Tag: Max Tegmark

The Goldilocks Complexity Zone

Mark Davis1 Comment

FractalSince my time in the early 90s at Santa Fe Institute, I’ve been fascinated by the informational physics of complex systems. What are the requirements of an abstract system that is capable of complex behavior? How do our intuitions about complex behavior or form match up with mathematical approaches to describing complexity? For instance, we might consider a snowflake complex, but it is also regular in it’s structure, driven by an interaction between crystal growth and the surrounding air. The classic examples of coastlines and fractal self-symmetry also seem complex but are not capable of complex behavior.

So what is a good way of thinking about complexity? There is actually a good range of ideas about how to characterize complexity. Seth Lloyd rounds up many of them, here. The intuition that drives many of them is that complexity seems to be associated with distributions of relationships and objects that are somehow juxtapositioned between a single state and a uniformly random set of states. Complex things, be they living organisms or computers running algorithms, should exist in a Goldilocks zone when each part is examined and those parts are somehow summed up to a single measure.

We can easily construct a complexity measure that captures some of these intuitions. Let’s look at three strings of characters:

x = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

y = menlqphsfyjubaoitwzrvcgxdkbwohqyxplerz

z = the fox met the hare and the fox saw the hare

Now we would likely all agree that y and z are more complex than x, and I suspect most would agree that y looks like gibberish compared with z. Of course, y could be a sequence of weirdly coded measurements or something, or encrypted such that the message appears random.… Read the rest

Entanglement and Information

Mark DavisLeave a comment

shannons-formula-smallResearch can flow into interesting little eddies that cohere into larger circulations that become transformative phase shifts. That happened to me this morning between a morning drive in the Northern California hills and departing for lunch at one of our favorite restaurants in Danville.

The topic I’ve been working on since my retirement is whether there are preferential representations for optimal automated inference methods. We have this grab-bag of machine learning techniques that use differing data structures but that all implement some variation on fitting functions to data exemplars; at the most general they all look like some kind of gradient descent on an error surface. Getting the right mix of parameters, nodes, etc. falls to some kind of statistical regularization or bottlenecking for the algorithms. Or maybe you perform a grid search in the hyperparameter space, narrowing down the right mix. Or you can throw up your hands and try to evolve your way to a solution, suspecting that there may be local optima that are distracting the algorithms from global success.

Yet, algorithmic information theory (AIT) gives us, via Solomonoff, a framework for balancing parameterization of an inference algorithm against the error rate on the training set. But, first, it’s all uncomputable and, second, the AIT framework just uses strings of binary as the coded Turing machines, so I would have to flip 2^N bits and test each representation to get anywhere with the theory. Yet, I and many others have had incremental success at using variations on this framework, whether via Minimum Description Length (MDL) principles, it’s first cousin Minimum Message Length (MML), and other statistical regularization approaches that are somewhat proxies for these techniques.… Read the rest