Denis wrote: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.

Unfortunately, this proclamation may be based on an incomplete picture of what the Uniqueness assumption really brings to the table. Lets look at Denis's definition of confluence...

Definition: a resolution theory T has the confluence property if any two partial resolution paths can be completed to meet at a same knwoledge state.

Consequence: if a resolution theory T has the confluence property, then for any puzzle P there is a single final state and all the resolution paths lead to this state. In particular, if T solves P, you can't miss the solution by choosing the "wrong" rule at any time.

There is nothing wrong with Denis's definition and consequences. In fact I will use them to show that Uniqueness preserves the confluence property.

Many experts know that you cannot "destroy" a uniqueness deduction. In order to understand why that is, you have to understand what it is that Uniqueness is bringing to the table. Consider a UR in cells r12c34 which haven't been given as initial solved cells in the puzzle. What uniqueness actually tells you is that for any two digits, a & b, you cannot have a total of four a's and b's in those four cells. This is a prescription for a "weak link" for any two situations that together would imply four a's and b's in the cells. For example, there is a weak link between two a's in the cells and two b's in the cells. There is also a weak link between three a's and b's in three particular cells and either an a or b in the remaining cell. This is quite similar to two candidates representing the same digit inhibiting each other if they lie in the same house. Thus, uniqueness should be treated in the same way pattern-wise or theory-wise as the weak links in an xy-chain, etc.

These weak links are all that uniqueness is bringing to the party. So if these weak links exist both at the initial state of the puzzle and at the single final state of the puzzle then one has to conclude that uniqueness preserves the confluence property. The fact is that a puzzle with a unique solution must obey those weak links every step of the way, up to and including the final solution; and we must therefore conclude that theories using the Uniqueness axiom can preserve confluence (i.e. it won't be the uniqueness axiom that is responsible for any lack of confluence.)