Information
What is information? We know that "information" sometimes means "The meaning associated with data..." (Donald Knuth) but in this context we are referring to a scalar quantity that is a measure related to quantities like the log2(number of states) of some system. Finite Nature means that the world is made up of digital information. We understand "digital" exactly (made of, or making use of digits; symbols representing small integers) but we don't yet have a complete understanding as to how to measure or calculate with the scalar quantity named "information". This is not an easy problem, because information, like energy or work, comes in different forms and thus it is hard to give a simple definition. Kinetic, potential, heat, electrical, chemical and others are all forms of energy; any of which can be transformed to other forms. Momentum, on the other hand is much simpler. There is really nothing else that momentum can be transformed into.
Despite not having a good definition of what information is, we postulate a new law; Conservation of Information. In this context information is the stuff that enables a reversible system to, if time were reversed, proceed backwards along the exact path that it took going forwards. All reversible systems have a conserved quantity; log2(n), where n is the number of different states that the system progresses through in one cycle. (All finite state reversible systems evolve along a closed cycle.) This number is constant, independent of where the system is on that cycle and is equal to the quantity of information conserved.
Consider a reversible cellular automaton with a local neighborhood consisting of the seven cells which are east, west, north, south, up, down and past in relation to a particular cell. Finally, we have a transition rule, F, which defines for every possible combination of the states of the neighborhood cells, what state the particular cell should become.
Cx, y, z, t 1=F(Cx-1, y, z, t , Cx 1, y, z, t , Cx, y 1, z, t , Cx, y-1, z, t ,
Cx, y, z 1, t , Cx, y, z-1, t , Cx, y, z, t -1)
For the system to be reversible, there must be another function, G, so that:
Cx, y, z, t-1=G(Cx-1, y, z, t , Cx 1, y, z, t , Cx, y 1, z, t , Cx, y-1, z, t ,
Cx, y, z 1, t , Cx, y, z-1, t , Cx, y, z, t 1)
If these two criteria are met (the functions F and G exist), then we can say that the system conserves information, i.e., it is a law that information is conserved; no information ever gets lost. It may not be obvious as to what has happened to some information, but all of the information must be there in one form or another.
By observing the behavior of a non-trivial, reversible cellular automata, it becomes apparent that there is something almost magical about the way it evolves. What is hard to appreciate are the consequences of rules that appear to collapse information (a lot of information becoming just a little bit of information), yet nothing gets lost. If each cell has 3 states then 37 or 2187 possible states go into the computation of F. The resulting value of F is just one of three possible states. This seems to involve the loss of information; but each of the neighbors is also involved in the computation of the states of other neighbors.
In ordinary physics, we can easily imagine perfect reversibility based on the fact that no effect ever gets lost because no matter how infinitesimal it becomes, it is still there. This approach makes it clear that the reason that an asteroid might hit the earth 5 years from now might be a direct consequence of the gravitational effect of a rock that rolled down a hill on the moon 2 billion years ago, delicately affecting the orbits of the Earth and all of the asteroids. But with the extreme quantization of the digital process, we have something more akin to the "For the want of a nail, the shoe was lost; for the want of the shoe, the horse was lost..." The results can be similar.
