Explaining consequences obtained from ontological reasoning is an active research topic. Unlike the computation of the cause of the consequence, the explication of the cause has received little attention so far. However, as many ontologies are designed by experts, the terms and notions used in an explanation need not be known to the user of the ontology-based system, before they can attempt to understand the underlying logical process. In this paper, we address the task of making an explanation of a consequence more comprehensible to a variety of users by re-phrasing it in a vocabulary known to them. Assuming the existence of a dictionary, we attempt to rewrite technical jargon into expressions using a simpler vocabulary. We show that solving this problem requires taking several technical issues into account.
With the deployment of arti cial intelligence (AI) approaches in industrial applications, there
is a growing need to explain the decisions made by arti cial agents to external users who
might not be fully knowledgeable about the system and its internal mechanisms. This need has
given birth to the eld of explainable AI (XAI) [
What constitutes a good explanation from the point of view of the social sciences was recently
summarized in by Miller in [
Some approaches generate logical proofs of di erent shapes, to allow the user to understand
the derivation steps made [
We propose user-aware generation of explications for ontology consequences. This general task poses two challenges to be addressed. The rst is to use terminology that is appropriate for the explainee in terms of being neither too complex nor overly simplistic. The second challenge is to choose an appropriate structure for the explication that does not confuse the explainee—neither by the use of complex logical expressions nor by its sheer length. We deem the rst challenge more fundamental and address it in this paper. Our approach for user-aware generation of explications is based on a dictionary, which allows us to transform technical jargon appearing in a justi cation into logically equivalent, but easier to understand expressions.
representing their “level of technicality.” Intuitively, the dictionary is strati ed according to technicality levels for di erent users. Our approach is orthogonal to proof generation and translation to natural language. Indeed, an ideal explanation approach would combine all three of them, adding further user-aware exibility.
provides the vocabulary they are able to understand, and the explanation should be restricted
to this vocabulary only. From an abstract point of view, this idea is very similar to the task of
forgetting (or uniform interpolation) [
obvious at rst sight. We thus show that solving the problem is not only useful for the area of
in the knowledge representation and reasoning eld, is to take into account who the user is, and how to better approach them. There is certainly no universal solution to this, as di erent users have varying levels of understanding and explanatory needs. This motivates our interest in methods providing user-aware explanations; that is, methods to generate explanations targeting di erent user’s needs.
logic (DL) [
C ::= > | A | C u C | ∃r.C , with A ∈ NC , r ∈ NR, and its sublogic L0 which disallows the ∃r.C constructor. Knowledge in DLs is expressed through an ontology: a nite set of axioms, of which the most common are so-called general concept inclusions (GCIs) that are statements of the form C v D with C, D concepts. The semantics of this logic is de ned in terms of interpretations, which are pairs I = (ΔI , ·I ) where ΔI is a non-empty set called the domain and ·I is the interpretation function that maps each concept name A ∈ NC to a set AI ⊆ ΔI and every role name r ∈ NR to a binary relation rI ⊆ ΔI × ΔI . The interpretation function is extended to complex concepts by setting >I := ΔI , (C u D)I := CI ∩ DI , and (∃r.C)I := {δ ∈ ΔI | ∃η ∈ CI .(δ, η) ∈ rI }.
CI ⊆ DI . The GCI C v D is a consequence of the ontology O i every model of O is also a
model of {C v D} and is denoted by O |= C v D. We often use the term “entailment” to refer
to consequences as well. For full details on DLs see [
GCIs in an ontology act as constraints on the class of relevant interpretations, i.e. on the class of models. In particular, this means that consequences are monotonic; that is, if ϕ is a consequence of the ontology O, then ϕ is necessarily a consequence of any ontology containing all the GCIs in O. In the following, we also use two special kinds of GCIs. A concept equivalence is of the form C =. D where C, D are concepts. It abbreviates the two GCIs C v D, D v C and hence is satis ed whenever CI = DI . A concept de nition is a concept equivalence A =. D where A ∈ NC ; that is, the left-hand side consists of a concept name only.
One early approach to answer this question was to provide a so-called justi cation; that is, a (subset) minimal sub-ontology of O from which ϕ still follows. The existence of one or more of such minimal subsets is guaranteed by the monotonicity of the logic. The set of axioms forming a justi cation can be seen as a crude explanation for the entailment of ϕ. It provides su cient information about the derivation (the justi cation still entails the consequence) and avoids any super uous information given by irrelevant GCIs. Importantly, a single consequence may have several (even exponentially many) justi cations [10].
original ontology. Developed and maintained by experts and with di erent agendas in mind, the GCIs in an ontology tend to use a quite technical, or even private, terminology which might not be intelligible to the user of the ontology requiring an explanation of the consequence.
the attribution (or information extraction) step of explanation development. Our goal, instead, is to provide an explication: a variant of such a justi cation which is understandable by the user and information preserving. In slightly more formal terms, we are interested in the following problem.
