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Justification Oriented Proofs In Justification Oriented Proofs in OWL Matthew Horridge, Bijan Parsia, and Ulrike Sattler School of Computer Science, The University of Manchester Abstract. Justifications — that is, minimal entailing subsets of an on- tology — are currently the dominant form of explanation provided by ontology engineering environments, especially those focused on the Web Ontology Language (OWL). Despite this, there are naturally occurring justifications that can be very difficult to understand. In essence, justifi- cations are merely the premises of a proof and, as such, do not articulate the (often non-obvious) reasoning which connect those premises with the conclusion. This paper presents justification oriented proofs as a poten- tial solution to this problem. 1 Introduction and Motivation Modern ontology development environments such as Prot´eg´e-4, the NeOn Toolkit, Swoop, and Top Braid Composer, allow users to request explanations for entail- ments (inferences) that they encounter when editing or browsing ontologies. In- deed, the provision of explanation generating functionality is generally seen as being a vital component in such tools. Over the last few years, justifications have become the dominant form of explanation in these tools. This paper examines justifications as a kind of explanation and highlights some problems with them. It then presents justification lemmatisation as a non-standard reasoning service, which can be used to augment a justification with intermediate inference steps, and gives rise to a structure known as a justification oriented proof. Ultimately, a justification oriented proof could be used as an input into some presentation de- vice to help a person step though a justification that is otherwise too difficult for them to understand. 1.1 Justifications as Explanations A justification is a minimal subset of an ontology (a set of axioms) that is sufficient for a given entailment to hold. As an example, consider the small ontology O = {A B,A(i),C D},whichentails B(i), written O|= B(i)1.A justification J for O|= B(i)isaminimal subset of O that entails B(i), in this case J = {A B,A(i)}. The major benefit of justifications is that they pinpoint and isolate the hand- fuls of axioms, in what could be a very large ontology, that cause the entailment 1 B(i)meansi is an instance of B. P.F. Patel-Schneider et al. (Eds.): ISWC 2010, Part I, LNCS 6496, pp. 354–369, 2010. c Springer-Verlag Berlin Heidelberg 2010 Justification Oriented Proofs in OWL 355 InverseProperties(hasPet, isPetOf) isPetOf(Rex, Mick) Domain(hasPet, Person) Male(Mick) reads(Mick, DailyMirror) drives(Mick, Q123ABC) Van(Q123ABC) Van Vehicle WhiteThing(Q123ABC) Driver ≡ Person ∃drives.Vehicle Person ¬Movie Driver Adult RRated CatMovie Man ≡ Adult Male Person CatMovie Movie WhiteVanMan ≡ Man ∃drives.(Van WhiteThing) RRated ≡ (∃hasScript.ThrillerScript) WhiteVanMan ∀reads.Tabloid (∀hasViolenceLevel.High) Tabloid Newspaper Domain(hasViolenceLevel, Movie) Fig. 1. A justification for Person ⊥ Fig. 2. A justification for Newspaper(DailyMirror) to hold. For example, the SNOMED medical ontology contains roughly 400,000 axioms, but a justification for an entailment in this ontology is on average less than ten axioms in size [2]. Unlike full blown proofs, justifications are conceptually simple structures with a natural relation to the ontology development process—they are directly related to what has been asserted or stated in an ontology. Justifications require very little additional knowledge beyond the semantics of the language. This concep- tual simplicity, coupled with the fact that the computation of justifications for real ontologies tends to be practical [10], and the fact that off-the-shelf imple- mentations of justification finding services exist, has most likely lead to the large uptake of justifications as a type of explanation. 1.2 Problems with Justifications However, despite the fact that justifications are a popular form of explanation in the OWL world, observations show there are justifications that people find difficult or impossible to understand. Indeed, the justifications shown in Figure 1 and Figure 2, both from real ontologies, gave many users trouble when trying to understand how they lead to their respective entailments. Indeed, some people questioned whether the justification shown in Figure 1 was a justification at all. In the case of the justification shown in Figure 1, which is a justification for Person ⊥2 spotting that the justification entails Movie ≡3 is key to under- standing how the justification works. Since everything is entailed to be a Movie, and Person is disjoint with Movie, Person is disjoint with , hence Person is 2 ⊥ is read as “bottom” and is the same as owl:Nothing. 3 is read as “top” and is the same as owl:Thing. 