David P Bird wrote:you wrote:But you should write that all the chains (whether reversible or not) are "XOR/NAND".

I've always considered that theoretically it's possible to start chains from an OR inference where both arguments could be true. These have been called strong-only links because it's impossible for both terms to be false. For example if two cells contain three candidate digits there is a strong-only link between two of them as they would both be true if the third one is false.

I can't see your point. For the two cells, say (r, c) and (r, c'), you have XOR(n1rc, n2rc, n3rc) XOR(n1rc', n2rc', n3rc'). I don't see how you turn this into a chain.

In any case, the fact that all the chains are XOR/NAND is the fundamental of AICs, or it should be if AIC is intended to mean anything.

David P Bird wrote:Some people do use 'memory chains' by adding asterisks to remembered candidates and the ones they affect later in the chain which simulates your approach to some extent.

What I called above "

pretending not to be aware of previous work and thus re-inventing the wheel", all this being dissimulated by a different vocabulary and obscure notation.

BTW, they provide additional examples that "my" oriented chains are used by manual solvers, contrary to

DonM's claims.

David P Bird wrote:BTW I would still love to see what you could achieve in finding the optimum solution path for a puzzle. There is a balance between the number of steps taken and their complexity to be considered which could provide interesting variations.

This is totally unrelated with the above discussion.

Unfortunately, in spite of the multiple tries about rating the global resolution path, nothing consistent has ever been written on this topic, and I'm no better than anyone else on this very difficult point. Moreover, as I have no reasonable idea of how to deal with it, I'm not at all working on it.

Given a resolution path, some post-processing can be considered in order to clean it: proceeding backwards from the end of the path, delete some eliminations not necessary for the subsequent parts of the path.

I have done this in PBCS2 when I used CSP-Rules as an assistant theorem prover to show that many "classical" rules of Slitherlink are equivalent to sequences of short whips. But this is purely cosmetic. Generally, I'm too lazy to do it manually.

To give you a slight idea of the basic questions that should be solved: how do you compare a path using 5 patterns of size 4 and a path using 3 patterns of size 6 (all the rest being singles, to make the question simpler)? (You can replace "pattern" by any fixed pattern, among your preferred ones.)

More generally, I'm not even sure the question is meaningful. I have shown in great detail that there cannot be a single hardest-step rating, but that one must choose a rating consistent with their resolution model and adapted to their purposes or personal preferences. I think this applies with still more strength to potential ratings of the global path.

As an illustration, I gave a very easy solution of the first puzzle in this thread, based on 4 or 5 elementary chains (the most basic AICs) plus a single whip[3]. Other people gave what they call "one-step" solution - using much more complicated and longer patterns. They are so artificial that I can't see by which rational standard they could be claimed simpler than mine. In any case, not just because they can be written in only one line of hermetic symbols in (pseudo-)AIC notation: this only line required pages of explanations to be understood (even by people supposed to be proficient in AIC notation).