There are several proposals to deal with inconsistencies in DL ontologies through argumentation. Di erent from existing approaches, in this paper, we consider the scenario that our knowledge is both uncertain and inconsistent and/or incoherent, and we propose a logic-based argumentation framework to deal with incomplete and con icting DL ontologies. We do so by adopting a distinct notion of attack [3, 4] among arguments to encompass di erent forms of con icts in DL ontologies. The paper presents the following major contributions: (1) a general framework for reasoning with uncertain, inconsistent and/or incoherent ontologies with the use of logic-based argumentation; (2) a general labelling method, sensitive to the numbers of attacks and the weights of arguments, with di erent interesting instantiations to identify the justi cation statuses of each argument; and (3) a number of inference relations derived from our framework in order to obtain meaningful answers without increasing the computational complexity of the reasoning process compared to classical DL reasoning. We also study the logical properties of these new entailment relations. We consider, in this abstract, the possibilistic ALC , denoted by ALC , as an adaptation of ALC within a possibility theory setting [5]. A possibilistic axiom is a pair ( ; w) where is an axiom and w 2 [0; 1] is a weight for the con dence degree of . We say that an ALC ontology O is con ict-free if O is consistent and coherent. The maximal con ict-free subontologies of O are de ned as MC(O) = fO1 O j O1 is con ict-free and 8 O O2 O1; O2 is not con ict-freeg. De nition 1. A prudent argument for an axiom w.r.t. an ALC ontology O is a triple h ; ; !i such that (1) is coherent, (2) is a justi cation for w.r.t. O 0, where O 0 = f j ( ; !) 2 Og, and (3) ! = minf!i j ( i; !i) 2 g. De nition 2. The argument structure for an axiom is a pair of sets hP; Si with P the set of argumentation trees [3] for and S the set of argumentation trees for : . De nition 3 (`8MC, `9MC, `nMoC, `A). Given an ALC We rst extend some classical inference relations proposed by [2, 1] to ALC . ontology O and an axiom >8(R; a) if a > 0 Lab(A) = <(A; a) if a < 0 >:U if a = 0
{ O `8MC { O `9MC
no { O `MC { O `A if MC(O) 6= ; and for every O0 2 MC(O), O0 ` ; if there exists O0 2 MC(O) s.t. O0 ` ; if O `9MC and 8 O0 2 MC(O), 6./ O0; if there exists an argument for , and there is no argument for : in O.
Next, we introduce several notions of consequence relations in the light of the
argument structure and the labelling functions. A labelling function is to decide the state
of an argumentation tree among accepted (A), rejected (R), or undecided (U ). Let T
be an argumentation tree for an axiom and A = h ; ; !i be a prudent argument
in T . A prudent argument labelling is a total function Lab : A ! f(A; !); (R; !); U g
de ned as follows: (
Jabbour et al. where a = f (N1; : : : ; Nm) for Ni(1 i m) 2 C(A) is computed by a function f : An 7! [ 1; 1].
A possible initialization of f is by the normalization of the di erence between the number of the rejected children of a node and that of the accepted ones. The intuition is that if a prudent argument is attacked by more accepted defeaters than those labelled as rejected, this argument should be rejected because its defeaters are more often accepted. Otherwise, it can be accepted, unless if it has the same number of accepted and rejected defeaters where its labelling should be U . Due to space limit, we do not initialize f in this abstract.
First, for a given argument structure hP; Si for w.r.t. an ALC ontology O, let us consider the following conditions: C1. P 6= ; and S = ;.
C2. 9 T 2 P, J udge(T ) = (W arranted; !).
C3. 8 T 2 P, J udge(T ) = (W arranted; !), and P 6= ;.
C4. maxT 2P f! j J udge(T ) = (W arranted; !)g d, d 2 [0; 1].
C5. 8 T 0 2 S, J udge(T 0) = (U nwarranted; !0).
C6. 8 T 2 P, J udge(T ) 6= (U nwarranted; !).
De nition 4 (`c; `s). We say an axiom is credulously (resp. skeptically) inferred from an ontology O with degree d, denoted O `c ( ; d) (resp. O `s ( ; d)), i . the argument structure hP; Si for satis es C1, C2, and C4 (resp. C1, C3, and C4).
It is important to stress that the above inference relations `A; `c; and `s for a conclusion are conservative, hence rather unproductive. To relax such constraint, we propose in the following another reasoning type via three logical consequence relations, namely `a8rg, `arg and `anrog.
9 De nition 5 (`a8rg; `a9rg; O `anrog). Given an ALC ontology O and an axiom
Proposition 1. We have `c; `s, `A, `8MC, `9MC, `nMoC, `a8rg; `a9rg; and `anrog satisfy Soundness; `c,`s, `A, `8MC; `nMoC, `a8rg, `a9rg, and `anrog satisfy Consistency. And, `c; `s, `a8rg, `a9rg, and `anrog satisfy Monotonicity w.r.t. degree.
Proposition 2. The productivity comparison among the nine inference relations is given bellow. A ) B means the entailement relation A is more productive than B.
no `arg `a9rg `a8rg `9MC `c `s `A `8MC
no `MC Proposition 3. For all the inference relations in the gure above, the entailment problem is in EXPTIME.