The New Sudoku Players' Forum

[denis_berthier]

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In [HCCS2], I announced that I'd publish "soon" all the data I computed in order to build the numerous tables in the book. Thats' now done (not as soon as I wanted, because I did some cleaning, but finally done). I've created a new GitHub repository with all the collections analysed in the book, all the results and (almost) all the commands necessary to reproduce the results.
Each directory corresponds to one of the collections and contains all the calculations for it.
The presentation is uniform: same file names for all the collections, same basic statistical analyses, same "RESULTS.clp" for each collection.
I've also added to SudoRules global variables for the names of each collection, so that you can easily reproduce all the calculations.
See the README.md file in https://github.com/denis-berthier/Sudoku-classif for all the details.

Note 1: if I analyse more collections along similar lines, this is where the detailed results will now be published.
Note 2: if you have limited disk space, the whole repository requires 1.3 GB.
Note 3: the previous two repositories (https://github.com/denis-berthier/Classifications-of-TE3-Sudokus and https://github.com/denis-berthier/Classification-of-TE2-Sudokus) have been rejuvenated and have become parts of the new one; the old versions will soon be archived.

[Edit]: OOPS, I had forgotten to make the new repository public. It's now done.
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[denis_berthier]

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Erratum:
in HCCS2, section 4.3.2, there's a typo. You should read:
let us define a 1+BRT-expand(P) as any puzzle that is the BRT-expand of a 1-expand of P.
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[denis_berthier]

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In [HCCS2] and in a previous post, I defined the set P and its subset M, and I defined the BRT-topology on them.
Continuity wrt this typology means invariance under BRT-equivalence.
All the universal classifications in [HCCS] are continuous wrt the BRT topology.

However, there's another fundamental property expected of any rational classifcation system: monotonicity wrt to puzzle expansion. I'll show that it can also be expressed as a continuity property.
This will be one more occasion to show that BRT-equivalence and (1+BRT)-expansion must be taken seriously.


1) The sets PE and ME
Define
PE as the set of BRT-equivalence classes of consistent puzzles
ME as the set of BRT-equivalence classes of minimal puzzles

By choosing the BRT-expand common to all the puzzles in a class, one can also say that:
PE is the set of self-expand puzzles
ME is the set of min-expand puzzles


1) The BRT-inc topology on the PE and ME sets
For puzzles in PE or ME, define the partial order relation by
P < Q is P is a strict sub-puzzle of Q.
This turns PE and ME into posets (partially ordered sets).

Definition:
The BRTinc topology on PE (ME) is the topology associated to their poset structure.

(This is a classical concept in maths.)
(as before "BRT" stands for the universal "Basic Resolution Theory", "inc" stands for "inclusion", "BRTinc" stands for "inclusionn modulo BRT-equivalence"

To make it short, a basis of open sets is defined by all the P- for P in PE (resp. ME):
P- = the set of self-expand [resp. min-expand] puzzles Q with Q ≤ P.



2) Fundamental result:
Theorem: continuity wrt the BRTinc topology means monotonicity (decreasing, with this choice of the basic open sets)
Proof : this is classical for the topology associated to a poset.

All the universal classifications in [HCCS] are continuous wrt the BRTinc topology.



3) Intricacies of the poset structure and the BRTinc topology
It is quite obvious that one can have long strictly descending chains in PE (see any of the expansion path in the 1+BRT-expand thread).
But it is far from obvious in ME. The reason is, one needs to find minimal puzzles of which the puzzles in the chain are BRT-expands. One already knows chains of length 1 in the T&E(3) databse, but longer lengths are not obvious.
Note that a descending chain is also a sequence of striclty decreasing non-empty open sets.

I launched a chalenge here: http://forum.enjoysudoku.com/1-brt-expansion-paths-within-t-e-n-and-beyond-t45647-85.html
In the terms of this post, the challenge can be summarised as
find a sequence of strictly decreasing non-empty open sets (with some stricter conditions)
and the answer to it is, one can indeed find quite long chains.

This shows that the BRTinc toplogy is much more subtle than one might think and that the study of (1+BRT)-expansions has still a lot to deliver.


[Edit]: note Notice that it would also make sense on P or M to define a strict partial order relation between two puzzles P and Q by: P < Q if and only if BRT-expand(P) is a strict sub-puzzle of BRT-expand(Q). But beware that this definition of P < Q wouldn’t make any difference between puzzles with identical BRT-expands and that, in the associated non-strict order, P ≤ Q wouldn’t mean that P is a sub-puzzle of Q; indeed, Q could well be a strict sub-puzzle of P, as long as they have the same BRT-expand. I shall therefore avoid to write this order relation on P or M. This is a clear example of the necessity of taking BRT-equivalence seriously.

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[denis_berthier]

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One more point of interest wrt to the above-defined topology is about its isolated points.
Theorem: a puzzle is isolated in the BRTinc topology on PE or ME iff it is a min-expand (a member of ME) and it is the common BRT-expand of all its minimal puzzles.
The proof is quite easy, knowing that the isolated puzzles in the topology associated to a poset as above are its minimal elements (in the sense of the order).

This is interesting, because it gives a topological meaning to a class of puzzles encountered by mith in his search of T&E(3) puzzles.

[Edit]: one more interesting point about isolated puzzles in T&E(3).
In the whole collection of 4,634,101 known T&E(3) minimals, there are 667,767 min-expands and 449,209 (i.e. 67.27 %) non isolated min-expands.
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