic writing accessible to me. Anyway, I kept digging, this time into the base of the equations I wanted to detail, the Bellman Equations, which led me to this document: http://www.wu.ac.at/usr/h99c/h9951826/bellman_dynprog.pdf, this is the main topic of this post, why? because this genius named Richard Bellman, who writes "At Stanford I had a chance to do analytic number theory, which I had wanted to do since I was sixteen." !! touched on many of the topics I am worried about and that I described here and constantly trying to understand better, it was extremely enlightening, Level up, here are some quotes and their relation to this post:

“An interesting question is, ‘Where did the name,This is related to the desire of people to research things, and that in the end come up with useful things because of that, despite it seeming pointless to the 'management'. Is it pointless looking into RL for locomotion?

dynamic programming, come from?’ The 1950s were not

good years for mathematical research. We had a very interesting

gentleman in Washington named Wilson. He was

Secretary of Defense, and he actually had a pathological

fear and hatred of the word, research. I’m not using the

term lightly; I’m using it precisely. His face would suffuse,

he would turn red, and he would get violent if people used

the term, research, in his presence."

"Let’s take a word that has anThere are many ways to present the same idea, it is ok to choose the one that fits the target. This is not directly related and not new, but it shows that even something as cool as RL needed this to start off.

absolutely precise meaning, namely dynamic, in the classical

physical sense. It also has a very interesting property

as an adjective, and that is it’s impossible to use the word,

dynamic, in a pejorative sense. Try thinking of some combination

that will possibly give it a pejorative meaning.

It’s impossible. Thus, I thought dynamic programming was

a good name. It was something not even a Congressman

could object to. So I used it as an umbrella for my activities”

“I could either be a traditional intellectual, or a modernThat's also one of my ongoing concerns that no one else seems to worry about, finding the right balance, and even Bellman used to think about it.

intellectual using the results of my research for the

problems of contemporary society. This was a dangerous

path. Either I could do too much research and too little

application, or too little research and too much application."

“My first task in dynamic programming was to put it onAha, so that's where it all comes from! intuition! and not equation 3.10 with cryptic expansion steps. Relieving...

a rigorous basis. I found that I was using the same technique

over and over again to derive a functional equation.

I decided to call this technique “The principle of optimality.”

Oliver Gross said one day, ‘The principle is not rigorous.’

I replied, ‘Of course not. It’s not even precise.’ A good

principle should guide the intuition."

"This isCoding-wize, I tended to be on the 'ties a chair to his legs' type but that was long time ago, and it actually is useful to be in this state for a limited amount of time, checking out the possible extremes is always good, even Buddha checked the extremes before finding the golden middle, it is only logical, how can you know where the middle is if you have never seen where the extremes are! I am therefore happy I have been at both extremes and have developed a good eye for the Golden middle, not only in code.

pertinent to a comment made by Felix Klein, the great

German mathematician, concerning a certain type of mathematician.

When this individual discovers that he can jump

across a stream, he returns to the other side, ties a chair

to his leg, and sees if he can still jump across the stream.

Some may enjoy this sport; others, like myself, may feel

that it is more fun to see if you can jump across bigger

streams, or at least different ones"

“What is worth noting about the foregoing developmentI found this one great as well, equations are thrown at us in lectures, in papers, in tutorials, and we are supposed to just say yes it makes sense, this is usually not enough for me, I prefer to be able to get into the context which allowed the 'inventor' to come up with those ideas, based on what he knew, the problem he faced, and how he thought to come up with what he came up with.

is that I should have seen the application of dynamic programming

to control theory several years before. I should

have, but I didn’t. It is very well to start a lecture by saying,

‘Clearly, a control process can be regarded as a multistage

decision process in which... ,’ but it is a bit misleading.

Scientific developments can always be made logical and

rational with sufficient hindsight. It is amazing, however,

how clouded the crystal ball looks beforehand. We all wear

such intellectual blinders and make such inexplicable blunders

that it is amazing that any progress is made at all."

A lot of work and time goes into what becomes a one line 'obvious' equation, it was not always obvious! not even to the person that came up with it, treating it as obvious is almost a crime. However I also understand that there is the danger of tying one's legs to the chair if one wants to do this for every tiny bit of theory, not getting any new progress done is a crime as well, balance is key, as usual. This exact dilemma is what started me on today's enlightenment Journey, and again Bellman touches on that :), using a writing style 100x superior to mine of course, but the idea is there ...

