1/ A thread on the brain as a computer (tl; dr: 🧠=💻):

2/ I think many neuroscientists are confused by the terms "algorithm" and "computer". Very few seem to understand how Turing machines (TMs) play a part in defining those words. In fairness, it can be hard to understand if your background is in physiology and psychology.

3/ Briefly: "algorithm" can be understood intuitively as any set of instructions for solving some task. A TM is just a mathematical tool to make this intuitive definition formal.

4/ In particular, the Church-Turing thesis states that our intuitive definition of "algorithm" can be formalized by TMs. We define "computable fns" to be those fns which a TM can solve, and an "algorithm" is *any* recipe for solving computable fns.

5/ A "computer" is then simply anything that can implement algorithms for computable fns. Also, any device which solves computable fns is running an algorithm, by definition. NB: an algorithm does not need to be symbol based or discrete-time (that's not in the definition)!

6/ Of course, all objects implement some fn, and we don't wanna call a stone or a feather a "computer". So to be more stringent we can speak of "Turing complete" computers. A Turing complete computer is one which can implement *any* computable fn if programmed correctly.

7/ A stone is not Turing complete, but what we call "computers" today (the machines in our pockets and on our desks) are Turing complete. Also, it has been proven that neural networks with multiple layers or recurrent connections are Turing complete.

8/ Thus, the brain *is* a Turing complete computer running algorithms (if we assume it has capabilities >= artificial NNs). As such, saying the brain is a computer running algorithms is not a metaphor, it's a *fact* borne of the definitions of computer and algorithm.

9/ Note: this also all means that the fact that the brain is a continuous-time, dynamical system does not invalidate any of this. The role of the TM here is not to say what a computer must look like in the real world, it is only a tool for defining the set of computable fns.

10/ If you're interested in this stuff, I was exposed to it via Stephen Cook's amazing classes at U of T ( http://www.cs.toronto.edu/~sacook/ ). I also noticed he references this textbook in one of his lectures, and at a brief glance, it seems pretty good ( https://www.amazon.ca/Introduction-Theory-Computation-Michael-Sipser/dp/113318779X …).

11/ Whether all this matters practically for neuroscience is another interesting Q... Time will tell whether computability theory has any impact on our understanding of the brain, beyond giving windbags like me a reason for frustration with neuroscientists. 😉

You can follow @tyrell_turing.

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