Dealing with inconsistencies is a long time studied topic. The current paper introduces a mechanism for reasoning with conflicts in knowledge bases. The conflicts are addressed through a form of logicbased argumentation. We explore argument structures to define various inference relations that can rationally draw meaningful answers to queries even from inconsistent knowledge bases. We investigate the productivity of these entailments and their relationships with other well-known existing relations.
Additionally, it might be possible to construct only a part of the tree related to the conclusion in question, thus having a better representation.
After briefly introducing the necessary notions in Section 2, we study in Section 3 an approach that can be employed to define multiple conflict-tolerant inference relations in order to draw meaningful answers, while preserving the non-conflicts of the derived conclusions. We also study the productivity landscape of these new entailment relations.
In this section, we briefly sketch some basics in propositional logic and deductive argumentation. We assume an arbitrary finite set of propositional variables . We use the set with the classical connectives (¬, ∨, ∧, →) as well as the two constants ⊤ (true) and ⊥ (false) to build in the usual way the propositional language ℒ(). Well-formed formulas from ℒ() are denoted by Greek letters , , , etc. We also denote by ⊢ the classical consequence relation. Two formulas , ∈ ℒ() are called equivalent, denoted as usual by ≡ , if { } ⊢ and { } ⊢ . A knowledge base is a finite set of propositional formulas. We write K to denote the set of all knowledge bases built over . Also, is said to be inconsistent if there exists a formula such that ⊢ and ⊢ ¬ . Otherwise, is consistent.
Given a knowledge base ∈ K , a subset ⊆ is a maximal consistent set (MC, in short) of if is consistent and ∀ ∈ ∖ , ∪ { } is inconsistent.
The following consequence relations are typically defined on the basis of maximal consistent subsets of a given knowledge base ∈ K : Definition 1. ⊢∀MC if for all ∈ (), ⊢ . ⊢∃ if for some ∈ (), ⊢ .
Now, we introduce the logical argumentation based on propositional logic studied by Besnard
and Hunter [
Definition 2. For ∈ K , an argument is a pair ⟨Φ , ⟩ where Φ ⊬ ⊥, Φ ⊆ , Φ ⊢ and ∄Φ ′ ⊂ Φ s.t. Φ ′ ⊢ .
Definition 3. ⊢ if there exists an argument for in , and there is no argument for ¬ . Arguments are not necessarily independent in the sense that an argument can be contained within another one. Formally, ⟨Φ , ⟩ is more conservative than an argument ⟨Ψ , ⟩ if Φ ⊆ Ψ and ⊢ .
Since information may be conflicting, arguments may also be conflicting among them. Various
attack relations based on inconsistency have been studied (see [
Next, we assume that there exists a canonical enumeration of all maximally conservative undercuts for an argument ⟨Φ , ⟩. Such canonical enumeration allows us to avoid some amount of redundancy among counter-arguments.
Definition 4. ⟨Ψ , ¬( 1 ∧ . . . ∧ )⟩ is a canonical undercut of ⟨Φ , ⟩ if ⟨Ψ , ¬( 1 ∧ . . . ∧ )⟩ is a maximal conservative undercut of ⟨Φ , ⟩ and ⟨ 1, . . . , ⟩ is the canonical enumeration of Φ .
To merge arguments and counter-arguments for/against a given conclusion, the authors of
[
As many diferent argument trees for a given claim can exist, one can represent all these trees in a global manner by using the argument structure defined as follows: Given a formula , the argument structure for is a pair of sets ⟨, ⟩ s.t. (resp. ) is the set of argument trees for (resp. ¬ ). 3. Novel Argumentation based Inference Relations Given an argument tree, we want to determine whether the root argument (i.e., goal) wins (i.e., it is undefeated) or whether it loses (i.e., it is defeated), or undecided. This is called argument labelling. Let be an argumentation tree for a conclusion and A = ⟨Φ , ⟩ be an argument in . An argument labelling is a total function : A → {′, ℛ, }.
Various studies in literature rely on some given qualitative or quantitative information to
determine, or give more refined accounts of, the justification status of arguments. Other works
refine the notion of acceptability based on the number of (counter)-attacks on arguments (see
[
C1. ̸= ∅ and = ∅. C2. ∃ ∈ , ( ) = . C3. ∀ ∈ , ( ) = , and ̸= ∅. C4.∀ ∈ , ( ) = .
