Propositional Argumentation Systems and Symbolic Evidence Theory

84 6 IMPLEMENTING ASSUMPTION–BASED SYSTEMS
If N 1 and N 2 are CNFs, then the union N 1 ∪ N 2 has to be interpreted as
conjunction N 1 ∧ N 2 . In contrast, if N 1 and N 2 are DNFs, then N 1 ∪ N 2
corresponds to the disjunction N 1 ∨ N 2 .
The intersection of two normal forms N 1 and N 2 is obtained according to
the following equation (Davey & Priestley, 1990; Ngair, 1992):
µ(N 1 ∩ N 2 ) = µ{L 1 ∪ L 2 : L 1 ∈ N 1 , L 2 ∈ N 2 } (6.19)
Thus, intersection corresponds to expansion. Two embedded loops and a
procedure ls-union that computes the union of two literal sets according
to Equation 6.11 are needed to implement the intersection:
PROCEDURE nf-intersection(nf1,nf2);
VAR result := the-empty-list;
BEGIN
FOR EACH ls1 IN nf1 DO
FOR EACH ls2 IN nf2 DO
result := add-element(ls-union(ls1,ls2),result);
END FOR;
END FOR;
RETURN eliminate-subsuming-literal-sets(result);
END;
Again, the interpretation of this procedure depends on the interpretation
of the given normal forms: if N 1 and N 2 are CNFs, then the intersection
N 1 ∩ N 2 corresponds to N 1 ∨ N 2 ; if N 1 and N 2 are DNFs, then N 1 ∩ N 2 has
to be interpreted as N 1 ∧ N 2 .
Using the procedures nf-union and nf-intersection, it is possible to compute
conjunctions and disjunctions of CNFs and DNFs. Similarly, other
operations for normal forms like negation, projection, transformation from
CNF into DNF and vice versa, etc. can easily be implemented. An extensive
workbench for propositional logic exists as Common Lisp program. It assumes
a fixed order of the propositional symbols and uses the bit–sequence–
representation from Subsection 6.2.1. The toolbox includes the procedures
described above, as well as a variety of other helpful routines. It can be
considered as the kernel of Evidenzia.