Abox Abduction

Abduction is qualified in [EKS06] as an ampliative service since it provides for more knowledge than can be obtained deductively. For example, from the following knowledge base (Σ),

T : C vD, DvE, H vE,E vF A :∅

and a concept assertion γ : i:E.

The assertion i :F is entailed Σ∪γ |=i: F due to deductive reasoning (in every model in which i: E is true, i :F is also true). In abductive reasoning, the assertion γ : i: E might be explained through some hypothesis (∆) such that Σ ∪ ∆ |=γ holds. For the previous knowledge base, various hypotheses such as ∆₁ =i :E, ∆₂ =i:H, ∆₃ =i :D and ∆4 =i:C are possible, since for each of them Σ∪∆i |=γ holds. As can be noticed from this example, criteria for hypothesis selection is required to reduce the number of possible explanations as much as possible.

Abduction is formalized in this work as a type of non-standard retrieval inference service. More formally, for a given Abox assertion γ (observation) and a knowledge base Σ, the abductive retrieval inference service aims at deriving all sets of Abox assertions ∆ (explanations) such that the following holds

Σ∪∆|=γ (2.1)

In this way, the abductive retrieval inference service has the task of determining what should be added (hypothesized) to the knowledge base such that the formula above holds.

Remind that Σ = (O,R, A_x). As explained in Section 2.1.5, the AboxA_xof the knowledge base contains the interpretation results of a specific media object. Therefore, in the formula aboveγcorresponds to one assertion inA_x that should be explained. Determining which assertion in Ax should be explained is described in Section 2.5.4.

50 CHAPTER 2. DEEP-LEVEL INTERPRETATION OF TEXT Moreover, various ∆s can be obtained, therefore a strategy to choose preferred expla-nations should be developed. In this context, there is a set of constraints that should be considered as criteria in order to reduce the set of possible explanations to one that containspreferred explanations. The constraints are classified asmust and recommended, and are suitable in the context of media interpretation.

• Must-constraints:

– Consilience: sup(poset(∆s, λ(x, y)•S(x)> S(y))), S =| {α∈∆| O ∪A_x |=α)} |

– Informativeness: ∆ is preferred if there exists no other explanation ∆⁰ such that Σ∪∆γ with Σ = (O,R,A) and O ∪∆⁰ ∆

Elsenbroich et al. [EKS06] provide a thorough discussion of abductive reasoning tasks in DLs including Abox abduction and provide also a set of constraints which overlap with the ones presented above. The overlap applies to consistency and relevance.

Consistency is a criterion that must be fulfilled and is application independent.

Thus, according to classical logic, an inconsistent knowledge base (in this case Σ∪∆) can entail anything. In other words, without satisfiability any explanation fulfills Σ∪∆|=γ.

Relevance is considered as optional in [EKS06], but in this work it is proposed (similar to consistency) as a necessary and application-independent constraint. Relevance means that the observations (γ) only should not be the explanation (∆) (∆ 2 γ). An explanation that does not fulfill this constraint is irrelevant. For example, in a medical diagnosis scenario an irrelevant explanation is one that explains sneezing as symptom by the same sneezing as its cause.

Simplicity means that we want to minimize the number of assertions (α) in the explanation (∆) that are not entailed in (O ∪A_x 2α) and therefore are hypothesized.

From a poset of all possible explanations (∆s), ordered according to the criteria S, minimum ∆s are preferred. In other words, the criteria S of simplicity is to prefer the explanation ∆ with the least number of hypotheses necessary to explainγ.

This criterion is application-dependent. For example, for image interpretation, sim-plicity is recommended. If Figure 2.20⁶ is explained as a dinner table or as a dinner

6The image is taken from [NM06]

Figure 2.20: Which is the best interpretation? A dinner table or a dinner table for two?

table for two, the first explanation is preferred since the second explanation needs to hy-pothesize a second set of object configurations. In medical diagnosis it might be more reasonable not to fulfill simplicity since one is interested in knowing all possible explana-tions including those in which the number of hypothesis are high, since discarding some explanations (causes) due to simplicity is risky.

