Notes on Computational Equivalence Principle

Can simple programs produce very rich and complicated behaviors? Could it be possible that all of the amazing things we see in the universe be the result of a simple program? That would be very exciting to have a program that is an ultimate precise model of our universe. If you run it long enough it would produce complex models. Stephen Wolfram asks how would this program be like.

The principle of computational equivalence says that “systems found in the natural world can perform computations up to a maximal ("universal") level of computational power, and that most systems do in fact attain this maximal level of computational power.” With Wolfram's own words "When it comes to computation—or intelligence—we are in the end no more sophisticated than all sorts of simple programs, and all sorts of systems in nature. But from the Principle of Computational Equivalence there emerges a new kind of unity: for across a vast range of systems, from simple programs to brains to our whole universe, the principle implies that there is a basic equivalence that makes the same fundamental phenomena occur, and allows the same basic scientific ideas and methods to be used."

The structure of a system need not be complicated for its behavior to be highly complex. Cellular Automata (e.g rule 30) or reaction-diffusion models that Alan Turing developed just before his death can provide an example to provide models for a wide variety of complex systems. Even Turing machine model with one cell updating would be a model for complex behaviors. It is easy to see the reflections of the Computational Equivalence Principle in Richard Dawkins Biomorphosis, Thomas Ray’s Tierra project, and Karl Sim’s virtual creatures.

Evolutionary biologist Thomas Ray makes a comparison as the genetic language consists of an alphabet of 20 letters and a computer language has many. He takes inspiration from natural science and uses computer science to solve problems. Not only algorithms, but he also distinguishes the inside of a computer as a physical system by making an analogy of the sun, the source of energy, as CPUCentral Processing Unit, an →IC, generally refered to as the processor in a computer and creatures living in the memory. This also supports Wolfram’s idea of a close correspondence between physical processes and computations. Like the Computational Equivalence principle says, Thomas Ray seems to build very complex ecological phenomena with his very simplified computer program.


Wolfram says “Universe is not mathematical but computational” Maybe he sincerely believes that everything is computable or maybe there is another meaning into it. Scientists and mathematicians always look for a simple answer like the ‘theory of everything’ or a simple elegant equation. Thinking in terms of simple programs Wolfram thinks that it is possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known features of our universe will emerge. And this immediately suggests that the phenomenon of universality is vastly more common and important in both abstract systems and nature.

“Organisms are algorithms” says Harrari. In biology, so many organisms exhibit such great complexity but they are all similar calculators constantly processing biochemical processes of calculation. Feelings are a process of biochemical calculation shaped by millions of years of natural selection. Organisms collect the data with their senses, calculate the probabilities very fast and the answer appears not as a number but as a feeling or emotion.

For me, computation would be a dealing strategy or another language to allow us to understand or solve problems. We can compute natural phenomena and create models in some extend but this doesn’t mean we can compute everything. This could only be a tool for understanding or looking things. I think the urge to look for simple rules are the result of the desire to make sense of the world we live. We assume that that there is a meaning somewhere and we try to find it but I think this seeking is nonsensical.

The idea of simple algorithms generating complex structures is very similar to compression algorithms. All the models and abstractions can be seen as compression algorithm. It compress a general structure into a simple fascinating rule. Every abstraction, at the same time, reject something about reality. Just like simulation allegory in Borges tale; they tried to draw the best map for the Empire and draw up a map so detailed that it ends up exactly covering the territory. If the map didnt cover the territory then it had to ignore something.

At the end,

"All models are wrong, but some are useful". George Box.