"All this contributes to the misleading nature of conventional

history, whether it be analysis of a scientific discovery

or of a political movement. We are always looking at

the situation from the wrong side, when events have already

been frozen in time. Since we know what happened, it is

not too difficult to present convincing arguments to justify a

particular course of events. None of these analyses must be

taken too seriously, no more than Monday morning quarterbacking."

Conclusion

Level up.

.

## Sunday, May 24, 2009

### Sandbox Release #8: Collision Detection, Running/Racing Demo

Release #8 is here.

This time it was about physics integration for LOS, collision detection and distance queries, a bit of tactical cover analysis, and mostly as usual, animation / motion planning.

I made many improvements to the motion planner and refactored it into components compatible with our MVC framework.

We did not start working with parametric animations yet, this means that the motion planner cannot reach every possible point in space without some error correction or replanning to alternative locations.

Both error correction and robust plan failure handling and alternative planning were therefore built into two components: Locomotion and Navigation.

Additionally a form of collision avoidance was added, all that was integrated into a small racing prototype.

The new release also includes other improvements and features, for the full details, head to http://aigamedev.com/premium/releases/sandbox-v8-running-game/

## Sunday, May 17, 2009

### johndcook: RT @aycangulez: How do startups morph from agile, can-do companies to ones that have lost their edge? A: New building. http://bit.ly/10N459

http://steveblank.com/2009/05/15/supermac-war-story-11-the-curse-of-a-new-building/

"That’s when things went south.

The thing I like about this is that I think it also applies to individuals! so watch out :)

.

"That’s when things went south.

**Lets Fix Everything that Was Broken**"The thing I like about this is that I think it also applies to individuals! so watch out :)

.

## Saturday, May 16, 2009

### Board game (Keltis) AI micro post-mortem.

Keltis

I finally got a copy of Keltis PC, I had to learn the game, design and write the AI all in just 6 weeks, obviously, not enough time for polishing. Additionally, Keltis itself is pretty tricky and apart from it being an incomplete information game, it is also based on a delicate balance of risk vs. reward, short-term vs. long-term gameplay, which is what it's inventor Dr. Reiner Knizia's games are famous for.

AI design

Still, I managed a well playing AI, I based it on a careful mix of statistics, heuristics, board evaluation, opponent modeling and planning.

The heuristics were useful for the start and middle phases where planning is difficult and way too many options are possible. In contrast, the AI's planning abilities become more and more important as the game approaches it's (difficult to time and predict) end phase and allows it to take risks, catch you off guard and snatch the victory in front of your nose.

The AI took both 'approaches' into consideration during the whole game ensuring no strange changes in strategy and smooth purposeful decisions.

The opponent modeling component was used to predict the end phase of the game, decide what cards to discard or not, when to take risks. This was one part I really wished I had more time to work on, it would have definitely improved the AI's quality.

Probabilistic model

The probabilistic model for this game happens to be a negative hypergeometric distribution however I started prototyping with a much simpler model, which turned out to be very inaccurate for a range of inputs, because of that I resorted to using the precise model, for that I unfortunately needed a so called big-number library because the calculations involved huge factorials which are then divided for the final 0.0-1.0 range probabilities.

After trying some of them I settled on TTMath because of its small size and because it fit my needs very well. However when it came to porting to NDS and mobile phones, this turned out to be a problem because the library used x86 assembly to do some of it's work, in the short time I had, the best thing I could do was to reimplement those features in c++.

Planning with probability

To plan based on probabilities, The planner would choose a fixed probability, per example 0.7, with that a precise non-probabilistic plan gets built based on the probabilistic model. This plan can then be evaluated and scored.

In the case of Keltis this envolves many things including planning what pieces to start or not (which depends on again different estimations including opponent modeling), how to use the bonus points (which are crucial to playing well) to state a few.

It might turn out per example that such a plan will not get the AI enough points to win the game if the player is already ahead, in that case the AI can choose to play with more risk and recursively test plans with lower probabilities until one that has a good risk/reward ratio is found.

Playing it risky in this case means hoping to get cards that have low probability. This was the approach I used to adaptively choose the AI's risk readiness based on the situation (comfortably winning, close game, hopeless ....)

Lessons learned

In retrospect improving the simple probabilistic model I had for better accuracy on the problematic input range, or finding an simpler approximate model would have probably been a much better idea and would have saved me some time that I could have used for improving the AI.

