2 ms·

Regarding XOR: This is sloppy quoting of Frankel, who writes about the monotonic functions "And and (nonexclusive) Or" – so, yes, inclusive Or only. However, as

Regarding XOR: This is sloppy quoting of Frankel, who writes about the monotonic functions "And and (nonexclusive) Or" – so, yes, inclusive Or only. However, as described, both the normal and the inverted (negated) outputs of the flip-flops ("toggles") are fed into the network, so there's X and Y and negated X and negated Y available.

kragen 7y ago
Right, and as I explained at https://news.ycombinator.com/item?id=20488446 https://news.ycombinator.com/item?id=20488446, if you have negated versions of all the literals available, you can make not just XOR but any arbitrary Boolean function as a sum of products (or a product of sums, CNF), just like a PLD does.
- masswerk 7y ago
  Notably, you need XOR (single-bit add) for an ALU anyway… (So the somewhat weak exclusion of "exclusive" in Frankel's paper is a bit luxurious.)
  - kragen 7y ago
    "Luxurious"? I'm not sure what you intend to say. Do you perhaps mean "unnecessary" or "essential" or "very strong" or "very weak" or "well-thought-out"? As I read it, he's pointing out that since the diodes and resistors can't do things like XOR, because it's not monotonic, it's necessary for "[e]ach input variable" to be "presented in duplicate". Another example that is in some sense more fundamental than XOR is the MUX function b if a else c. If at some point b = 0 and c = 1, the output is the negation of A, which diodes and resistors alone cannot achieve.
    - masswerk 7y ago
      "Luxurious" is a term from psychology, where it's used to describe "ornamental" behavior, which may be hinting at something deeper. ;-) In the particular case, I just found it interesting, since the duplicate outputs are already well established, before the text "withdraws" to monotonic functions (for which both inputs are used). A bit of an understatement. Edit: I think, "luxurious" as used this way is a great term, because it doesn't attempt to apply any classification. It's just pointing out that there's a somewhat independent process, which consumes energy.

kragen 7y ago

Right, and as I explained at https://news.ycombinator.com/item?id=20488446 https://news.ycombinator.com/item?id=20488446, if you have negated versions of all the literals available, you can make not just XOR but any arbitrary Boolean function as a sum of products (or a product of sums, CNF), just like a PLD does.

masswerk 7y ago
Notably, you need XOR (single-bit add) for an ALU anyway… (So the somewhat weak exclusion of "exclusive" in Frankel's paper is a bit luxurious.)
- kragen 7y ago
  "Luxurious"? I'm not sure what you intend to say. Do you perhaps mean "unnecessary" or "essential" or "very strong" or "very weak" or "well-thought-out"? As I read it, he's pointing out that since the diodes and resistors can't do things like XOR, because it's not monotonic, it's necessary for "[e]ach input variable" to be "presented in duplicate". Another example that is in some sense more fundamental than XOR is the MUX function b if a else c. If at some point b = 0 and c = 1, the output is the negation of A, which diodes and resistors alone cannot achieve.
  - masswerk 7y ago
    "Luxurious" is a term from psychology, where it's used to describe "ornamental" behavior, which may be hinting at something deeper. ;-) In the particular case, I just found it interesting, since the duplicate outputs are already well established, before the text "withdraws" to monotonic functions (for which both inputs are used). A bit of an understatement. Edit: I think, "luxurious" as used this way is a great term, because it doesn't attempt to apply any classification. It's just pointing out that there's a somewhat independent process, which consumes energy.