Problem 1 (justi cation explication). Given a justi cation J and a user u, construct an ontology E equivalent to J using only terms understandable by u.
to a user u, their available terminology, and the capability of rewriting technical concept and role names into more comprehensible ones. Depending on the speci c scenario, an ontology E as required by Problem 1 might not exist. This issue can be slightly alleviated by recalling that a given consequence may have several justi cations. Thus, we consider also a more general version of the problem.
Problem 2 (consequence explication). Given an ontology O, a consequence ϕ, and a user u, construct an ontology E , equivalent to some justi cation of ϕ on O, using only terms understandable by u.
all possible justi cations, and try to solve Problem 1 on them, until an adequate ontology E is found, or it is deduced that none exists. Such an approach introduces a polynomial overhead on the number of justi cations (which, as mentioned, may be exponential) in the case of L0. For
many justi cations [11]. In the following, we focus on Problem 1 only.
understandability of each individual user. In this way, adequate terms to present to the user guaranteeing that they are understood can be selected automatically.
of the vocabulary that they can understand. Hence, for example, depending on their expertise, users may understand better or prefer to use the (equivalent) terms “monotremata,” “oviparous mammals,” or “egg-laying, milk-producing animals.” We thus consider two sources of knowledge. First, we have a domain ontology OD, which encodes all the knowledge of the domain. This ontology, which is usual encountered in knowledge-aware applications, is maintained by experts and hence assumed to use as precise and technical a terminology as needed for an adequate modelling of the domain knowledge. Second, we have a technical ontology T , which models relationships between more technical (or more complex) terms, and colloquial terms often used at lower levels of expertise. Before formally de ning the technical ontology, we need to specify the technicality classes.
is a partially ordered vocabulary (Voc, ) where α β expresses that α is at most as technical as β. Thus, someone with greater expertise in the area would use terms that are larger w.r.t. . Importantly, the use of a partial order admits to have incomparable terms (where none is more technical than the other) and terms that are equally technical. Each element α ∈ Voc de nes a technicality class dαe := {β ∈ Voc | β α} containing all the terms from Voc which are at most as technical as α. The idea is that a user capable of understanding α can also understand all terms in dαe. Given a concept C, the expression V(C) denotes the set of all concept and role names appearing in C; i.e., the vocabulary of C. This notion is extended in the obvious manner to GCIs and ontologies: V(O) is the set of all names appearing in the ontology O. Since users need to understand all the terms appearing in a GCI, justi cation or ontology, we associate their expertise (or technicality level) to technicality classes.
Example 3. Figure 1 depicts a Hasse diagram for a technical vocabulary. For instance, Mammal Oviparous means that Mammal is not more technical than Oviparous. Since these notions do not appear together in the diagram. Mammal is in fact less technical. Animal and Milk are equally technical, and incomparable to Egg. The technicality class of Mammal is
dMammale := {Mammal, EggLaying, Egg, MilkProducing, Animal, Milk}.
De nition 4 (technical ontology). Let (Voc, ) be a technical vocabulary. A technical ontology T is a nite set of concept de nitions using only terms in Voc, such that for every concept de nition A =. C ∈ T it holds that V(C) ⊆ dAe.
based on terms of lesser or equal technicality level. In simpler terms, T is a strati ed dictionary of concept names. The idea is that through those de nitions, one can “rewrite” terms that a user may not understand into expressions using a “simpler” vocabulary. Ideally, this rewriting
Monotreme Oviparous, Endothermic
Mammal EggLaying
Egg MilkProducing Animal, Milk should contain expressions which use only concept and role names intelligible for the user, while preserving equivalence.
Example 5. Using the technical vocabulary from Example 3, we can de ne the technical ontology T := { Monotreme =. Oviparous u Mammal,
.
Mammal = Animal u MilkProducing,
Oviparous =. Animal u EggLaying }, which expresses that monotremata are oviparous mammals; mammals are milk-producing animals; and oviparous are egg-laying animals. Note that the condition of decreasing technicality is satis ed by these de nitions.
in Section 2. To specify the level of technicality, we use the technicality classes de ned by the technical ontology. That is, a user selects a term α ∈ Voc such that they understand all terms in dαe.
Problem 6. Let J be a justi cation, (Voc, ) a technical vocabulary, T a technical ontology, and α ∈ Voc. Construct an ontology E such that 1. E ∪ T and J are equivalent, i.e. have the same models and 2. V(E ) ⊆ dαe.