Another thing I learned is that if time is very short, you can go a long way with heuristics, and cover the rest with special case code... Andreas Epple my ex-colleague senior programmer helped me get myself to actually accept going the special-case way to bend the last problems before shipping, it worked much better than I expected.

## Monday, May 11, 2009

### A new (and cool) AIGamedev video + article, showing some of my work

Hard-Earned Insights from The AI Sandbox Development

You can see here some of the work I previously have not mentioned here including Hierarchical Pathfinding, terrain analysis, skeleton mirroring.

## Thursday, May 7, 2009

### The most AMAZINGLY detailed pac-man article I have ever seen

I must admit I did not read the whole thing but man!!! just have a look at it, I am sure it is more detailed than some AI designs for more 'complex' games

http://home.comcast.net/~jpittman2/pacman/pacmandossier.html

As an additional bonus, it seems that the author, Jamey Pittman, also suffers Game Developer/Guitarist schizophrenia just like me... (bottom of the article)

(Thanks to http://twitter.com/JurieOnGames for twittering the link)

### 'In Defense of Eye Candy' Article, a must read.

Please choose one button to start your search:

The nice but still professional-looking one on the right? choose an AI SDK for your next game, choose your next car, choose your next employee based on his portfolio... I'm afraid aesthetics will always play a role in all of this, no matter how hard we try to be objective.

My personal way of looking at things since a couple of years is that feelings are objective as well, they are the objectivity of your subconscious, whenever you get overwhelmed and your logic circuits overflow, fall back to the feelings, and you know what? it's absolutely the right thing to do, but I digress...

http://www.alistapart.com/articles/indefenseofeyecandy/ is about the importance of aesthetics, it was written on listapart.com, a web design site (I think). It is an excellent read, throw away your coder art, there are good free models out there!

This reminds off the GDC AI demos (which I watched on the AIGamedev GDC report videos), some companies had great looking assets, while others had wireframe blocky shapes, although this does not tell much about AI, psychology again played it's role in making an opinion about them. Now of course, it might not totally fool one, but it will still tend to tip the balance in the depths of your brain, this article explains why.

The nice but still professional-looking one on the right? choose an AI SDK for your next game, choose your next car, choose your next employee based on his portfolio... I'm afraid aesthetics will always play a role in all of this, no matter how hard we try to be objective.

My personal way of looking at things since a couple of years is that feelings are objective as well, they are the objectivity of your subconscious, whenever you get overwhelmed and your logic circuits overflow, fall back to the feelings, and you know what? it's absolutely the right thing to do, but I digress...

http://www.alistapart.com/articles/indefenseofeyecandy/ is about the importance of aesthetics, it was written on listapart.com, a web design site (I think). It is an excellent read, throw away your coder art, there are good free models out there!

This reminds off the GDC AI demos (which I watched on the AIGamedev GDC report videos), some companies had great looking assets, while others had wireframe blocky shapes, although this does not tell much about AI, psychology again played it's role in making an opinion about them. Now of course, it might not totally fool one, but it will still tend to tip the balance in the depths of your brain, this article explains why.

### Framework1 / Quake3 level Bezier patches

Today I got the Bezier patches working in the renderer.

Quake3 uses the simplest patches possible, bi-quadratic, which means 9 control points per patch.

A face in Quake3 can be composed of multiple patches arranged in a grid, when this is the case, each patch shares a line of 3 control points with it's horizontal and vertical neighbor.

To render such a face, all what needs to be taken care of is to correctly setup and share the control points, the resulting tesselated vertices however, are not shared and also not stitched together. The fact that they are not shared is obvious since they will tesselate to different vertices, not having to stitch them together however (based on the references mentioned in previous articles), sounded a bit strange to me, but it seems this really is the case, since the renders look ok.

I implemented two version of the Bezier tesselator, a simple one (as a reference for debugging and testing) that makes a mesh directly out of the control points, and another real tesselator.

The whole implementation took roughly 8 hours of work.

As usual, I took some screenshots, some are quite dark (despite a bit of gimp post-processing), but if you download the image you can see sufficient detail.

I grouped all the shots in one image, the ones the left show the 'reference' control point renderings, while the ones on the right show a tesselation of level 5 (6x6=36 vertices per patch).

In some places, small artifacts due to texture coordinates are visible, (the arched gate), I checked the level with another Quake3 bsp renderer and I observed the same artifacts, so this might be a level design problem, in any case it is not a big deal.

