[ddmd]

One interesting question is how to implement linear algebra for the case of binary variables x_i in {0,1} or x_i in {-1,0,+1} as opposed to having continuous variables. Of course, it will be somewhat different theory but the general properties should be analogous to those of standard linear algebra. For example, dot product should express the notion of orthogonality, there should be well defined distance etc. I am not sure that such a theory exists at all but if yes then I am curious if there exist implementations.

[dragontamer]

Discrete mathematics theory is possibly more difficult than the set of complex numbers. Case in point: linear optimization across the reals/complex field is solved with the simplex algorithm.

Boolean optimization in contrast, is NP-complete since its equivalent to the knapsack problem (0 for "don't pack the object" and 1 for "do pack the object". Each object has a space it takes up + a value. Optimizing on value is NP-complete). Literally more difficult from a computational perspective than its Real/Complex analogue, and __provably__ so.

I'm enjoying my personal studies into Binary Decision Diagrams: the data-structures that are used in Verilog / VHDL synthesizers, libraries, CPU-design, verification (etc. etc.) to encode boolean truth tables and yes... tackle NP-complete and even #P-complete problems (#P complete is "determine the number of solutions" to an NP complete problem. Knights Tours aren't NP complete but... see the paper for an example of #P-like counting problems: "The Number of Knight's Tours Equals 33,439,123,484,294": https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45...).

The boolean truth table for a 64x64-bit multiplication is 128-bits of input and 128-bits of output. That's 2^253 bytes of storage under a naive scheme, which is clearly unworkable. Binary Decision Diagrams make it possible to store, compute, and "think" about these functions at a high level. (Ex: counting solutions, finding the 1st solution, combining truth tables together, etc. etc.)

Knuth's TAOCP Volume 4A has been a good introduction to the theory, but Knuth's writing style is so difficult sometimes. I'm probably going to buy a supplemental textbook on the subject...

[ddmd]

> Discrete mathematics theory is possibly more difficult than the set of complex numbers.

Probably it is so because in continuous case there is the luxury of having enough points between any other points. In discrete case you are much more constrained, for example, you cannot choose a point between 0 and 1, and what is the length of the diagonal of a binary cube or angle between its two hyper-planes? Sometimes these notions can be naturally defined but in other cases the formal theory is not so natural.

By the way, an interesting question is how complex boolean numbers could be defined (naturally).

[tryingtopost]

It exists. Linear algebra is defined over finite fields. In the case of {0, 1} that is Z_2 and the case of { -1, 0, + 1 } that is Z_3.

[ouid]

There are algorithms which fail over GF(2). Gram-schmidt, for instance.

[tlarkworthy]

Mixed integer programming?

Here is a browser version https://observablehq.com/@tomlarkworthy/mip

[yobbo]

You might find what you're looking for if you google "linear systems over finite fields" or "integer linear systems".