Proposition 7. Let J be a justi cation, (Voc, ) a technical vocabulary, T a technical ontology, and α ∈ Voc. There exists an α-bounded explication of J if • every role name r ∈ V(J ) is such that r α, and • for every concept name A ∈ V(J ) there exists a concept C such that T |= A =. C and
V(C) ⊆ dαe.
can be constructed by simply substituting every concept name in J by their equivalent C. Note that the converse is not necessarily true: there could exist α-bounded explications which are not constructed through substitutions of concept names, but by substituting complex concepts.
De nition 8 (concept explication). Let A ∈ NC , (Voc, ) a be technical vocabulary, T a technical ontology, and α ∈ Voc. A concept C such that T |= A =. C and V(C) ∈ dαe is called an α-bounded explication of A.
Animal u MilkProducing u EggLaying is indeed an Oviparous-bounded explication of Monotreme—in fact, it is even a Mammalbounded explication. However, for a user who understands the term Oviparous, expanding out the de nitions to the point of EggLaying may be counterproductive as it makes the explication verbose. Not to mention the possibility of taking o ence from an overly simplistic explanation.
nd one which is as technical as possible, while remaining understandable. In our example, a more adequate explication (from this point of view) of Monotreme is Oviparous u Mammal. In general, we will prefer optimal explications that are as technical as possible within the selected technicality class.
De nition 9 (optimal explication). We extend the ordering to complex concepts by setting C D i for every term v ∈ V(C) there is a w ∈ V(D) such that v w. C is more technical than D (denoted C ≺ D) i C D but D 6 C.
An α-bounded explication D of C is optimal if there exists no α-bounded explication D0 of C which is more technical than D.
A rst idea to try to nd (optimal) explications of concepts would be to follow a term rewriting approach [12]. Indeed, each concept de nition A =. C in the technical ontology can be seen as a rewriting rule which substitutes the concept name A with a complex expression using the symbols from C. These symbols may be further rewritten with other complex expressions, using other concept de nitions from the technical ontology. Under this view, all symbols in dαe are terminals; that is, terms that cannot not be further rewritten using the technical ontology T . Such terminals are already understandable to the user. All other symbols in Voc are non-terminals, and should be rewritten into simpler, potentially more understandable terms.
3.1. Non-determinism Induced by Multiple Definitions
it is not clear a-priori which de nition should be used in the rewriting of a concept. Indeed, a wrong choice may result in a concept that is not an α-bounded explication, but which cannot be further simpli ed, even in cases where adequate explications exist.
Example 10. Suppose that the technical ontology from Example 5 contains also the concept de nition Monotreme =. Oviparous u Endothermic. If we want to nd a Mammal-bounded explication of the concept name Monotreme, we can substitute the name Monotreme by either (i) the term Oviparous u Mammal or (ii) the term Oviparous u Endothermic. In the former case, further expanding the de nition of Oviparous yields an understandable concept at the Mammal level of technicality. The latter case, however, yields the concept Oviparous u Endothermic which contains a concept name (Endothermic) that does not belong to the technicality class dMammale, nor has an associated concept de nition. Hence no adequate explication can be derived from it.
at least one α-bounded explication. A naive construction of all possibilities would generate exponentially many such concepts, thus requiring exponential time and space to nd such an explication (or decide that it does not exist).
which guesses, for every substitution step, which de nition to use. This means that we can decide whether an α-bounded explication exists—and, indeed, construct it—in non-deterministic polynomial time. While this solves the issue of multiple-de nitions, it does not guarantee that the constructed explication is optimal. 3.2. Sub-optimality by Limited Use of Concept Equivalences
only terms that are at most as technical as the concept name in the left-hand side. However, beyond this requirement there is no restriction about the ordering of the terms in di erent de nitions. Moreover, we are ultimately interested in constructing equivalent concepts, for which the semantics of concept de nitions need to be carefully considered.
Recall that the semantics of A =. C guarantee that A and C are equivalent. So far, we have used this as a unidirectional “rule” to substitute an occurrence of A by the concept C. However, logical equivalences also hold in the converse direction: substituting the concept C by the concept name A would be equally correct. At rst, this may seem as an irrelevant insight as it implies increasing—rather than decreasing—the technicality level of the resulting concept; we are substituting a simple description with technical jargon. Still, optimal solutions might require such an interplay of technicality decrease and increase.
Example 11. Consider once again the technical vocabulary from Example 3, but now using the technical ontology
.
T := { Monotreme = EggLaying u MilkProducing u Animal,
.
Mammal = Animal u MilkProducing,
.
Oviparous = Animal u EggLaying }; that is, we have modi ed the rst concept de nition. Note that there is no non-determinism here, since every concept name is de ned only once. If we want to produce a Mammal-bounded explication for the concept name Monotreme, we expand its de nition, obtaining the complex concept EggLaying u MilkProducing u Animal. This is already an explication on the right level of technicality and can be understood by a user who understands Mammal. However, it is not optimal.