356 M. Horridge, B. Parsia, and U. Sattler unsatisfiable. People who fail to realise that Movie ≡is also entailed generally fail to understand how the justification gives rise to the entailment. Similarly, the justification shown in Figure 2, is also rather difficult for people to work through. There are fifteen axioms of many different types in the justifica- tion. It is far from obvious how these axioms interplay with each other to result in the entailment Tabloid(DailyMirror). When a user works through this justification, they have to spot intermediate entailments, for example, WhiteVanMan(Mick)and Person(Mick), in order to arrive at the conclusion Tabloid(DailyMirror)). In a exploratory study [5], it was observed that many justifications for entail- ments of interest in real ontologies can be understood by people with a variety of backgrounds, and these kinds of justifications serve extremely well as explana- tions. However, it was also observed that there are justifications that are difficult or impossible for people to work through. Two obvious reasons for this are: (1) People do not spot key entailments within justifications, that are necessary for them to understand how the justification works (as is the case with the justifi- cation in Figure 1, and (2) People find large justifications, with many types of axioms, tedious and therefore difficult to work through (as is the case with the justification in Figure 2). In other words, when people fail, or find it difficult, to spot intermediate entailments, conclusions or steps they can fail to under- stand why a justification supports the entailment in question, and hence fail to understand why the entailment in question holds in their ontology. 1.3 From Justifications towards Proofs The above notion of “intermediate steps” that could guide a person through understanding a justification, raises the question of whether full blown proofs, such as natural deduction style proofs with inference rules, should be used for explaining entailments in OWL ontologies. One of the typical claims about natural deduction is that it mimics human reasoning—that is, it has a strong cognitive adequacy [18]. However, there is ongoing debate in the field of cognitive psychology about how human reasoning actually works. Some camps favour a “logic” or rule based account [16], while others favour a “model” based account [8]. Even for simple cases of natural language based deduction, it is unclear which account is correct. Moreover, other research [14] shows that relatively untrained people—clearly without having a complete set of deduction rules at their disposal—can successfully work through surprisingly complex reasoning puzzles. It is therefore impossible to say whether or not natural deduction and similar proof systems mimic human reasoning. What is clear, is that representations that have a strong cognitive adequacy are not necessarily useable. Hence, even if natural deduction has a strong cognitive adequacy, there is no guarantee that it is usable as a form of explanation for entailments in ontologies. In summary, it is likely that natural deduction style proofs are not necessarily the best form of explanation. On the other hand, justifications are an appealing type of explanation. It is known that that a wide range of people can cope with justifications [5]. This includes domain experts who have very little training or Justification Oriented Proofs in OWL 357 background with the Description Logics that underpin OWL. Justifications ap- peal to these kinds of people because they are conceptually simple—very little training is needed in order to understand how justifications work. The same can- not be said about natural deduction style proofs. Additionally, people are used to seeing axioms, albeit in a frame-based style of presentation, and justifications reflect this familiarity. If natural deduction style proofs were presented to people such as domain experts, they would require special training in order to read the proofs. 1.4 Justification Oriented Proofs What is needed, is something that lies between justifications and proofs. Given the popularity and conceptual simplicity of justifications, the work presented in this paper uses them as building blocks for structures that begin to look like proofs, but are independent of any calculus or deduction rules. In essence, inter- mediate steps are introduced into a justification, which are themselves explained with justifications. This results in a directed acyclic proof graph of the form shown in Figure 3. Ultimately a justification is extended with “lemmas” into a justification oriented proof. The main idea behind a justification oriented proof is depicted in Figure 3. The numbered lozenges represent axioms, with the leftmost lozenge, labelled η, representing the entailment of interest. The white lozenges labelled with “1” – “6” represent exactly the axioms that appear in the original justification J for the entailment (and are therefore in the ontology as asserted axioms). Grey shaded lozenges represent lemmas that are entailed by the deductive closure of J but are not in J as asserted axioms. For a given node, its direct predecessors con- stitute a justification for that node. This produces a weakly connected directed acyclic graph, with one sink node that represents the entailment of interest and a source node for each axiom in the justification.
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