Propositional Argumentation Systems and Symbolic Evidence Theory

86 6 IMPLEMENTING ASSUMPTION–BASED SYSTEMS
given normal form to an initially empty list. Each time a literal set is added,
the resulting list has to be transformed into a prime normal form. Thus,
the main problem is to adjoin a literal set to an existing prime normal form,
such that the result again is a prime normal form.
The main procedure is called generate-prime-normal-form. Its argument
nf is an arbitrary normal form that is not necessarily minimized. It computes
and returns the corresponding prime normal form. A procedure called
adjoin-literal-sets is used to sequentially adjoin all literal sets of nf to
the empty list.
PROCEDURE generate-prime-normal-form(nf);
BEGIN
RETURN adjoin-literal-sets(nf,the-empty-list);
END;
The procedure adjoin-literal-sets has two parameters: a set nf of literal
sets to be adjoined and a prime normal form pnf. To adjoin the elements
of nf sequentially, a loop over nf is needed:
PROCEDURE adjoin-literal-sets(nf,pnf);
BEGIN
FOR EACH ls IN nf DO
pnf := adjoin-literal-set(ls,pnf);
END FOR;
RETURN pnf;
END;
Finally, the main problem of adjoining a new literal set L into an existing
prime normal form πN remains. Two cases have to be distinguished:
(1) If the new literal set L subsumes one or more literal sets L ′ ∈ πN of
the existing prime normal form, i.e. if L ⊇ L ′ , then the literal set to
be adjoined is irrelevant and πN remains unchanged.
(2) If this is not the case, then the literal sets L ′ ∈ πN that subsume L,
i.e. L ′ ⊇ L, are eliminated from πN and L is added to the remaining
set. Finally, all possible resolvents ρ(L, L ′ ) found for the new literal
set are adjoined in the same way (using adjoin-literal-sets).