To nd an optimal explication, observe that MilkProducing u Animal is equivalent to Mammal in T . Thus, we can simplify the previously constructed concept into EggLaying u Mammal, which is in fact an optimal explication for Monotreme.
it may still be impossible to nd an optimal explication simply by increasing the technicality through a “converse” concept rewriting. As the following example shows, it may be necessary to further decrease the technicality of a concept, before it can be rewritten into an optimal explication.
Example 12. Consider again the technical vocabulary from Example 3, along with the technical ontology
.
T := { Monotreme = EggLaying u Mammal,
.
Mammal = Animal u MilkProducing,
.
Oviparous = Animal u EggLaying }, and suppose that we want now an Oviparous-bounded explication of Monotreme. We rst have no choice but to expand Monotreme to its unique de nition. The resulting concept EggLayinguMammal is already an understandable explication to the desired level of technicality, but as we shall see, it is not optimal. In contrast to the previous example, it is not possible to substitute part of the explication with the concept Oviparous yet: although we have the conjunct EggLaying, we are still missing the term Animal in it. Notice that Mammal can be substituted by Animal u MilkProducing yielding the (less technical) explication EggLaying u Animal u MilkProducing, which can now be equivalently rewritten into the optimal explication (at the requested level of technicality) Oviparous u Mammal. These simple examples highlight a very general limitation of the approach that uses TBox axioms as rewrite rules to nd explications. In order to guarantee optimality of the resulting concept, it may be necessary to cycle through several rounds of increased and decreased technicality. It is also possible to encounter cases where the technicality of a concept needs to be increased beyond the user-speci ed technicality level before it is later rewritten into an adequate explication; we leave the construction of such an example to the interested reader.
the idempotency of the conjunction. Note that in the last step of Example 12, we used the fact that Animal and Animal u Animal are equivalent, in order to construct the two terms Oviparous and Mammal; each of them requires a mention of Animal in their de nition. This means that some terms may need to be “recalled” after being used in a rewriting step to allow for another rewriting. The following example makes the issue explicit.
Example 13. Using the technical vocabulary from Figure 1, suppose that we are interested in explaining the concept name Monotreme to the technicality degree of Oviparous based on the technical ontology
.
T := { Monotreme = EggLaying u Mammal,
.
Monotreme = Animal,
.
Oviparous = EggLaying u Animal }.
Using these de nitions, we can transform Monotreme either to EggLaying u Mammal or to Animal. At this point, no other rewriting step is possible. However, since Monotreme is equivalent to EggLaying u Mammal, the latter concept is also equivalent to Monotreme u EggLaying u Mammal which can now be rewritten as Animal u EggLaying u Mammal, yielding the optimal explication Oviparous u Mammal.
Our goal is to explain to non-expert users the reasons why a consequence follows from a domain ontology. In contrast to other approaches where the emphasis is in the presentation of the axioms and explanation of the logical steps followed, we are more interested in explicating given justi cations using terms that are understandable to the user. To this end, we assume the existence of a dictionary, which de nes technical jargon through equivalent, less technical expressions. Even at the restricted setting of this work, we have seen that nding optimal explications requires specialised techniques, which still need to be developed.
All our examples are based on the very simple logic L0. Clearly, additional issues arise if we extend the language to EL or beyond. For instance, it should be clear that deciding whether a given concept C is an optimal α-bounded explication of A is in co-NP for L0: if it is not, simply guess another concept D (which is a conjunction of concept names in dαe) and verify that it is equivalent to A, and that C ≺ D. This method does not work in EL because, due to the potentially nested existential restrictions, we have no guarantee that optimal explications are even of polynomial length.
related to user education: if a term cannot be explained to the user’s level of technicality, we want to propose a vocabulary—as close as possible to the user’s own—which will allow them to understand it. Afterwards, we will look at EL and beyond to expressive DLs and other KR formalisms and entailments.
and give up the requirement that it is equivalent to the original concept, but that it is similar enough. For instance, some of the conjuncts in the explication could approximate the original ones from above while others could be approximated from below. Finding out how to deal with such similarity will require to explore several alternatives.
[9] F. Baader, S. Brandt, C. Lutz, Pushing the EL envelope, in: Proceedings of IJCAI’05,
Professional Book Center, 2005, pp. 364–369. [10] F. Baader, R. Peñaloza, B. Suntisrivaraporn, Pinpointing in the description logic EL+, in: Proc. of KI’07, volume 4667 of LNCS, Springer, 2007, pp. 52–67. doi:10.1007/ 978-3-540-74565-5_7. [11] R. Peñaloza, B. Sertkaya, Understanding the complexity of axiom pinpointing in lightweight description logics, Arti cial Intelligence 250 (2017) 80–104. doi:10.1016/j.artint. 2017.06.002. [12] F. Baader, T. Nipkow, Term Rewriting and All That, Cambridge University Press, 1998.