Preference classification

One of the possible reasoning task involving partial preference orders consists classify a given w. p. s.  $\langle\preccurlyeq,\prec\rangle$ w.r.t. the axioms seen above

We solve this problem by designing an ASP-specification.

We start by defining in ASP the predicates prec($\cdot$,$\cdot$), precneq($\cdot$,$\cdot$), and equiv($\cdot$,$\cdot$), to render the relators $\preccurlyeq$, $\prec$, and $\sim$, resp.

Moreover, the fact of ``being an event'' (i.e. a member of $\mathcal{E}$) is stated through the monadic predicate event($\cdot$).

Auxiliary predicates/functions are used to render set-theoretical construcs, such as $\cap$, $\cup$, and $\subseteq$, whose implementations rely on user-defined C-libraries.

The characterization of potential legal answer sets is done by asserting properties of prec($\cdot$,$\cdot$), precneq($\cdot$,$\cdot$), and equiv($\cdot$,$\cdot$), as follows:
prec(A,B) :- event(A;B), equiv(A,B).
prec(B,A) :- event(A;B), equiv(A,B).
:- precneq(A,B), event(A;B), equiv(A,B).
prec(A,B) :- event(A;B), precneq(A,B).
equiv(A,B) :- event(A;B), prec(B,A), prec(A,B).

Also axioms (A1')-(A3') must be imposed. For instance (A3') is rendered by weeding out all answer sets where $A\subseteq B \wedge B\prec A$ holds for some events $A$ and $B$:

:- event(A;B), subset(A,B), precneq(B,A).

Consider now one of the axioms, say (B'), for simplicity. To test its satisfaction, we use a rule of the form:

failsB1 :- event(X;Y;Z;W), subset(X,Y), X!=Y, subset(Z,W), Z!=W, subset(W,Y), W!=Y, subset((W$\setminus$Z),(Y$\setminus$X)), equiv(X,Y), precneq(Z,W).

Having in mind (B'), this clause is of immediate reading: the fact failsB1 is true (i.e. belongs to the answer set) whenever there exist events falsifying (B'). All other axioms have been treated similarly.

When smodels is fed with such program and a description of an input preference relation, different outcomes may be obtained:

a)

If no answer set is produced, then the input weak preference structure violates some basic requirement, such as axioms (A1')-(A3').

b)

Otherwise, if an answer set is generated, there exists a numerical (partial) model representing the input weak preference structure. Moreover, the presence in the answer set of a fact of the form fails$\mathscr{C}$ (say failsL1, for example), witnesses that the corresponding axiom ((L') in the case) is violated by the given preference order.