Pattern and Randomness

(Oct 22, 2019)

Everytime we talk about Emergence, there is a notion of a pattern, something that we consider favourable, that eventually comes into existence by the complex interactions and dynamics of the system in question. On the other hand, the word ‘random’ typically represents the opposite of ‘pattern’. Or does it?

I argue here, randomness is an emergent property. Say I got {H,H,H} on 3 successive coin tosses, I can interpret the coin as 100% biases. But it can also happen to be one of those 8 possibilities that showed up in this Universe, while the other 7 cases were swept under the rug of the many-World’s interpretation of a measurement (albeit classical). Whether it is actually unbiased cannot be understood from a few trials unless the law of large numbers comes into play, i.e. until the prefect ideal probability distribution is at least captured in some approximation in the statistics. Randomness is a statistical parameter, making no sense for a single experiment, like the temperature of an individual atom. Often, randomness is associated with the entropy of the microstate. 3 Heads has higher order and less surprise than 2 Heads and 1 Tail. But that assumes the coin is unbiased as a prior. What if we want to understand the property of the system itself? For example, if we are looking for radio signals from extra-terrestrial life, or decodings the heiroglyphs of an ancient civilization? How would we distinguish a random signal from a non-random one? The entropy of a bitstring also deals with how much information can be communicated via it, or in the Kolmogorov sense, if it can be compressed and later decompressed with a wrapper semantic overhead. Let’s assume a situation where I tell a friend that I would either send a string of 1s if the answer if yes, or a string of equal 1s and 0s if the answer is no. Assuming no noise in the channel, now, the meaning of the word random loses it’s entropic context, as here, a string with 75% 1s would be more near to a random message.

Is pattern also an emergent parameter? Is it a statistical low entropy configuration or a collection of semantically meaning states?

  • Arguments against the 1st idea: based on how we semantically understand something, a higher entropy system can show more pattern. E.g. a program in BrainFuck printing 1s forever will have less algorithmic entropy than a program in C++ generating the Fibonacci series due to the inherent non-rationality of te golden mean; or a C++ code for 1s would have lower entropy than a BrainFuck code for golden mean; even though it should depend on the semantics of the language for the compiler, like an english sentence has lower entropy than a japanese sentence due to the higher number of japanese alphabets.
  • Arguments against the 2nd idea: if something has semantic meaning, it should be reducible to a cost function for which a pattern would give a higher score than a random input. For a program/language, it would be syntactic correctness, e.g. grammarly. But still the association to the application is missing, the same problem as with shannon information metric.

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Fascination with Fractals

(Oct 22, 2019)

Why are fractals so ubiquitous in Nature than Euclidian geometry? What property of fractals make them so favourable for these blueprints? I like to approach this from 2 different angles.

God is a lazy programmer. Imagine you have to render the graphics of fire or clouds with triangles or ovals! Hell of a task, right? Indeed, a few iterations of a simple yet elegant fractal equation can generate these on your game world. It is not so difficult to drive home the point that fractals are the generator equations of the world we see around us, so fractal equations can easily generate models of them - low algorithmic complexity - lazy programmer. But, that’s a bit of ouruboros logic. The real equation is, why do we see fractal generator equations in the blueprints of the Universe? Why can’t clouds just be oval or fires as triangles like in the computer games of the early 1980s?

This has to do with compressing. Fractals are the edge of chaos, where the system transitions from a periodic attractor to a chaotic randomness. This also goes hand in hand with class 4 Wolfram automata which are universal computers which has enough expressive power to program everything in an unified structure, yet, the rules are simple enough and don’t get lost in chaos. Fractals are also great data compressors that can be prioritized with respect to the iteration level, working exactly like a Discrete Wavelet Transform, where the larger amplitudes and low frequency terms are captured in the lower iterations whereas the finer details can be compressed in the higher iterations allowing viewing the final product at different levels of approximation without losing the big picture, to interpret the general law behind them. Thus, there is a very subtle difference between a fractal of 100 iteration (say a Koch curve) and a fractal of 100 iteration with a small variation allowed at each level (say the coastline of Britain). In the later, an enormous amount of information can be encoded at different level of approximations. A little child can build an encoded message with pebbles on a particular beach without changing the overall fractal dimension much.

So fractals in a way allows us to start with a vague design and then periodically tweek it with small modifications to reach the design of interest. The question remains: is that how the Universal laws emerged? Chunks of smaller and smaller sized phenomena adding higher order refinements to the evolution of the universe.

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GUT from It

(Mar 27, 2019)

Recently I was reading this article on constructing space-time from computation which opened the flood gates of correlating theories in my head.

Before I describe my proposition, let’s list down the ingredients:

  • Plancherel’s theorem which states the integral of a function’s squared modulus is equal to the integral of the squared modulus of its frequency spectrum.
  • Kolmogorov/Algorithmic complexity
  • String length
  • Launderer’s principle
  • Fourier transform
  • It from Bit
  • Thermodynamics
  • Measurement in Quantum Mechanics

The equation: \[\biggl\Vert \int_0^{t_u} |f(x)|^2dx - \int_{-\infty}^\infty|f(\xi)|^2d\xi \biggl\Vert \equiv \bigl(len(S) - K_\mathcal{U}(S|X)\bigl)kTln2\]

The interpretation:

Let $f(x)$ be the state of the Universe encoded as a bit string. The absolute difference between the integral of the function’s squared modulus and the integral of the squared modulus of its frequency spectrum gives us the amount of new information generated by the Universe in the time duration of the integral of the function, i.e. $[0,t_u]$. This is equivalent to the work value of the bit string given by the fuel value of the string scaled by the Boltzmann constant and the temperature, following reversed Launderer’s principle. The fuel value is the difference between the length of the string and the conditional Kolmogorov complexity of the bit string, given the Fourier transform of it. This transform represents the derivable physical laws given the bit pattern of the Universe.

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