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https://ceur-ws.org/Vol-2540/FAIR2019_paper_63.pdf
Explanation for defeasible entailment
Victoria Chama1 and Thomas Meyer1,2
1
Department of Computer Science, University of Cape Town
chmvic006@myuct.ac.za
2
Center for Artificial Intelligence Research
tmeyer@cs.uct.ac.za
Description Logics (DLs) are well-known formalisms for reasoning about in-
formation in a given domain. DLs have many advantages such as being decidable
fragments of First-Order Logic, and having a clear semantics and well-defined
reasoning procedures which can be automated [7, 2]. Take the classic penguin
example, and consider a knowledge base containing the statements: “penguins
are birds”, “robins are birds”, “penguins do not fly”, “birds fly” and “birds have
wings”.We can use the well-defined syntax and semantics of DLs to define en-
tailment which allows us to derive implicit knowledge that can be made explicit
through inferences [2]. For example using the information above we can query
the knowledge base, ask “do robins have wings”, and the answer would be YES.
DLs employ various reasoning services such as concept satisfiability, sub-
sumption, consistency checking and instance checking which can be used to
derive useful implicit information from knowledge bases. Reductions between
reasoning services also enable only one reasoning procedure to be implemented
which alleviates the need of creating tools to perform each and every reason-
ing service [9, 10]. Various reasoning techniques/algorithms have been developed
to solve some of the reasoning problems highlighted above. The most widely
used technique, the tableau-based approach, have been shown to be efficient in
practice for real knowledge bases [2].
The DLs services mentioned above can be more useful by adding explana-
tions to the conclusions that DLs systems can draw. Using the example above,
the answer to our query “do robins have wings” was YES. However, it is more
beneficial to users if the DL system can also provide an explanation of how it
came to the conclusion. In this example an explanation to the query is that “we
know that robins are birds, and birds have wings, therefore we can conclude
that robins have wings”. Explanation facilities are useful in understanding en-
tailments, debugging and repairing information declared in knowledge bases and
also knowledge base comprehension. In our example above our knowledge base
is very small with only five statements. In reality knowledge bases can contain
ten of thousand of statements and without automated support for explanation,
it can be difficult to identify the statements that give rise to entailments [3,
6]. There are various algorithms to compute justifications, and implementations
of these algorithms for the DL case are available through the ontology editor
Protégé [6].
Classical DLs cannot deal with exceptional cases. For this reason, there have
been numerous proposals to define non-monotonic reasoning systems. One such
approach is the KLM approach to defeasible reasoning, which was originally
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2 V Chama, T Meyer
defined for propositional logic, but has been lifted to the case for DLs [5] .
Rational closure is a specific method within the KLM approach that has a well-
defined semantics [4] and an implementation [8] for the ontology editor Protégé.
The rational closure algorithm performs reasoning by first constructing a
ranking where every defeasible condition has a rank [4]. A defeasible conditional
is exceptional if its antecedent is exceptional with respect to the knowledge
base. In order to determine the rank of a statement, you first have to check how
exceptional it is.
Let us extend the example above to the defeasible case. Suppose the knowl-
edge base contains the following the statements: “penguins are birds”, “robins
are birds”, “penguins do not fly”, “birds typically fly” and “birds typically have
wings”.
When we query this knowledge base and ask “do penguins typically have
wings”, the answer to the query using rational closure is NO. The reason the
system returns NO is not as straightforward as one might think. Initially, a user
might use the same arguments that were used in our initial version of our example
but that justification leads to errors. Thus, it is useful if defeasible reasoning is
extended to include explanations.
In this work, we combine explanations with defeasible reasoning. Generally,
when you look at classical DLs, the explanations can be derived from using DL
reasoning services. But here we need to take defeasibility into account, and the
way in which the explanations are obtained are more complex because we have
to consider the ranking of statements.
For instance, in order to obtain the explanation to the query “do penguins
typically have wings”, we first look at all justifications that support the answer
YES. The statements “penguins are birds” and “birds typically have wings”
can be used to justify the answer. However, according to the information in our
knowledge base we know that penguins can not fly thus the logical consequence
of this is that there are no penguins since the existence of penguins will cause a
conflict. As a result using rational closure, the statements “birds typically fly”
and “birds typically have wings” are discarded meaning we can no longer use
them in our justification, thus giving the explanation of why the answer to the
query is NO.
Generally from an algorithmic perspective what is happening is if the answer
to a query is YES, look at all the minimal sets which include only the statements
required for the entailment to hold. Then check for the ones that occur as part of
the ranked statements. That will be the real explanation. However, note that an
entailment can have more than one justification and other justifications can be
used to build up the explanation. If the answer is NO, then check to see if there
are any minimal subsets that entail the negation [1] of the query and build the
explanation from there which is how we came up with the explanation for the
query above. Thus, looking at the example above it is clear that explanation for
defeasible reasoning will have the same advantages explanation has for classical
reasoning systems.
Explanation for defeasible entailment 3
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