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Composing complete and partial knowledge.
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*S. Verbaeten and A. Bossi.
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The representation of knowledge in the logic OLP-FOL is split in two parts:
writing definitions for known concepts, and writing constraints, expressing
partial knowledge on other concepts. An OLP-FOL theory Th =(Th_{d}, Th_{c}),
divides the set of predicate symbols in two disjoint subsets: the defined
predicates, which occur in the head of a clause of Th_{d}, and the open
predicates, which occur at the most in the body of the clauses of Th_{d}. The
definition part Th _{d} contains the definitions for known predicates in the
form of a normal open logic program (OLP), whereas the first order logic
(FOL) part Th _{c} is a set of FOL axioms, expressing partial knowledge on
other predicates. The semantics of OLP-FOL is a generalisation of the
well-founded semantics.

In previous works, the composition of two OLP-FOL theories, with non-intersecting sets of defined predicate symbols, was studied. It was argued that their composition is given by the set of common models. Here, we investigate the possibility of composing two OLP-FOL theories, which define the same predicate. Therefore, we introduce two operators on theories: the p-opening operator, which opens the definition of the predicate p in a theory completely, and the conditional p-opening operator, which maintains the definition of p in a theory if a certain condition holds, and opens p in the other cases. We show that we can compose two theories, which both have an open definition for the same predicate, or which both have a conditional open definition for the same predicate, with non-overlapping conditions.