The relation is called Substitutable Values or SV relation iff the following three propositions hold:
Any SV relation is reflexive, symmetric and transitive thus being an equivalence relation.
Having a weak order preference as equivalence relation this prerequisite yields the so-called REGULAR SV semantics.
Preference SQL uses REGULAR SV semantics by default.
If the identy relation as smallest equivalence relation is chosen the identity yields the so-called TRIVIAL SV semantics.
Since only x = x is valid the propositions (1)-(3) get trivial.
User-defined SV semantics is enabled by clauses of the kind like SV((valuei, ... , valuej), ... ,(valuex, ... , valuey)) by postponing this clause at the end of any preference clause. The ANTICHAIN preference constructor in combination with complex preference constructors offers the easiest way to demonstrate the effects of user-defined SV semantics.
by attaching this clause behind any grouping attribute. Also each newGroupingAttribute must have a unique attribute name. Obviously, aggregation functions in conjunction with newGroupingAttribute take care of the user-defined SV semantics.
SV((value i, ... , value j) [AS 'Class_1'] , ... ,
(value x, ... , value y) [AS 'Class_n'] ) AS newGroupingAttribute
By using the REGULARIZED keyword behind any complex preference in brackets, the outcome of the complex preference is inherenty transformed to a layered preference . Each layer corresponds to a BMOi which is generated by a muli-level BMO evaluation ( EP15for the special case of a Pareto preference) where BMO0 is the standard BMO set . Obviously, the level function of each layer is defined by f(x) = i. For simplicity, the REGULARIZED keyword is at present restricted to just the combintation of (p1) REGULARIZED PRIOR TO (p2) ... REGULARIZED PRIOR TO (pn). Thus, this special case of regularization is implemented as a LAYERED1 preference, because the generation of BMO0 is sufficient and correct with respect to prioritization.
Indifferent values, which are not substitutable values, are called Alternative Values.