4 ms·

Do you know if there's been any thought on extending this to arbitrary algebras? Seems like you've already got the mechanism to define a partition of the space,

by DavidSJ 11y ago

Do you know if there's been any thought on extending this to arbitrary algebras? Seems like you've already got the mechanism to define a partition of the space, and the lookup table describing the behavior of an operation with respect to the subsets of that partition, so this might have utility beyond arithmetic.

jacobolus 11y ago
Huh. I wonder how fast the lookup tables for various operations could be if they were built using some kind of programmable logic. Then you could theoretically just interpret the bit patterns to be whatever kind of number-like object you want, and program the lookup tables accordingly.
- DavidSJ 11y ago
  Right, now that I think about it, it's only coincidental that the objects in question represent the subsets in a partition of the reals. It seems what we have here is a finite set A (in arithmetic, A would be a finite collection of sets which constitute a partition of the reals) and two operations: f, g: P(A) × P(A) -> P(A), taking a pair of subsets of A to another subset of A, with the property that f(W ∪ X, Y) = f(W, Y) ∪ f(X, Y), and so on. This can all be represented as a bitmask table of size |A|^3, or half if commutativity is enforced. We also have the "inverse" which takes X and Z, and returns the set {a ∈ A : f(X, {a}) ∩ Z ≠ Ø}. If I'm not mistaken, the mechanisms for doing all of this in hardware would come for free, and it represents quite a variety of problem domains beyond arithmetic. Perhaps some arithmetic-specific optimization would be ruled out (e.g. the bit twiddling to get additive or multiplicative inverses).
- trsohmers 11y ago
  So, instead of ROM, it would be SRAM... no "reprogrammable logic" required. The difference between ROM bit cell and a RAM bit cell is around ~3x larger, and uses a lot more power wise for read and writes.
dnautics 11y ago
Agreed. Interesting algebras like z/n[i] or z/n[√5] exist, but I'm not sure what the utility outside of theoretical algebra is.
- jacobolus 11y ago
  The algebraic integers in Q(√5) — namely, numbers of the form aτ + b, where a and be are integers and the golden ratio τ = (1+√5)/2 — are quite interesting. If you make vectors of them, you can make a dodecahedron, iconsahedron, 120 cell, 600 cell, etc. This is great practically for modeling quasicrystals, as well as things like buckyballs, certain types of viruses, architectural structures, ... If you want to play with the symmetries, I recommend buying some Zometool construction toys, https://www.zometool.com https://www.zometool.com

by DavidSJ 11y ago

jacobolus 11y ago

Huh. I wonder how fast the lookup tables for various operations could be if they were built using some kind of programmable logic. Then you could theoretically just interpret the bit patterns to be whatever kind of number-like object you want, and program the lookup tables accordingly.

DavidSJ 11y ago
Right, now that I think about it, it's only coincidental that the objects in question represent the subsets in a partition of the reals. It seems what we have here is a finite set A (in arithmetic, A would be a finite collection of sets which constitute a partition of the reals) and two operations: f, g: P(A) × P(A) -> P(A), taking a pair of subsets of A to another subset of A, with the property that f(W ∪ X, Y) = f(W, Y) ∪ f(X, Y), and so on. This can all be represented as a bitmask table of size |A|^3, or half if commutativity is enforced. We also have the "inverse" which takes X and Z, and returns the set {a ∈ A : f(X, {a}) ∩ Z ≠ Ø}. If I'm not mistaken, the mechanisms for doing all of this in hardware would come for free, and it represents quite a variety of problem domains beyond arithmetic. Perhaps some arithmetic-specific optimization would be ruled out (e.g. the bit twiddling to get additive or multiplicative inverses).
trsohmers 11y ago
So, instead of ROM, it would be SRAM... no "reprogrammable logic" required. The difference between ROM bit cell and a RAM bit cell is around ~3x larger, and uses a lot more power wise for read and writes.

dnautics 11y ago

Agreed. Interesting algebras like z/n[i] or z/n[√5] exist, but I'm not sure what the utility outside of theoretical algebra is.

jacobolus 11y ago
The algebraic integers in Q(√5) — namely, numbers of the form aτ + b, where a and be are integers and the golden ratio τ = (1+√5)/2 — are quite interesting. If you make vectors of them, you can make a dodecahedron, iconsahedron, 120 cell, 600 cell, etc. This is great practically for modeling quasicrystals, as well as things like buckyballs, certain types of viruses, architectural structures, ... If you want to play with the symmetries, I recommend buying some Zometool construction toys, https://www.zometool.com https://www.zometool.com