Hi
Serg,
You're right, of course. I've edited the original post.
As soon as I got out the door and headed down the road, I remembered the proper definitions for strong-vs-weak minimals
The rest of what I wrote is correct, in the sense that if it doesn't match the "original defintion" (wherever that might be found (?)), it's "equivalent" to the original definition. I tried to frame it in a way that conformed to
Mathimagics' notion of "P := S - U", and it's solutions.
[ The (set of) "permutations of U (with the same footprint as U)" mentioned elsewhere, and the set of "solutions to P, restricted to the cells in U", are (of course) the same sets. ]
Serg wrote:blue wrote:Here's a proper definition.
(...)
I think UA set definition should be more complicated (see original
Red Ed's
definition).
Let's call solutions of P
unavoidable set permutations (in accordance with
Red Ed's terminology). If every cell of unavoidable set's permutation contains unique value (comparing it with all other permutations of considered unavoidable set), this permutation is called
minimal.
If an unavoidable set contains at least one minimal permutation, such unavoidable set called minimal. If all unavoidable set's permutations are minimal, unavoidable set is
strongly minimal, otherwise unavoidable set is
weakly minimal.
Everything seems correct, except the part that I underlined.
In the example(s) with valency 3, if S = (P + U), S2 = (P + V), and S3 = (P + W) are the 3 solutions to P, then V and W are unavoidable sets in S2 and S3 (respectfully). Neither one is a minimal UA for its respective grid, though, since each of them contains a (smaller) U4.
On the other hand, {U,V,W} is the set of permutations for U, and for V and W as well, and that set includes U, which
is minimal.
The "contains at least one minimal permutation" concept, then ... as applied to V, for example ... isn't a guarantee that V itself, is minimal (in the usual sense). Check the "BTW:" at the bottom of this post, if that doesn't make sense.
Serg wrote:Usually strongly minimal unavoidable set has valency 2 (contains 2 permutations), but nobody proved, that strongly minimal unavoidable sets with valency 3+ don't exist. Usually weakly minimal unavoidable set contains 1 minimal permutation only, but nobody proved, that it cannot contain 2+ minimal permutations.
All true, AFAIK ... no "non-existence" proofs, and no evidence to contradict the non-existence hypotheses.
--
BTW: You do understand (please, I hope) ... that above, {U,V,W} is a just set of permutations (of a single "seed", in principle) ... and that only U,V or W,
individually, could be an "unavoidable set" ... minimal, strong, weak, or otherwise ?
You use, twice, phrases like "
contains a minimal permutation", where "
has a minimal permuation" should appear instead.
For example: "Usually weakly minimal unavoidable set contains 1 minimal permutation only".
It should be: "Usually [a] weakly minimal unavoidable set
has 1 minimal permutation only" ... that being the set itself.
