Ask HN: Solving problems by mapping to other problems that we know how to solve

I own a book that may be relevant: "Bypasses: A Simple Approach to Complexity" by Z. A. Melzak, (New York, Wiley, 1983) 0-471-86854-X, or maybe it's too theoretical.

A borrowable digital copy is at the Internet Archive: https://archive.org/details/bypasses00zdzi

From “The Origin of Consciousness in the Breakdown of the Bicameral Mind”:

“Understanding a thing is to arrive at a metaphor for that thing by substituting something more familiar to us. And the feeling of familiarity is the feeling of understanding.”

“Understanding in science is the feeling of similarity between complicated data and a familiar model.”

“The Bohr model of the atom is that of a proton surrounded by orbiting electrons. It is something like the pattern of the solar system, and that is indeed one of its metaphoric sources.”

This is very fundamental not only in maths but in theoretical computer science in general. Even to prove how "hard" a problem is. E.g. look into (Turing/many-one) reductions [1]

Many problems can be reduced to SAT and then you can employ an off-the-shelf SAT/SMT solver to solve it for you. Etc etc.

But in general, being able to reduce problem A to problem B implies that A is not harder than B. I.e., to show that a problem P is undecidable, you can reduce the Halting problem to P.

TSP is NP-hard and can be reduced to SAT, then solved with a SAT solver. But something like determining whether a player has a winning strategy in unrestricted chess (with an nxn board, for arbitrary n) is in EXPTIME and can't be reduced to an "easier" problem.

Edit: Someone ITT also mentioned ILP (integer linear programming). Also a good example. E.g., many optimization programs can be mapped to ILP.

[1] https://en.wikipedia.org/wiki/Turing_reduction

Another category theory resource you may find interesting is Category theory for programmers by Bartosz Milewski. He also has a great video course on youtube. If you want a taste of what category theory is, you can checkout his interview on corecursive.

https://github.com/hmemcpy/milewski-ctfp-pdf

You might be interested in the concept of "duality" often used in e.g. theoretical physics: https://en.wikipedia.org/wiki/Duality_(mathematics)

If you want to be even more general, check out a field of mathematics called category theory (https://en.wikipedia.org/wiki/Category_theory). Its' central idea is to describe mathematical structures in terms of "objects" and their relationship with each other (called morphisms). I'm not extremely knowledgeful in this subject, but I believe mostly everything you get out of categorical theoretical treatment is too generic to be useful, but has resulted in some real results in e.g. algebraic geometry.