A final state is a state in which no rule can be applied. Normally, most puzzles with a unique solution should have a single final state, if T is sufficiently powerful: the solution. But this is not always the case, depending on T.

I believe I see where the arguments are coming from.

Are you attempting to have a rating in general for all puzzles? I don’t think you should be looking at all puzzles that are unique or not.

It should be a rating system based solely on Unique Solutions. (1 possible final grid state).

This is a direct requirement for any “Puzzle” to be considered a “Puzzle” by definition of what a puzzle is considered.

Sudokus to the public: are not a mathematically 9 dimensional constraints allocations problem and mathematician’s bequeth uniqueness due to the mathematical speaking affects of such patterns. If a puzzle is none unique uniqueness patterns will remove some of the possible solutions from the total count.

A player is not trying to prove mathematically a given problem at hand has 1 pattern only. The creator of the grid should be the one verifying this.

We are told in the beginning that “Sudoku” is a “Puzzle”.

now find it given the basic lay out of constraints. (formulate patterns and resolution techniques that work)

To the public in general it is by definitions a “puzzle”. Must have 1 solution!

Any rating program should test that the grid is firstly a unique completion. If it is not a single solution don’t rate it. As it is no longer a “puzzle” but instead it is a mathematical problem of how many solutions does it contain?

most puzzles with a unique solution should have a single final state

All puzzles with a unique solution have only 1 final state.

In practice, this means that, if you have missed some elimination while you were using a more complex rule (to assert values or eliminate candidates), the rules in T will always allow you to do this elimination later.

One unpleasant property of the rules based on the axiom of Uniqueness is that any theory which contains any of them looses the confluence property. BUGs or URs may be applied in some knowledge state but not later.

You just caught your self in your own Confluent rules as well with these paragraphs.

I have had puzzles that involve: XY-wings. Sword-fish (variants), ALS-XY, and a few other moves at one stage of solving where valid. But a found a more complex eliminations that solved some singles. I chose the more complex manipulation:

But the affect of the move I chose disabled the application of all the aforementioned moves.

The candidate state changes, but certain techniques become no longer applicable; the rules of “T” allow specific candidates to be removed by a choice of a solver.

Variance in technique(choice) is what always changes to remove that candidate later on.

Even normal techniques become no longer valid.

In particular, if T solves P, you can't miss the solution by choosing the "wrong" rule at any time.

Correct any combination of moves that solves a puzzle.

The specific removals are the parts of “T” that can be in any order.

(It’s a question of what kinds of moves are valid at each stage of solving)

{T}+ {T}… = P

If you are generalizing the above to every problem may not be unique then uniqueness should not be included for mathematical reasons, this I can agree with but I do contest your quote of that it is not “pure logic” as I mentioned below they are constructed logically but used as a limitation.

But I do believe that sudoku is a “Puzzle” and being such shall have only 1 solution

{T}+ {T}… = P (unique)

P is limited to being 1.

Thus:

T can be any move including Uniqueness limitation arguments.

To me specific moves like muti -coloring x-wings and some other patterns are actually zero placement eliminations where the cover set of the pattern is locked to prevent the zero, but the contradiction viewed is the elimination placed forces a zero the pattern to display a zero solution or multiple candidate on a field (B,R,C). therefor any basic pattern rule could also be seen as a confluent as well by avoiding the zeros or contradictions states.

what makes thes rules any better if the pattern could actually incorrectly reduce a gird to a contradiction state when it is actually a zero state instead? that is confluente as well.

i have seen this happen in testing mutiople solution grids.

…And there's no guarantee that another rule in T will do the job in their stead

This is incorrect as well. Any given move can be replicated with a more complex technique later when a removal eliminates that simpler move earlier on.

Edit: {But depending, If I miss interpret this and you are in fact saying that all sudoku problems are not unique, and should be proved that they are instead a “puzzle”, where mathematically uniqueness arguments may remove possible solutions from a final solution count, or potentially leave a multiple solution grid with zero solutions.}

{In this alternative view point I can agree that mathematically uniqueness arguments remove clues other wise non removable when proving that it is a unique solution. } end Edit}

Meaning that Specific incorrect placements are always removed {question of how}, in the path leading to completion {P}

Examples if you want them are from the same page you quoted me on. A more complex chain instead of the bug, mugs, URs is required to solve.

http://forum.enjoysudoku.com/viewtopic.php?t=3907&postdays=0&postorder=asc&start=0,[quote]

The motives for one's choice may be varied but they can't be based on pure logic. My personal main motives for not using such rules are:[quote]

It is pure logic,

One can logically formulate arrange of clues with x many cells and create a variance of patterns that would always yield 2+ solutions in all arrangements of combinations.

Example: {basic UR}

Ab –ab

| |

ab – ab

2 formulated solutions from all points of view. Constructed logically

Then apply this directly to the definition of a “Puzzle” must have 1 solution.

Thus logically all the arrangements of candidates yielding to 2 solutions are invalid; as these are logically a limitation of being unique.

Ab –ab

| |

ab – abx--------- Ax

Being a “puzzle” l it cannot have any pattern that leaves more than one solution thus logically puzzle must have limitations of uniqueness Leaves this as the only valid path.

Ab –ab

| |

B – x--------- A

For your point of view Denis:

If I listed more path diversity a player could solve this pattern with an alternative move if they wish to.

{This is where I say there is variance in applicable techniques to apply at any given candidate state.}

Some do, some teach, the rest look it up.