Thursday, 25 September 2008

How to measure chaos

This is to propose the way to measure chaos in a computational system.

I studied the reversible algorithms of arithmetic multiplication (factorization problem). I have found a very interesting fact that when the information describing the ensemble grows fast, this exact process is difficult to reverse, i.e. find feasible computation reverse algorithm. Let me explain by example of computation.

Suppose there is an initial set of N bits {x}. If all bits are independent and can initially be set to either 0 or 1, then this is the ensemble of 2N states. Now suppose we have a function G describing all forbidden states of the ensemble. [The same up to inversion arguments would go with all allowable states.] Initially G=0, since all values of x are allowed.

Next suppose that we have a computational process which is done in atomic steps. Each step can be one of two kinds: either creating a new bit or forgetting an old. For example, add a new bit y1=x1&x2 (let us signify & as AND operation, | as OR, and ^ as XOR). The G function becomes y1^(x1&x2) if previous G=0, or in general case Gnext=Gprevious|y1^(x1&x2). Forgetting a bit x2 would make Gnext=Gx2=0&Gx2=1, where Gx2=0 is G with x2 replaced with 0.

It seems easy to keep G function defined while doing the calculation process. This is important because while we know G function the calculation is immediately reversible – it is easy at each backward step find out the previous forgotten bit. But the problem appears when G function becomes impractically huge in representation. This happens, for example, with arithmetic multiplication. The only way to attack this problem is to find symmetries in the function so to make it computationally practical, smaller.

Suppose that with a given intellectual strength we find the smallest representation of G function. This function can be written as a sequence of the atomic steps A=B&C, where A,B,C are bits or their negations (say A=B|C is ~A=~B&~C). The negation is not an operation, rather a perception of the bit. With this representation the function becomes measurable in number of atomic computational steps.

That makes it very natural to define the chaos as the increase of the size of the function G. Note that the definition includes the intellectual strength of the observer, which seems natural for the definition of chaos. Also it qualitatively resembles Lyapunov exponents as chaos measure.

Monday, 15 September 2008

Precise theory for Artificial Intelligence

Every research area has its own language. But there is a distinction between precise and non-precise theories describing the areas of interest. In physics, for example, the language is mathematical formulas. And this is a precise formal language; because formulae help, in general, to solve physical problems even if people applying the formulae have different interpretation of them. We can argue on the definition of Time and Enegry. But when it comes to the formula with E and t, everyone will get the same result regardless of personal (internal) understanding of these concepts.

Completely different situation is with Artificial Intelligence research area. Here are concepts of information, memory, awareness giving completely different philosophical theories depending on how the concepts are being understood. There is so big variety of opinions that it is difficult to call this area of research scientific. Nevertheless some groups of people sharing the same beliefs are able to communicate and even to make some progress in ideas. But with no practical results. Obviously now the understanding of the problem in question is different than ten years ago. Hopefully a simple model can be built in some near future, just to give the common basis for terminology used in this field.