1 Introduction
The aim of abductive reasoning is to generate explanatory hypotheses for new observations, enabling the discovery of new knowledge. Abduction was identified as a form of reasoning by C.S. Peirce [Peirce1878]. It has also become a recurring topic of interest in the field of AI, leading to work such as abductive extensions of Prolog for natural language interpretation [Stickel1991, Hobbs et al.1993]
, the integration of abduction with induction in machine learning
[Mooney2000] including work in the fields of inductive [Muggleton and Bryant2000]and abductive logic programming
[Kakas, Kowalski, and Toni1992, Ray2009] and statistical relational AI [Raghavan and Mooney2010].This paper focuses on abduction in description logic (DL) ontologies. These consist of a TBox of information about general entities known as concepts and roles and an ABox of assertions over instances of these concepts known as individuals. DL ontologies are widely used to express background knowledge and as alternative data models for knowledge management. They are commonly used in fields such as AI, computational linguistics, bioinformatics and robotics. The need for abductive reasoning in ontologies was highlighted by [Elsenbroich, Kutz, and Sattler2006]. Use cases include hypothesis generation, diagnostics and belief expansion for which most current reasoning methods for ontologies are not suitable. This has led to a variety of work on abduction in DLs, including studies in [Bienvenu2008] and applications such as ontology repair [Lambrix, Dragisic, and Ivanova2012, WeiKleiner, Dragisic, and Lambrix2014] and query explanation [Calvanese et al.2013]. For ABox abduction, methods in more expressive logics such as and its extensions have been proposed [Klarman, Endriss, and Schlobach2011, Halland and Britz2012, Pukancová and Homola2017]. Similarly, a variety of work exists on TBox abduction [WeiKleiner, Dragisic, and Lambrix2014, Halland, Britz, and Klarman2014]. However, few implementations and evaluations are available for abductive reasoning. Exceptions include the ABox abduction method of [Du, Wang, and Shen2014] in Datalog rewritable ontologies and a TBox abduction method using justification patterns [Du, Wan, and Ma2017].
The aim of this paper is to investigate the use of forgetting for ABox abduction in DL ontologies. Forgetting is a nonstandard reasoning method that restricts ontologies to a specified set of symbols, retaining all entailments preservable in the restricted signature. It is also referred to as uniform interpolation or secondorder quantifier elimination, and has been proposed as a method for abduction in different contexts [Doherty, Łukaszewicz, and Szałas2001, Gabbay, Schmidt, and Szałas2008, Wernhard2013, Koopmann and Schmidt2015b]. However, so far the forgettingbased approach has been insufficiently studied or applied, particularly in terms of preferred characteristics of abductive hypotheses and in the setting of large DL ontologies.
This work investigates the hypotheses obtained using forgettingbased abductive reasoning. These hypotheses are weakest sufficient conditions [Lin2001], related to the DL literature notion of semantic minimality [Halland, Britz, and Klarman2014], meaning that they make the fewest assumptions necessary to explain an observation given the background knowledge. However, without additional steps, these hypotheses are not guaranteed to be consistent and are likely to be mostly redundant when the forgetting based approach is applied to large ontologies. In this work, additional constraints are investigated to capture these redundancies and practical methods for their removal are presented.
The main contributions of this paper are: (1) Forgettingbased ABox abduction in DL ontologies is explored and formalised. The aim is to compute hypotheses that do not contain unnecessary assumptions nor misleading, i.e. redundant, explanations. The need to eliminate redundancies from uniform interpolants is motivated and solved. (2) A practical method for this task is presented for . It computes hypotheses containing only abducible symbols. Nonabducibles are specified by a forgetting signature consisting of any set of concept, but not role, symbols. Both the observations and hypotheses may contain any atomic or complex (or ) concepts, but cannot contain role assertions. An efficient annotationbased filtering method is proposed to eliminate redundancies from uniform interpolants. The method uses the forgetting tool LETHE which is shown to be applicable to ABox abduction, thereby answering an open question in [Koopmann and Schmidt2015b]. However, the general framework could use any forgetting method designed for . (3) The method is evaluated empirically over a corpus of realworld ontologies. An approximate and a full approach to redundancy elimination are compared.
Proofs and additional examples can be found in the appendix.