C5. ∀ ∈ , ( ) ̸= .
C6. ⋂︀=1 Φ |= , where = {1, . . . , } with A() = ⟨Φ , ⟩, and () = for 1 ≤ ≤ .
C7. ∃ ∈ with A( ) = ⟨Φ , ⟩, and Φ ⊆ ⋂︀∈MC() , ( ) = . C8. ∀ ∈ with A( ) = ⟨Φ , ⟩ and Φ ⊆ ⋂︀∈MC() ̸= ∅, ( ) = .
These logical constraints give rise to various entailment relations based on argumentation
which are all conflict-tolerant. Next, we discuss two sorts of semantics: the family of classical
semantics and the family of IAR semantics. Other inference relation that approximates the IAR
argumentation semantics, the cardinality based semantics, and the agent preferences based
semantics can be further defined as in [
Now, we revisit various classical inference relations in light of the previous constraints over argument structures. For the first type of reasoning, we suggest that a conclusion follows from a knowledge base if the latter has an argument structure that supports this conclusion but no no ⊢arg
∃ ⊢arg argument structure against that conclusion. Let be a knowledge base and is a formula. We say is credulously (resp. skeptically) entailed from , denoted ⊢ (resp. ⊢ ), if the argument structure ⟨, ⟩ for satisfies C1 and C2 (resp. C1 and C3).
Note that the above inference relations ⊢, ⊢, and ⊢ for a conclusion are conservative due to the requirement that there must be no argument against , hence rather unproductive. To relax such constraint, we propose in the following another reasoning type via four logical consequence relations, namely ⊢a∀rg, ⊢a∃rg, ⊢anrog, and ⊢IaArgR.
Definition 5 ( ⊢a∀rg). We say ⊢a∀rg if the argument structure ⟨, ⟩ for satisfies C3 and C4. Definition 6 ( ⊢a∃rg). We say ⊢a∃rg if the argument structure ⟨, ⟩ for satisfies C2 and C4.
The intuition behind ⊢a∃rg is that there exists at least one warranted argumentation tree for . Both ⊢a∃rg and ⊢a∀rg require that the arguments against should be labelled as rejected.
Next, we investigate a new argumentative inference relation based on the notion of
nonobjection [
Definition 7 ( ⊢anrog). ⊢anrog if the argument structure ⟨, ⟩ for satisfies C2, C4 and C5.
Other types of entailment relations may be defined by imposing further constraints on the argument structure. The following three conflict-tolerant entailments evaluate queries over the supports in the root nodes of the argumentation trees for the considered query. This gives rise to a family of IAR argumentation semantics defined as follows:
The first inference relation, called ⊢IaArgR, is based on the intuition that a valid conclusion has to be a consequence of the intersection of the supports of the root nodes of all accepted argumentation trees.
Definition 8 ( ⊢IaArgR). ⊢IaArgR if the argument structure ⟨, ⟩ for satisfies C4 and C6.
To relax the quasi-equivalence requirement among the root nodes of warranted argument trees, we focus on two variants of IAR argumentation semantics defined through the selection of warranted argument trees, for a formula, that belong to the intersection of maximal consistent sets of . Formally:
IAR ⊢arg,∀
IAR ⊢arg,∃ ⊢IMACR Definition 9 ( ⊢IaArgR,∃). ⊢IaArgR,∃ if the argument structure ⟨, ⟩ for satisfies C4 and C7. Definition 10 ( ⊢IaArgR,∀). ⊢IaArgR,∀ if the argument structure ⟨, ⟩ for satisfies C4 and C8. Intuitively, the ⊢IaArgR,∃ semantics requires that the intersection of all the maximal conflict-free sub-bases has to contain a warranted argument for . The ⊢IaArgR,∀ semantics is based on the idea that a given conclusion can be considered as valid if the support of each root node for is involved in the intersection of all the maximal conflict-free sub-bases.
Properties. Figure 1 and 2 show the productivity comparison among the diferent inference
relations presented above. The proofs are omitted due to space limitations and can be found in
[
Discussion. This paper presents several fine-grained inference relations based on argument structure and labelling functions and their comparison. We will explore more aspects of argument semantics, e.g. ranking semantics or extensions, to investigate new inference relations.