3 ms·
FWIW, in ternary the x->-x is a new 'uninteresting' symmetry, so for any non-constant-zero operation F(x, y) you've got a paired -F(x, y). If these pairs are un
by sdenton4 6y ago
FWIW, in ternary the x->-x is a new 'uninteresting' symmetry, so for any non-constant-zero operation F(x, y) you've got a paired -F(x, y). If these pairs are uninteresting (eg, you can run the operation -F for free, given F), you can cut the number of operators by another factor of 2.
More generally, if you allow composition of a permutation of the trit values 'for free', (eg, F(x,y) + 1 is 'free' given F) you get to reduce the 'atomic' operations by a factor of 6. This is the same as collapsing 'G' and 'not G' among binary operators.
But I guess you've got to implement these in circuits at some point, so the symmetries aren't necessarily helpful (eg, NAND is a thing, rather than AND composed with a symmetry?). (caveat: i am, obviously, more a mathematician than a circuit designer. :P )
- petschge 6y agoSo far I have imposed mainly F(x,y) = F(y,x). You could also enforce F(-x,-y) = -F(x,y), but I have not done that. The second reduction that I mention is actually to require every F to satisfy either F(-1,y) <= F(0,y) <= F(1,y) OR F(-1, y) >= F(0,y) >= F(1,y). And no the symmetries are not that useful to implement the circuitry, but are nice to select operators that compose nicely and allow many simplifications (think De Morgan's laws)