Tuesday, August 12, 2008

It's all so easy... (Brain stack dump)

For some time now, I have the feeling that all problems seem to carry no difficulty for me.
So I took some time to think about to find out if it's an illusion, a realtiy, or pure naive stupidity.

What I will begin with is not the first idea link in the chain of thoughts that I went through while thinking about this, but I still found it would be good to start with.

Of course, as with all topics, it is important to first define the used terms.
one dictionary definition of 'difficult' is: "not easy to do, understand, or solve". this definition uses the word easy, so this does not really help much, since I am searching for a deeper definition of difficult.
Looking up the definition for easy I get: "not difficult; simple", which is of course, also not helpful, since it uses the word 'difficult'. Instead of looking up simple (to probably find it using 'difficult' or 'easy'), I stop here and decide that the problem is really worth investigating!

So let us finally start: do you think the question 'what is the result of 1 + 1' is a difficult question for a 2 year old? yes? I think no...
I think there is a clear method to tackle any problem, and that there is no such thing as a difficult problem. The method is: identify the problem, search for a known method to solve the problem, if non is found, search for a way to solve the problem.
Pffff you say, we all know that and it does not make any problems less difficult. Sure it does.
Going back to the '1 + 1' question, let us apply our method. 1st "Identify the problem", ok the problem consits or the parts '1' '+' and the answer we are thinking of is '2'. Can we explain to a 1 year old what '1' is? no. so the problem is not difficult at all, we cannot even 'feed' the problem into the 1 year old, who's brain does not have yet any way of internally representing the 1st symbol '1', so the problem is non existent from the point of view of the 1 year old, but not 'difficult'.
Now what about asking the same question to a 6 year old who is good at math. easy u say? I disagree as well. when we ask what is '1 + 1' we get the answer '2' yes, but if we ask why ... we are stuck. the 6 year old did not solve the problem, he just looked up the answer in his memory, or counted on his fingers, but I do not consider the problem solved. To really solve the problem let us ask ourselves why does 1 + 1 = 2. Stupid question? not at all, 1 + 1 could be 10 like we see printed on popular nerd T-shirts, this would be the case if we are talking binary. 1 + 1 could also be 0, this would be if we are asking how many apples is 1 orange + 1 banana. So again, using our method, we 1st have to identify the problem. what does 1 + 1 actually mean, before we know that, it is useless to start solving. so, painful but necessary, what is '1' ? 1 could be anything, but abstracted it is a number, this is still not enough, what is a number? ... it is also a number using the decimal system, not binary, not hexadecimal. so 1 is one unit of 'anything', and also very important when we say 1+1 we do not many 1 of anything + 1 of anything, no, we mean 1 of anything + 1 of the same anything. of course we also have to define + (for which I do not have a clear definition without using a concrete 'something' like 1+1 oranges is having 1 unit of orange in some container and then adding on more unit of orange into the same container). And all of this is useless is we do not define what 2, 3, 4, 5, 6, 7, 8, 9, 10, (which I also do not find easy to explain in abstract terms without using oranges) and a method to count in decimal. Only after all this can we say that we have identified the problem. After having the problem identified, and by using the definition of how the decimal system works, and what it's symbols and their combinatorial representations mean (99, 1054), we find that 1+1 is basically by definition, 2. So the problem is not 'difficult' in itself, there is a clear method for solving it.

Now what if we try to explain this whole definition to a 15 year old, for 1 whole year, and that person still does not understand it? that's interesting point to think about. It means that people are like Computers, they have processors with capacity and speed, the processors are analogue but they are still processors, they are flexible and can grow and change, but they are still processors, and so I think that when a problem's definition is above the capacity of the brain that is supposed to 'solve' it, it is useless to talk about problem difficulty, the problem simply can't even be 'fed' into the brain.
There is also the case where that 15 year old would understand the problem, but needs 5 seconds to come with an answer, instead of 1 second, this would simply mean the capacity to accept the definition is there, but the processing speed is slow.
It still doesn't me the problem is 'difficult', not even relatively.

(Speaking about processors, here are some very cool mechanical ones)

Now after identifying the problem, our 2nd step says 'search for a known method to solve the problem', this could be using a method that we already know ourselves (saved in the brain), or doing research to find that people already found solutions to our problem or similar ones.
This would be per example the case when asking 'write a skeletal animation library' to someone who just started to learn '3D graphics', after identifying the problem, he would do some research and find that other people have done this countless times and written about it, he would then use his research to write the library. Does this mean the problem is 'difficult' for a 3D beginner ? again, no, we would be tempted to say that it is difficult for the beginner and easy for an experienced 3D Graphics Software engineer, but no, there is no 'difficulty' involved here, there is simply search time envolved that the experienced engineer need not do, but did at some point in the past, again it is not 'difficult'.

Another example is asking 'write a very robust game physics engine', is this difficult? again, I say no, because of the same reasons, defining what is exactly meant by a physics engine is, and what makes it robust, could fill a several papers, to someone who has never written a physics engine or used one, a lot of time would be needed working on step 1, does this mean that the step is difficult ? no, it simply needs, time, additionally, if that someone has no math experience, more papers would be added, and there are also many topics that would also justify a good amount of papers until reaching the end of the 1st step, but we are still following a clear recursive method, there is no magic nor difficuly envoled, it just takes time depending on the brain speed, assuming the capacity is there. after that would come step 2 also consuming lots of time, and if robust means more robust than the best current engines, we would finally come to step 3, if there are no known solution to the 'very robust' problem for problematic cases like huge world dimensions, very fast movements and rotations, huge mass ratios, etc... then we would need to 'search' for solutions, and a search is really a search, it is like an A* or similar in AI, there is no way to circumvent searches when the problems are new, it could also be that the problem is NP hard ( there are many places to read about P=NP problems, one I recommend though is the book "Artificial Intelligence, A Modern Approach, Stuart J. Russell, Peter Norvig"). Simplfied, NP hard means that until now, nobody managed to prove that we can do anything better than 'searching' to find a solution for such problems.
Searching would be done by using all known heursitics, and theoretical information, and trying plausible solutions until one is found or not found, finding out there is no solution is also a solution. Some robustness problems are per example very easy to solve if we use much better data types, like huge 512 bit floating or fixed point numbers instead of our usual floats or doubles, but then we hit the limits of our current technological limits (speed and memory of our current computers) and the real time constraints a game physics engine must satisfy, but all this still does not mean the problem is difficult, there is an obvious way to solve it, the solution might end up with a search that has no time bounds, or with a conclusion that the problem is not solvable given the constraints, but this does not mean 'difficult'. What is important though is finding out that sometimes, a search is needed, and that is of course the case for all 'new' problems, which are usually generated by solutions to old problems, or requiring improvements, and this is how the beautiful train of technology evolution rolls by, taking all of us enthusiasts on a nice ride.

Being the brain dump it is, there is no conclusion for this post, this topic still needs more thinking, and maybe after a few more related posts, a structured conclusion will come out.

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