2 Problem Definition
Concepts in the description logic have the following forms: , where denotes a concept name, and are arbitrary concepts and is a role name. Atomic concepts are concept names, while concepts such as are said to be complex. A knowledge base or ontology in consists of a TBox and an ABox. The TBox consists of a set of general concept inclusions of the form , where and are any concept. The ABox contains axioms and role assertions of the form , where is any concept and and are individuals. The signature of , denoted as , is the set of all concept and role names occurring in , where can be any ontology or axiom.
The aim of abduction is to compute a hypothesis to explain a new observation. This paper focuses on the following form of the ABox abduction problem.
Definition 1.
Abduction in Ontologies. Let be an ontology and a set of ABox axioms, where does not contain role assertions, such that , and . Let be a set of symbols called abducibles which contains all role symbols in . The abduction problem is to find a hypothesis as a disjunction of ABox axioms, without role assertions, that contains only those symbols specified in such that: (i) , (ii) , (iii) does not contain interdisjunct redundancy i.e., there is no disjunct in such that and (iv) for any satisfying conditions (i)–(iii) where , if then .
The set of abducibles defines the subset of symbols in the ontology that may appear in the hypothesis . Here, must contain all role symbols in and both the observation and may not contain role assertions. For our approach, the language of must be extended to include disjunctive ABox assertions over multiple individuals, and in some specific cases fixpoints [Calvanese, De Giacomo, and Lenzerini1999] to represent cyclic results. These will be discussed alongside the proposed method.
The rationale for the problem conditions is to focus efforts on computing informative hypotheses. Otherwise, the search space for hypotheses would be too large. Defining the set of abducibles allows a user to focus on explanations containing specific information represented as symbols, utilising their own knowledge of the problem domain.
Conditions (i) and (ii) of Definition 1 are standard requirements in most abductive reasoning tasks. Condition (i) requires that all generated hypotheses are consistent with the background knowledge in the ontology . Otherwise would be entailed from which everything follows. Condition (ii) ensures that explains the observation when added to the background knowledge in .
Conditions (iii) and (iv) capture two distinct notions. Condition (iii) ensures that each of the disjuncts in the hypothesis are independent explanations [Konolige1992] for the observation . That is, there are no disjuncts in that express information that is the same or more specific than that which is already expressed by the other disjuncts in . This also excludes disjuncts that are simply inconsistent with the background knowledge as a special case, since if for a disjunct we have then everything follows. Condition (iii) is referred to as interdisjunct redundancy. The example below illustrates its use:
Example 1.
Let , and . Consider the hypotheses: and . Both satisfy conditions (i), (ii) and (iv). However, contains two redundant disjuncts: and . provides no new information over the other disjuncts: , while is inconsistent with . While not stronger than , is unnecessarily complex. A user may have the false impression that is a valid explanation for , or that is an independent avenue of explanation compared to . Condition (iii) excludes these redundancies: for it is the case that and also and thus everything follows. As a result, is excluded and is returned as the solution.
As condition (iii) requires that each disjunct be consistent with the ontology , condition (i) is not strictly needed: follows if condition (iii) is satisfied. However, as consistency is a key condition in most abduction contexts it is useful to emphasise it as a separate characteristic.
Condition (iv) captures the notion of semantic minimality [Halland, Britz, and Klarman2014] under the background knowledge . It restricts hypotheses to those that make the fewest assumptions necessary to explain the observation given . This is shown in the example below.
Example 2.
Let , and . Consider the hypotheses and . Both satisfy the conditions in Definition 1(i) and (ii). However, hypothesis does not satisfy (iv), since , but the reverse does not hold. Thus, is a stronger or “less minimal” hypothesis than .
From this, it can be seen that condition (iv) rejects semantically stronger hypotheses. It should be noted that, unlike some other settings such as [Halland, Britz, and Klarman2014], here can contain disjunctions. Thus, redundant disjuncts must be considered separately, as in condition (iii), since condition (iv) does not account for these.
With these conditions, the aim of this work is to compute a semantically minimal hypothesis consisting of all disjuncts that each represent an independent explanation of the observation , none of which overlaps with either the background knowledge or the other disjuncts.
Definition 1 does not remove all choices between or redundancies in the forms taken by each disjunct in if they are equivalent under . For example, condition (iv) does not account for conjunctively joined redundancies that follow directly from . If Example 2 is extended so that the axiom is in and the signature of abducibles also contains , then is also a valid hypothesis under conditions (i), (ii) and (iv). While is not stronger than , it contains a form of redundancy: .