I really got into C# generics to make the tesselator be able to tesselate any kinds of vertices without needing to know their structure, I did this using Generics with 'where' constraints which was needed to be able to weight and add vertices together using functions of a required interface.

I really liked that, specially that implementing an interface is very cheap and does not mean that the functions implemented are 'virtual' and therefore less performant that non-virtual functions. It is simply a 'compile-time' constraint, nice!

This might be premature optimization, but since the whole design is based on streaming, all loading operations (such as loading a face that just became visible and tesselating in on the fly) better be fast enough.

## Sunday, May 3, 2009

### Article related to my work on Locomotion Planning

Alex Champandard just released the article 'Motion Planning for Fun and Profit!' which features some of my work with him at AIGamedev.com.

It is an insider article, so if you do not have an insider membership yet (which is free), go and get one, it is priceless for anyone interested in game AI.

It is an insider article, so if you do not have an insider membership yet (which is free), go and get one, it is priceless for anyone interested in game AI.

## Monday, April 27, 2009

## Sunday, April 26, 2009

### Framework1 / Quake3 level lightmaps

It is sunday, instead of doing something fun (which I am regretting now that I see that it's 23:41 ...) I added lightmap support to the Quake3 rendering part of the framework. 20 years too late, but still, it looks kinda cool...

Can't wait to halfway finish the framework to start using it for 'not old hat' graphics related experiments.

Lightmapped:

Detail texture only:

Lightmap texture only:

Detailt+Lightmap textures:

The Lightmaps of the level (all 128x128):

Can't wait to halfway finish the framework to start using it for 'not old hat' graphics related experiments.

Lightmapped:

Detail texture only:

Lightmap texture only:

Detailt+Lightmap textures:

The Lightmaps of the level (all 128x128):

## Friday, April 24, 2009

### Rendering Framework1 first video, Streaming Quake3

Aside from working on Animation and AI related things at AIGamedev.com, which by the way just went live with a brand new website :), I am slowly working on an XNA rendering Framework.

I figured it would be a good and time efficient idea to write a Quake3 map loader this time, since much content is directly available, All the nice geometry can then be used as a basis for graphics experiments.

This is the first version that successfully loads Quake3 levels, it took approximately 24 hours, it uses only colored vertices, but it already does something cool, leafs and their faces are streamed as needed and not pre-loaded, and the streaming usese Generics and Reflection and is Vertex format agnostic, of course streaming maps of this size (~100k vertices) does not really make sense since they could all fit nicely in one very small vertex/index buffer, but still, one of the goals of the framework is to focus on streaming and parallelization (think submission engine), so the code is being designed around that, with the Quake3 levels providing the data.

More technical details and source code will come as appropriate, for now, have fun with the (not very exciting colored vertices) video, at least it shows how only the pre-baked radiosity in vertex colors is already enough to create a sense of nice lighting.

References I used for the bsp format and loading:

I figured it would be a good and time efficient idea to write a Quake3 map loader this time, since much content is directly available, All the nice geometry can then be used as a basis for graphics experiments.

This is the first version that successfully loads Quake3 levels, it took approximately 24 hours, it uses only colored vertices, but it already does something cool, leafs and their faces are streamed as needed and not pre-loaded, and the streaming usese Generics and Reflection and is Vertex format agnostic, of course streaming maps of this size (~100k vertices) does not really make sense since they could all fit nicely in one very small vertex/index buffer, but still, one of the goals of the framework is to focus on streaming and parallelization (think submission engine), so the code is being designed around that, with the Quake3 levels providing the data.

More technical details and source code will come as appropriate, for now, have fun with the (not very exciting colored vertices) video, at least it shows how only the pre-baked radiosity in vertex colors is already enough to create a sense of nice lighting.

References I used for the bsp format and loading:

- http://graphics.cs.brown.edu/games/quake/quake3.html
- http://graphics.stanford.edu/~kekoa/q3/
- http://code.google.com/p/bsp-renderer/

## Thursday, April 23, 2009

### 3D Transform-Matrix multiplication reference

There is a known and well documented confusion about the order of matrix multiplication, row major, column major, c++ memory layout....

In general you just need to pay a bit of attention to detail to get it under control, once for each new library or SDK or combination thereof you get to deal with.

I dug up a small reference document I wrote some time ago, and added to it how XNA handles matrices since I am doing a bit of XNA development.

It is not complete so if you would like to add to it I will gladly send you the OpenOffice original.

