Regarding Deep Learning/NIM

[José Manuel Rodríguez Caballero]

>
> QUESTION. How can we design a program that can play perfect NIM with
> only being given the rules of play?


Here is a reference:

Lord, William, and Paul Graham. "Reinforcement Learning and the Game of
Nim." (2015).
https://www.diva-portal.org/smash/get/diva2:814832/FULLTEXT01.pdf

QUESTION. How good is this question?


According to the formalists, mathematics is a game, where the axioms are
the rules.

Statistical learning (including deep learning, decision trees, random
forests, linear regression, and many other techniques) can be combined with
proof assistants, such as Isabelle/HOL, Lean, and Coq, to suggest "good
mathematical moves" in the proof of theorems. This is similar to the way in
which statistical learning models can heuristically find good chess moves.
Archives of Formal Proofs can be used as the dataset for training the
statistical learning models, here is an example that I developed with Prof.
Unruh for quantum cryptography (grounded on the complex Hilbert spaces)

https://www.isa-afp.org/entries/Complex_Bounded_Operators.html

Of course, these "good math moves" suggested by the statistical learning
model will not be innovations, as they are just trends that worked in the
past. Assuming Stephen Wolfram's conjecture that there is macroscopic
mathematics that does not depend on axioms (microstates of mathematics in
analogy with a thermodynamic system), the statistical learning model should
provide good suggestions regardless of axioms. Another possibility is that
what the statistical learning model may be reflecting is not the structure
of mathematics, but the human way of doing mathematics, i.e., another
intelligent being, or even humans from another culture, may correspond to
different statistical distributions. In the latter case, the mystery is
neither in the statistical learning model nor in the structure of
mathematics, but in the human mind.

Returning to the idea of mathematics as a game, where the axioms are the
rules, the foundations of mathematics can be seen as a correspondence
between games. For example, the game ZFC will not produce a contradiction
if and only if the game ZFC with the continuum hypothesis added will not
produce a contradiction. The formulation of foundations of mathematics in
terms of games could be as follows: if something is impossible in game A,
something else is impossible in game B. In statistical learning we can
consider the weaker problem: if some event has probability zero in game A,
then another event has probability zero probability in game Y. Now,
returning to mathematics, we may imagine a hypothetical field known
statistical foundations of mathematics, where the goal is to consider the
following class of problems:

if the probability that a statistical learning model finds a contradiction
in theory A is zero, then the probability that another statistical learning
model finds a contradiction in theory B is zero.

The frequentist way to interpret the statement

"the probability that a statistical learning model finds a contradiction in
theory A is p"

is as follows: let a_n be the number of times that the statistical learning
model finds a contradiction in A during n invocations of the algorithm, the
limit of a_n divided by n is p as n goes to infinity.

This hypothetical branch of mathematics, statistical foundations of
mathematics, could be extremely useful for cryptography, e.g. if theory A
has contradictions but they are hard to be found by a statistical learning
algorithm, they can be used as proof of work:

Proof of work (PoW) describes a system that requires a not-insignificant
> but feasible amount of effort in order to deter frivolous or malicious uses
> of computing power, such as sending spam emails or launching denial of
> service attacks.

https://www.investopedia.com/terms/p/proof-work.asp


Kind regards,
Jose M.
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