To eliminate these redundancies and simplify the disjuncts themselves may require the use of preference criteria over the disjuncts in . As there are a variety of methods for defining and realising preference handling [Cialdea Mayer and Pirri1996, PinoPeréz and Uzcátegui2003, Delgrande et al.2004] we do not discuss this aspect. Here, the focus is on computing the space of independent explanations, rather than ensuring each takes the simplest form.
3 Forgetting and Uniform Interpolation
Forgetting is a process of finding a compact representation of an ontology by hiding or removing subsets of symbols within it. Here, the term symbols refers to concept and role names present in the ontology. The symbols to be hidden are specified in the forgetting signature , which is a subset of symbols in the ontology . The symbols in should be removed from , while preserving all entailments of that can be represented using the signature without . The result is a new ontology, which is a uniform interpolant:
Definition 2.
Uniform Interpolation in [Lutz and Wolter2011]. Let be an ontology and a set of symbols to be forgotten from . Let be the complement of . The uniform interpolation problem is the task of finding an ontology such that the following conditions hold: (i) , (ii) for any axiom : iff provided that . The ontology is a uniform interpolant of for the signature . We also say that is the result of forgetting from .
Uniform interpolants are strongest necessary entailments, in general, it holds that:
Theorem 1.
is a uniform interpolant of ontology for iff is a strongest necessary entailment of in .
This means that for any ontology , if and , then . Of the methods for uniform interpolation in , e.g., [Ludwig and Konev2014, Koopmann and Schmidt2015a], our abduction method uses the resolutionbased method developed by Koopmann and Schmidt [KoopmannFixpoints,KoopmannABoxes,KoopmannNonStandardReasoning].
Here, this method is referred to as . Motivations for utilising include the fact that it can perform forgetting for with ABoxes [Koopmann and Schmidt2015a], making it suitable for the setting in this paper. Furthermore, in theory the result of forgetting (and abduction) can involve an infinite chain of axioms. Using , such chains can be finitely represented using fixpoint operators. In practice, these are rarely required: in previous work only 7.2% of uniform interpolants contained cycles [Koopmann and Schmidt2013]. can also handle disjunctive ABox assertions which are not representable in pure . These will be needed for some abduction cases involving multiple individuals. In terms of efficiency, the size of the forgetting result is constrained to at most a double exponential bound with respect to the input ontology and is guaranteed to terminate [Koopmann and Schmidt2015a].
The method has two properties that are also essential to the proposed abduction method. (i) Soundness: any ontology returned by applying to an ontology is a uniform interpolant. (ii) Interpolation Completeness: if there exists a uniform interpolant of ontology , then the result of is an ontology such that . Thus, for any ontology and any forgetting signature , always returns a finite uniform interpolant.
The method relies on the transformation of the ontology to a normal form given by a set of clauses of concept literals. The inference rules of the forgetting calculus utilised in are shown in Figure 1. Definer symbols are introduced to represent concepts that fall under the scope of a quantifier. Resolution inferences are restricted to concepts in or definer symbols. Once all possible inferences have been made, any clauses containing symbols in are removed and the definer symbols are eliminated resulting in an ontology that is free of all symbols in . A discussion of this calculus and the associated method, including proofs, can be found in [Koopmann and Schmidt2015a].
We will also need the following notions. Each premise in an application of an inference rule in is referred to as a parent of the conclusion of the rule. The ancestor relation is defined as the reflexive, transitive closure of the parent relation. For example, the premises are expressed as the clauses: . For a forgetting signature , resolution between and gives . Resolution between and gives . The axioms and are the parents of the axiom and the ancestors of .
In this paper, we focus on ABox abduction where the set of abducibles includes all role symbols. Nonabducibles are specified by the forgetting signature which contains only concept symbols occurring in the ontology or observation . The proposed method utilises to compute semantically minimal hypotheses via forgetting and contrapositive reasoning, exploiting: iff where is an ontology and , are (ABox) axioms.
4 A ForgettingBased Abduction Method
The abduction algorithm takes as input an ontology , an observation as a set of ABox axioms and a forgetting signature .
Several assumptions are made regarding this input. The method does not cater for negated role assertions as can be seen in Figure 1, and the form of role forgetting in is not complete for abduction. As a result, cannot contain role assertions and is restricted to concept symbols in . Correspondingly, the signature of abducibles must contain all role symbols occurring in . Also, if does not contain at least one symbol in the observation , the semantically minimal hypothesis will simply be itself, i.e., . This is reflected in the fact that no inferences would occur between and under . To avoid this trivial hypothesis, should contain at least one concept symbol in the signature of . In the event that contains concepts that occur within a cycle in , the forgetting result obtained using may contain greatest fixpoints [Koopmann and Schmidt2013] to finitely represent infinite forgetting solutions. For our method, this means that the abduction result may contain least fixpoints due to the negation of greatest fixpoints under contraposition. In these cases, the output language would be .
