Propositional Argumentation Systems and Symbolic Evidence Theory

96 6 IMPLEMENTING ASSUMPTION–BASED SYSTEMS
WHEN NOT subset?(ps,new-ps)
result := add-element(ps,result);
END FOR;
RETURN add-element(new-ps,result);
END;
After µD P is computed, the decomposition of the knowledge base can be
started. Two embedded loops are needed to distribute the clauses among
the proposition sets of µD P :
PROCEDURE decompose(nf,p);
VAR dp,nf1,nf2,result := the-empty-list;
BEGIN
dp := generate-dp(nf,p);
FOR EACH ps IN dp DO
FOR EACH ls IN nf DO
WHEN subset?(propositions(ls,p),ps)
THEN nf1 := add-element(ls,nf1);
ELSE nf2 := add-element(ls,nf2);
END FOR;
result := add-element(nf1,result);
nf := nf2;
nf1 := the-empty-list;
nf2 := the-empty-list;
END FOR;
RETURN result;
END;
The hypergraph obtained from this procedure meets restrictions (1), (2),
and (3), but it disregards restriction (4). Possibly, the hypergraph may
have cycles. Therefore, it does not automatically form a hypertree. In this
case, a covering hypertree has to be found. Several algorithms exist for
that purpose. The method implemented in Evidenzia is called Tree–Alg
(Kohlas & Monney, 1995). The idea is to compute the cliques of a two–
section graph obtained from a perfect elimination sequence. To find good
elimination sequences, a heuristic called Maximum Cardinality Search
(MCS) (Tarjan & Yannakakis, 1984) is used. In general, results obtained
from MCS are satisfactory.
It is also possible to apply Tree-Alg and MCS on the hypergraph given
by the trivial decomposition D Σ = {{ξ 1 }, . . . , {ξ s }}. The resulting hypertree
determines another possible decomposition. In fact, the two–section