If you find I wrote anything wrong please let me know.

In general you just need to pay a bit of attention to detail to get it under control, once for each new library or SDK or combination thereof you get to deal with.

I dug up a small reference document I wrote some time ago, and added to it how XNA handles matrices since I am doing a bit of XNA development.

It is not complete so if you would like to add to it I will gladly send you the OpenOffice original.

If you find I wrote anything wrong please let me know.

## Saturday, April 18, 2009

### Rocket science Tic-Tac-Toe testbed

It has been some time since the last post,

I have been busy working on the AIGamedev.com sandbox, some of the work was related to A*, Hierarchical A*, terrain clustering, pathfinding, Animation mirroring, Locomotion planning ...

I also had the time to start my 'own private course' in academic Machine Learning thanks to the many good online resources and video lectures.

Remotely related, and since I decided to share much of the things I do, no matter how simple or meaningless I might find them, here is a Tic-Tac-Toe AI testbed, which started as an interview 3 hour coding test, but then received some fun additions, namely an Alpha-beta minimax AI, which is adapted from Steve Chapel's code and a Reinforcement learning value iteration AI (my own 1 hour exercise to apply one of the very basic machine learning techniques)

http://jadnohra.net/release/TicTacToeTestbed_rev28.zip

I have been busy working on the AIGamedev.com sandbox, some of the work was related to A*, Hierarchical A*, terrain clustering, pathfinding, Animation mirroring, Locomotion planning ...

I also had the time to start my 'own private course' in academic Machine Learning thanks to the many good online resources and video lectures.

Remotely related, and since I decided to share much of the things I do, no matter how simple or meaningless I might find them, here is a Tic-Tac-Toe AI testbed, which started as an interview 3 hour coding test, but then received some fun additions, namely an Alpha-beta minimax AI, which is adapted from Steve Chapel's code and a Reinforcement learning value iteration AI (my own 1 hour exercise to apply one of the very basic machine learning techniques)

http://jadnohra.net/release/TicTacToeTestbed_rev28.zip

## Friday, March 20, 2009

### Writing modular software / There is no Perfection only Beauty

I have been working for a while now on the AIGamedev.net sandbox.

For the sandbox, we decided to put extra weight on the 'modular' constraint. This of course puts it in conflict with other constraints I will try to explain why and how.

I like to base this on a nice and short but deep list summarizing good software design.

Taking modularity as our main constraint, we can explore the depth of the statement: Create a balanced distribution of responsibilities. We quickly notice that there is a catch here, and therefore depth. The depth comes from the world 'balanced'. It is pretty vague, and that is how it should be.

I remember dealing with exactly that problem of balancing during coding. At some point the code was so 'modular' that every single class almost did not do anything on it's own.

It was very modular but it was a strange feeling, and it also has its disadvantages like any other constraint, and balancing those conflicts is what it practically comes to, sometimes it is possible, but most of the times it is not.

To give an example we can take any physical piece of electronic equipment. Let us take a server as an example, or even better, a server rack.

In a server rack it is nice to think of one server as a module, and the first thing we see is ... the problem ... a big mess of cables, even if organized ... it is still pretty messy, but there is no way around it. This means, the higher the modularity the more cables, and small extra things to hold those cables nicely bundled and manageable.

In software this would mean taking those modules and using groups of them together to let them do something useful, this is where you have many options as well and flavors thereof: composition, aggregation, component based, virtual inheritance, delegates, functors... Each with it's own advantages and disadvantages.

Let us go one level deeper and examine the insides of one server, how modular is it? It is modular, we can simply see that from the fact that it has cables and removable components inside.

But we could ask for more, we could ask for the motherboard itself be more modular, by demanding to be able to replace The chipset chip, the secondary cache chips or for a graphics card we might to require the ability to replace the RAM easily. But why are these parts not modular. Of course, because of other constraints: performance, cost ...

In software it might be a bit more difficult to judge the limit where modular should stop, eventually, every single statement that is repeated somewhere in the code could be 'modularized' but then the amount of screws, pipes, cables and glue will become larger and much more complicated than the code that actually does something.

It is a tricky balance, and an important one. There is no recipe for it, but there are hints we get from the modules themselves ... does a module need too many cables (function parameters per example) to do something useful? Do I need 10 modules (classes) and a similar amount of pipes(each module having a pointer to some other module) to achieve the smallest module that actually achieves something on the level of detail I am working with? If yes, then it is time to redesign? The price can sometimes be duplication, but we have to learn to accept duplication sometimes.