The output is a hypothesis containing only the abducible symbols , that satisfies the conditions (i)–(iv) in Definition 1. Note that may be a disjunctive assertion over several individuals, again motivating the need to extend with these.
The algorithm reduces the task of computing abductive hypotheses for the observation to the task of forgetting, using the following steps:

[label=(0),leftmargin=7mm]

Compute the uniform interpolant of with respect to the forgetting signature .

Extract the set by omitting axioms such that .

Obtain the hypothesis by negating the set .
In more detail, the input observation takes the form of a set of ABox axioms: where the are concepts and the are individuals. The negation takes the form . The forgetting method is used to compute the uniform interpolant of by forgetting the concept names in , i.e., .
If forgetting was used in isolation, the negation of would be the hypothesis for under contraposition. However, this is only guaranteed to satisfy conditions (ii) and (iv) of Definition 1: since is the strongest necessary entailment of in as in Theorem 1, its negation would be the weakest sufficient condition [Lin2001, Doherty, Łukaszewicz, and Szałas2001]. Thus the hypothesis would be semantically minimal in , but would not necessarily satisfy condition (i), consistency, nor condition (iii), absence of interdisjunct redundancy. In practice most of the disjuncts will be redundant, as the experimental results show (Table 2). In the case that there is no suitable hypothesis, an inconsistent or “false” hypothesis will be returned since all of the axioms in would follow directly from .
Step (2) therefore omits information in that follows from the background knowledge together with other axioms in itself. This check is the dual of Definition 1(iii), and therefore eliminates interdisjunct redundancies such as those in Example 1. The result is a reduced uniform interpolant which takes the form where each is an concept.
If an axiom is redundant, it is removed from immediately. For the following disjuncts, the check is performed against the remaining axioms in . This avoids discarding too many axioms: if multiple axioms express the same information, i.e. are equivalent under , one of them should be retained in the final hypothesis . For example, if two axioms and are equivalent under , but are otherwise not redundant, only one of them is discarded. The order in which the axioms are checked can be random, or can be based on some preference relation [Cialdea Mayer and Pirri1996].
In Step (3) the reduced uniform interpolant is negated, resulting in the hypothesis . Thus, each disjunct in is the negation of an axiom in , i.e., .
The soundness and completeness of the method are made explicit in the following theorem.
Theorem 2.
Let be an ontology, an observation as a set of ABox axioms, excluding role assertions, and a set of abducible symbols that includes all role symbols in and . (i) Soundness: The hypothesis returned by the method is a disjunction of ABox axioms such that sig( and satisfies Definition 1(i)(iv). (ii) Completeness: If there exists a hypothesis such that and satisfies Definition 1(i)–(iv), then the method returns a hypothesis such that .
Theorem 3.
In the worst case, computing a hypothesis using our method has 3EXPTIME upper bound complexity for running time and the size of can be double exponential in the size of .
5 Practical Realisation
For redundancy elimination, Step (2) requires checking the relation for every axiom in . Since entailment checking in has exponential complexity and is in the worst case double exponential in the size of , this step has a 3EXPTIME upper bound which is very expensive particularly for large ontologies. Regardless, Step (2) is essential; without it there will be a large number of interdisjunct redundancies (Definition 1(iii)) in the hypotheses obtained. This is reflected in the experiments (Table 2).
To obtain a computationally feasible implementation of Step (2), the number of entailment checks performed must be reduced. Our implementation of this step begins by tracing the dependency of axioms in on the negated observation . An axiom is defined as dependent upon if in the derivation using it has at least one ancestor axiom in . The set of axioms dependent on is in general a superset of the reduced uniform interpolant and is referred to as , i.e., an approximation of .
In this paper, dependency tracing is achieved by using annotations, similar to [Kazakov and Skoc̆ovský2017, Koopmann and Chen2017, Penaloza et al.2017]. These take the form of fresh concept names that do not occur in the signature of the ontology nor the observation. Annotations act as labels that are disjunctively appended to existing axioms. They are then used to trace which axioms are the ancestors of inferred axioms. This relies on the fact that the annotation concept is not included in the forgetting signature . Thus, it will carry over from the parent to the result of any inference in , as formalised in the following property:
Theorem 4.
