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In order to formulate the constraints that will only license tree-shaped solved forms, we must first consider each individual case for
. For each case
and its negation
, we will formulate characteristic constraints involving the set variables that we introduced above.
Let's consider the case for which a solution looks as shown below:
For convenience, we define, for each variable , the additional set variables
and
as follows:
We write for the constraint characteristic of case
and define it as follows:
I.e. all variables equal or below are below
, all variables equal or above
are above
, and all variables disjoint from
are also disjoint from
. This illustrates how set constraints permit to succinctly express certain patterns of inference. Namely
precisely expresses:
The negation is somewhat simpler and states that no variable equal to is above
, and no variable equal to
is below
. Remember that
expresses that
and
are disjoint.
We can define the other cases similarly. Thus :
and its negation :
For the case we first introduce notation. We write
for the tuple defined as follows:
where when the constraint
occurs in
(more about this when presenting the problem-specific constraints). Now we can simply define
as:
and its negation as:
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