We can see from all of this that something that is modular and nice from one point of view (or level of detail) might become a pain to work with from another one, it is all really relative, and there is no one design that can look good at all levels of detail, I am convinced of that.

The more levels of detail we want to cover, the more general we want to be, the more we will loose the ability to be optimal for only one task or for only one level of detail, so optimization for the task at hand, the level of detail at hand, will always be inevitable, unless we have infinite resources, which I do not see coming in the next few infinitely many years, So remember: There is no perfection, only beauty. Or: only strive for perfection up to the visible horizon (beauty).

For the sandbox, we decided to put extra weight on the 'modular' constraint. This of course puts it in conflict with other constraints I will try to explain why and how.

I like to base this on a nice and short but deep list summarizing good software design.

- Create crisp and resilient abstractions.
- Maintain a good separation of concerns.
- Create a balanced distribution of responsibilities.
- Focus on simplicity.

Taking modularity as our main constraint, we can explore the depth of the statement: Create a balanced distribution of responsibilities. We quickly notice that there is a catch here, and therefore depth. The depth comes from the world 'balanced'. It is pretty vague, and that is how it should be.

I remember dealing with exactly that problem of balancing during coding. At some point the code was so 'modular' that every single class almost did not do anything on it's own.

It was very modular but it was a strange feeling, and it also has its disadvantages like any other constraint, and balancing those conflicts is what it practically comes to, sometimes it is possible, but most of the times it is not.

To give an example we can take any physical piece of electronic equipment. Let us take a server as an example, or even better, a server rack.

In a server rack it is nice to think of one server as a module, and the first thing we see is ... the problem ... a big mess of cables, even if organized ... it is still pretty messy, but there is no way around it. This means, the higher the modularity the more cables, and small extra things to hold those cables nicely bundled and manageable.

In software this would mean taking those modules and using groups of them together to let them do something useful, this is where you have many options as well and flavors thereof: composition, aggregation, component based, virtual inheritance, delegates, functors... Each with it's own advantages and disadvantages.

Let us go one level deeper and examine the insides of one server, how modular is it? It is modular, we can simply see that from the fact that it has cables and removable components inside.

But we could ask for more, we could ask for the motherboard itself be more modular, by demanding to be able to replace The chipset chip, the secondary cache chips or for a graphics card we might to require the ability to replace the RAM easily. But why are these parts not modular. Of course, because of other constraints: performance, cost ...

In software it might be a bit more difficult to judge the limit where modular should stop, eventually, every single statement that is repeated somewhere in the code could be 'modularized' but then the amount of screws, pipes, cables and glue will become larger and much more complicated than the code that actually does something.

It is a tricky balance, and an important one. There is no recipe for it, but there are hints we get from the modules themselves ... does a module need too many cables (function parameters per example) to do something useful? Do I need 10 modules (classes) and a similar amount of pipes(each module having a pointer to some other module) to achieve the smallest module that actually achieves something on the level of detail I am working with? If yes, then it is time to redesign? The price can sometimes be duplication, but we have to learn to accept duplication sometimes.

We can see from all of this that something that is modular and nice from one point of view (or level of detail) might become a pain to work with from another one, it is all really relative, and there is no one design that can look good at all levels of detail, I am convinced of that.

The more levels of detail we want to cover, the more general we want to be, the more we will loose the ability to be optimal for only one task or for only one level of detail, so optimization for the task at hand, the level of detail at hand, will always be inevitable, unless we have infinite resources, which I do not see coming in the next few infinitely many years, So remember: There is no perfection, only beauty. Or: only strive for perfection up to the visible horizon (beauty).

## Tuesday, January 27, 2009

### Beam me up scottie, update...

One small step for a man, one giant leap for teleportation:

http://news.cnet.com/8301-17938_105-10150272-1.html

http://news.cnet.com/8301-17938_105-10150272-1.html

## Friday, January 16, 2009

### You gotta read this one!! trust me...

Our world may be a giant hologram

http://www.newscientist.com/article/mg20126911.300-our-world-may-be-a-giant-hologram.html?full=true&print=true

I also found this fun thing:

http://www.einstein-online.info/en/index.html

http://www.newscientist.com/article/mg20126911.300-our-world-may-be-a-giant-hologram.html?full=true&print=true

I also found this fun thing:

http://www.einstein-online.info/en/index.html

Subscribe to:
Posts (Atom)