Let be an ontology, an observation as a set of ABox axioms, a forgetting signature and an annotator concept added as an extra disjunct to each clause in the clausal form of where and . For every axiom in the uniform interpolant , is dependent on iff .
Therefore, the presence of the annotation concept in the signature of an inferred axiom indicates that the axiom has at least one ancestor in . Since the aim is to trace dependency specifically on , only clauses that are part of need to be annotated. As it is not important which specific clauses in were used in the derivation of dependent axioms, only one annotation concept name is required. This will be referred to as . Using this technique, the process of extracting from the uniform interpolant is a matter of removing all axioms in that do not contain . Then, can be replaced with to obtain the annotationfree set .
Since this annotation based filtering is sound, i.e., it only removes axioms that are not dependent on , as these are directly derivable from and are thus guaranteed to be redundant, it can be used at the start of Step (2) to compute . To guarantee the computation of the reduced uniform interpolant , the entailment check in Step (2) must then be performed for each axiom to eliminate any redundancies not captured by the annotationbased filtering. Since some axioms may have multiple derivations, they can contain the annotation concept but still be redundant with respect to Definition 1. For example:
Example 3.
Let and . The annotated form of is . Using , the result of Step (1) is . Note: no inference is made with , since . In Step (2) extracting all axioms with annotations and setting gives the set . Despite being derivable using , it follows from the original ontology and is therefore redundant with respect to Definition 1(iii). This can now be removed via the entailment check in Step (2).
This method of filtering out redundancies has several advantages. First, it is not specific to and can be applied if the abduction method is later extended to more expressive logics. Second, by removing axioms that are not dependent on , the method reduces the cost of Step (2) since checking the signature of each axiom for the presence of is linear in the size of . In the worst case could be equal to and a double exponential number of entailment checks would still be required. In practice, this is unlikely as is usually a small fraction of as shown by the experiments (Table 2). In these cases, each redundancy eliminated from to replaces an exponential check with a linear one.
The entailment checks that must be performed on to compute may still be costly in the event that many axioms are dependent on in . Therefore, we propose that in some cases it may be pragmatic to relax the allowed hypotheses by negating instead of the reduced uniform interpolant itself. In this case, an additional check, , is required to rule out inconsistent hypotheses if all of the axioms in are redundant. This approximate approach results in a hypothesis which satisfies conditions (i), (ii) and (iv) in Definition 1, but not condition (iii). The results in Table 2 illustrate the effect in practice.
To summarise, we suggest two realisations of Step (2) of the proposed abduction method: (a) approximate filtering, which computes an approximation of the hypothesis by negating , (b) full filtering, which performs the entailment check in Step (2) for each axiom in to obtain and thus which is guaranteed to fully satisfy Definition 1. Note that for setting (b), the approximation step is still used to reduce the overall cost of Step (2).
6 Experimental Evaluation
Ontology  DL  TBox  ABox  Num.  Num. 

Name  Size  Size  Concepts  Roles  
BFO  52  0  35  0  
LUBM  87  0  44  24  
HOM  83  0  66  0  
DOID  7892  0  11663  15  
SYN  15352  0  14462  0  
ICF  1910  6597  1597  41  
Semintec  199  65189  61  16  
OBI  28888  196  3691  67  
NATPRO  68565  42763  9464  12 
Ont.  Mean Time Taken /s  Max Time Taken /s  Mean Redund. Removed  Size /disjuncts  Mean % of  

Name  no app.  no app.  Mean  Max  Redund.  
