# A question on the Kolmogorov Complexity of Human I/O behaviour

Note: From my Twitter poll I managed to get feedback from AI researchers and neuroscientists so far and I think it would be interesting to get input from theoretical computer scientists on this question as well.

The specific question I am interested in is the following: What is the ratio of the Kolmogorov Complexity of Human I/O behaviour, let's call this K(H), relative to K(S), the Kolmogorov Complexity of Slime Mould I/O behaviour?

Here's how I understand the problem:

1. By I/O behaviour I am referring to the agent-environment interaction loop which may be represented by coarse-grained state-action sequences that are objective, as viewed by an external observer, and potentially countably infinite.
2. Symbolically, we may refer to these sequences as follows: $$(a_i,s_i)_{i \in \mathbb{N}}$$ where $$s_i$$ represents the state of the observable Universe at instant $$i$$.
3. If we compress this sequence we will probably end up with a short program which describes the agent-environment interaction as a coupled dynamical system. For clarification, this sequence is obtained by sampling from:

$$$$p(a_t|s_t)$$$$

as well as from the state-transition probability distribution conditioned on the actions of our agent:

$$$$p(s_{t+1}|a_t,s_t)$$$$

Given this formulation I think it's fair to say, intuitively, that the causal influence of an individual organism $$\mathcal{A}$$ on the observable Universe(which is responsible for all possible input signals) is negligible relative to the causal influence of the observable Universe(which includes many other organisms) on $$\mathcal{A}$$. Now, considering the previous argument it follows that the Kolmogorov Complexity of the organism's environment alone, without that particular organism, is approximately equivalent to the Kolmogorov Complexity of the environment plus an individual Human or Slime Mould so we surely have:

$$$$\frac{K(H)}{K(S)} \approx 1$$$$

I must note that I don't think this question can be answered in a definitive manner but I'd like to know whether there are important conceptual arguments which I might have overlooked and whether the question itself might be fundamentally flawed.

References:

1. Thirty eight things to do with live slime mould. Andrew Adamatzky. 2015.

2. A three minute introduction to the slime mould

3. A brief introduction to Kolmogorov Complexity by Marcus Hutter

Update: Just for clarification, an input signal can take many forms. Examples include electromagnetic waves in the form of sunlight, liquid in the form of rain and even extra-terrestrial comets that may cause extinction.

• Out of curiosity I just googled for the information content of the human genome. Roughly 6 billions bits ( sequence of 3 billions A/T/C/G base-pairs ) are enough to encode the human genome; if you are able to "unzip" them into a virtual human you should be able to simulate his coarse-grained state-action sequence :-). Molds have ~$10^7$ base-pairs. Both sequences are highly compressible (4 megabytes are enough for a human genome) – Marzio De Biasi Feb 24 '19 at 22:26
• @MarzioDeBiasi I am not sure DNA is sufficient actually as an approximation of the state-transition probability distribution, p(s_t+1|s_t), is necessary to get state transitions. Now, in order to simulate p(s_t+1|s_t) you will need to compress the observable universe somehow. This is probably a lot more than 4 megabytes in my opinion. – Aidan Rocke Feb 24 '19 at 22:33
• In order to compare the two I think that you should define more accurately (mathematically) what you mean with "coarse grained state-action sequence". If the grain is very fine then the two $K$s are comparable (and almost incompressible); if the grain is very coarse then a mould should be very predictable if compared to a human; but these are only quick thoughts about your question. – Marzio De Biasi Feb 24 '19 at 22:35
• @MarzioDeBiasi I added a clarification regarding the origins of the state-action sequence. If the question intrigues you as much as it fascinates me we may also discuss it via email. – Aidan Rocke Feb 24 '19 at 22:59