Consilience in this work means that we want to maximize the number of assertions in the explanation (α∈∆) that are entailed by O ∪ A_x |=α.

In other words, we want to maximize the propositions from the background knowledge that are used to create an explanation ∆, in order to explain the highest number of ob-servations γ. For example, in the context of scene interpretation the image in Figure 2.20 can be explained as a western style dinning table or as anasian style dinning table (where cutlery is not used). One would prefer western style dinning table as explanation since it considers cutlery, while the second explanation ignores the cutlery and only consider the plate and the coffee set in its explanation.

To explaininformativeness consider the background knowledge presented at the be-ginning of this section (see page 49). From the following two explanations, ∆₁ ={i:D}

and ∆₂ ={i:C}, ifO∪∆₂ |= ∆₁ then ∆₂is preferred since it is the more specific one, and there is no other ∆⁰ such thatO ∪∆⁰ |= ∆₂. The more specific an explanation is, the more restrictions on the domain are imposed. This results in a more informative explanation.

The less specific an explanation is, the less informative it is. A more generic information is more likely to be correct since less assumptions are imposed. In other words, we want to find the ∆ ∈∆s that is subsumed by all the other explanations in ∆s w.r.t.O. For exam-ple, consider again the image in Figure 2.20. Provided a background knowledge base con-taining the following axioms T ={DinnerT T vT ableT op, Breakf astT T vT ableT op}, and two interpretations ∆₁ and ∆₂. ∆₁ explains the observations as a dinner table (DinnerT T) and ∆2 explains the observations as T ableT op. ∆1 provides more

informa-52 CHAPTER 2. DEEP-LEVEL INTERPRETATION OF TEXT

Figure 2.21: Formalizing deep-level interpretation of text as Abox abduction.

tion, while ∆₂ is more generic. Thus, informally speaking, ∆₂ has a higher probability of being correct than ∆₁ since less hypotheses are required. In [HSAM93], this is called the

“informativeness-correctness trade off”.

It should be noted that informativeness and simplicity compete with each other. Thus, in the example above, if one prefers simplicity, then ∆1 is preferred since the hypothesis is minimal compared to ∆₂. Considering both criteria might seam a contradiction. For this reason a preference order for the application of criteria is required. In the implemen-tation of the abductive reasoning service, the must-constraints are first applied, followed by simplicity and consilience to obtain a preference score, and finally, the criterion for

“informativeness” is applied to further reduce the number of preferred explanations in case more than one preferred explanation exists.

In the context of the deep-level text interpretation framework (see Figure 2.21) the elements of the abduction formula are characterized as follows. Γ represents the infor-mation extracted from a text document through SLI processes. SLI processes formally encode their results as a set of Abox assertions, such that, for every word recognized in the text, a corresponding concept assertionγ is made explicit in Γ, and for every relation between words that is recognized in the text, a role assertion γ is added to Γ.

∆ represents the result of the abductive process performed by DLI, which extends the Abox obtained from SLI, with new concept and role assertions describing the content of the text document (expressed in Γ) at a deep-level. In this work the representation of deep-level content semantics is done by instantiating DLC concepts and relating such instances with SLC instances through SLR roles (see Section 2.4.2 for a short description of SLC, DLC and SLR).

In the process of Abox abuction the O and R part of the knowledge base are used as follows

• The set of rules R are used to define the space of possible explanations also called abducibles. The rules are practical mechanisms used to produce new assertions that will compose an explanation. Moreover, they provide representation means to express the configuration of an aggregate concept (to capture constraints among parts of aggregates) to cope with the expressivity limitations in the ontology part.

• The ontology O is formally represented through DLs. The T part is used to model aggregate concepts through GCIs and complex concept descriptions, called DLC concepts, and exploits reasoning mechanisms to ensure that explanations are con-sistent. Thus, the ontology helps in reducing the number or possible explanations that result from the application of rules via abduction.