Next, we discuss the implication operator used in our semantics. As described in Sec- tion2.2, in case only the internal implication operator is used, the paraconsistent version of several tractable DLs can be translated into DLs under classical semantics where con- sequences and tractability can be preserved simultaneously, so that it is possible to use available standard reasoners. Since this property is crucial for potential implementations of our approach, we employ an implication operator (presented in Table3.1) in our se- mantics which is defined like internal implication in [MMH13] for the truth values b, t, f and u, apart from the assignment u ⊃ f2, which is no longer mapped to a designated
2Since we restrict our semantics to three truth values in the case of the ontology component, omitting the
truth value. ⊃ b st t f cf u b b t t f f f st b t t f f f t b f t f f f f t t t t t t cf t t t t t t u t f t f t t
Table 3.1: Semantics of the implication operator.
In order to justify the truth evaluation of the assignment u ⊃ f , as well as other partic- ular truth value assignments in the interpretation function of the implication operator, we have to take the minimization of models into account, which is a central component of the logic of MKNF. Knowledge minimization in the logic of MKNF enables non-monotonic default reasoning under the CWA by minimizing everything that cannot be derived to false. Therefore, the truth value f has to be the least element of the minimization order used for knowledge minimization in the semantics. As a result, the K-atom in the head of an MKNF rule with undefined body would always be minimized to f if the truth as- signment u ⊃ f was mapped to a designated truth value. However, this is not always intended as the following example shows.
Example 3.2. Consider the following ground hybrid MKNF knowledge base KG only
consisting of a program component.
KP (a) ← KQ(a) KQ(a) ← notQ(a)
In this knowledge base, KQ(a) should be assigned the truth value undefined like usual in the WFS due to the recursion through default negation. However, KP (a) should also be undefined as the only MKNF rule in which it occurs in the head has an undefined body. ♦ Consequently, only the truth value cf can be allowed for the consequent of an impli- cation with undefined implicant, and cf cannot be smaller in the minimization order of our semantics, so that KP (a) in the previous example is also not minimized to cf .
Furthermore, the minimization of models poses a problem for the propagation of inconsistencies as well. In the minimization order, there cannot be a designated truth value that is smaller than the truth value t because all facts, and the heads of rules whose body is evaluated to a designated truth value, would be minimized to this value, though they should normally be mapped to true. However, in the case of the truth value st, the same strategy we pursue in the case of cf , i.e. defining st to be greater in the minimization order and forbidding minimization to the truth value t in case a rule has an inconsistent body, does also not work as the following example shows.
Example 3.3. Consider the following ground hybrid MKNF knowledge base KG.
> v ¬P KQ(a) ← KP (a) KR(a) ← KQ(a) KP (a) ←
In this example, the modal K-atom K P (a) is clearly contradictory since the only fact in the program implies that K P (a) holds. At the same time, the classical negation of P is derivable from the ontology component. Consequently, the K-atom K Q(a) should be suspiciously true in an interpretation satisfying KG. Moreover, K R(a) should also be
mapped to st as it can only be derived by consulting inconsistent information. However, if the truth value t is smaller than st in the minimization order, K Q(a) and K R(a) are evaluated to be true in every model of KG (if this is allowed by the definition of
the implication operator). One strategy to solve this problem would be to forbid the assignment of the truth value t for the K-atom in the head of a rule with inconsistent or suspiciously true body. Yet, assume we add K Q(a) as a second fact to KG. Then, K Q(a)
can be inferred without relying on inconsistent knowledge. The same holds for adding the fact KR(a) to the knowledge base. Consequently, the assignments b ⊃ t and b ⊃ st have to be mapped to a designated truth value. Another strategy would be not to define t to be larger in the minimization order than st. In this case, the correct truth value would be assigned to KQ(a) and KR(a) in the original version of KG(without the additional facts).
Yet, there would still be a problem since all true knowledge would also be minimized to
suspiciously true. ♦
Our solution to the problem described in Example 3.3 consists in defining st to be smaller in the minimization order than b and t, but at the same time we do not define st to be a designated truth value and map implications with the truth assignment t ⊃ st to a non-designated truth value (cf. Table3.1). However, implications where the implicant is evaluated to b or st and the consequent is mapped to st in an interpretation are still satisfied by this interpretation. In this way, K-atoms which are only implied by rules with inconsistent or suspiciously true bodies are minimized to suspiciously true. Though, when the K-atom simultaneously occurs as a fact, in the head of a rule with true body, or can be derived from the ontology, it is still forced to be true (or inconsistent if its classical negation is also derivable) in an interpretation satisfying the knowledge base.
As we have discussed in Section2.3.3, Contradiction Support Detection fails in the case of W F SXpwhenever a program atom that can only be derived from contradictory
knowledge is also implied by a rule with undefined body. Because we aim to develop a semantics that corresponds to W F SXp w.r.t. the semantics assigned to the program
component alone, we also have to inhibit the propagation of inconsistencies in case a modal K-atom appears in the head of a rule whose body is interpreted to be undefined
in a model. On the basis of the previous considerations, this can easily be done by also mapping u ⊃ st to a non-designated truth value. As in the case of t ⊃ st, this inhibits the minimization to suspiciously true since the truth value st is not allowed for the head in this case (when considering interpretations satisfying these rules). In this way, the desired behavior of the program semantics can be achieved by utilizing the interaction of knowledge minimization and the definition of the implication operator.