BFO  0.01  0.01  0.09  0.01  0.07  0.14  52  0  1.97  4  0 
LUBM  0.02  0.03  0.30  0.11  0.16  1.21  90  0.80  2.73  11  29.30 
HOM  0.03  0.05  0.18  0.40  0.54  0.86  82  0.03  2.07  13  1.45 
DOID  0.44  1.09  1071.35  1.11  6.98  1095.07  7891  0  7.23  104  0 
SYN  0.95  3.92  2421.96  2.33  61.52  2593.13  15351  0.03  20.63  457  0.15 
ICF  0.30  0.56  t.o.  0.52  1.58  t.o.  8505  0  2.30  7  0 
Semin.  3.13  5.12  t.o.  9.29  15.36  t.o.  72827  0.03  3.60  10  0.83 
OBI*  3.82  32.17  t.o.  25.18  95.37  t.o.  29191  6.48  52.48  161  12.35 
NATP.  26.54  179.70  t.o.  39.51  544.50  t.o.  111318  0.03  48.70  204  0.06 
A Java prototype was implemented using the OWLAPI^{1}^{1}1http://owlapi.sourceforge.net/ and the forgetting tool LETHE which implements the method.^{2}^{2}2http://www.cs.man.ac.uk/ koopmanp/lethe/index.html. Using this, two experiments were carried out over a corpus of real world ontologies, which were preprocessed into their fragments. Axioms not representable in , such as number restrictions of the form where is a role symbol and is a concept symbol, were removed. Others were represented using appropriate axioms where possible. For example, a range restriction r is converted to , where is the inverse role of . The choice of ontologies was based on several factors. They must be consistent, parsable using LETHE and the OWL API and must vary in size to determine how this impacts performance. Since many realworld ontologies are encoded in less expressive DLs such as , the corpus was also split between and to determine if the performance over suffers as a result of the additional capabilities of the method for . The final corpus contains ontologies from the NCBO Bioportal and OBO repositories,^{3}^{3}3https://bioportal.bioontology.org/^{4}^{4}4http://www.obofoundry.org/ and the LUBM [Guo, Pan, and Heflin2005] and Semintec ontologies.^{5}^{5}5http://www.cs.put.poznan.pl/alawrynowicz/semintec.htm The corpus is summarised in Table 1. The experiments were performed on a machine using a 4.00GHz Intel Core i76700K CPU and 16GB RAM.
For each ontology, 30 consistent, nonentailed observations were randomly generated using any concepts from the associated ontology, some of which were combined using operators to encourage variety. The aim was to emulate the information that may be observed in practice for each ontology, while adhering to the requirements for expressed in Definition 1. As the current prototype uses the OWLAPI, which does not allow disjunctive assertions over multiple individuals, the experiments here are limited to observations involving one individual. For the filtering in Step (2), the preference relation used in these experiments was simply based on order of appearance of each disjunct.
For the first experiment, was set to one random concept symbol from . The assumption was that users may first seek the most general hypothesis, i.e., the semantically minimal hypothesis for the largest set of abducibles. This allows the user to pursue stronger hypotheses subsequently by forgetting further symbols from the initial hypothesis. This experiment is therefore also representative of incremental abduction steps using a small . The second experiment was performed over the DOID, ICF and SYN ontologies to evaluate the performance as the size of increases. These ontologies were used as they have a sufficiently large signature of concepts and LETHE did not time out when forgetting in any case. In all cases, at least one symbol from was present in to avoid trivial hypotheses.
In both experiments, the approaches based on (a) approximate and (b) full filtering were compared for the same observations and same random selection of . Thus, the tradeoff between the additional time for entailment checking and redundancy in the final hypothesis is evaluated. In all cases, LETHE was subject to a 300 second time limit.
Table 2 shows the results for the first experiment. For the smaller ontologies, the difference in time taken between the approximate and full filtering was small. For the larger ontologies the cost of the full filtering was more pronounced, taking 313%, 742% and 577% longer across the SYN, OBI and NATPRO ontologies respectively. In all cases, it can be seen that the annotationbased filtering eliminated the majority of redundancies. In three cases (BFO, DOID, ICF), for all 30 observations the result of the approximation, , contained no redundancies and thus . For the other ontologies, in most cases contained few redundancies in both absolute terms and relative to the size of the final hypothesis. For the LUBM and OBI ontologies, however, these redundancies made up a more significant portion of .
The full filtering setting still uses the annotationbased method as a preprocessing step. To assess the benefit of this preprocessing, results for applying the entailment check in Step (2) directly to instead of were collected and are shown in the “ no app.” columns. For the largest ontologies, the time taken increased significantly e.g. taking 98,189% longer for the DOID ontology. For all of the ontologies the experiments were terminated after several days runtime, i.e., it took at least several hours to compute a single hypothesis on average. This indicates that the annotationbased filtering significantly reduces the time taken, particularly over large or more expressive ontologies.
Figure 2 shows the results of the second experiment. The time taken for the forgetting step, Step (1), increased almost linearly with the size of . This was expected due to a higher number of inferences needed to compute . The time taken for filtering, Step (2), did not increase with the size of . However, for each ontology, maxima were observed for different sizes of . This implies that certain symbols increase the filtering time if they appear in . Forgetting commonly used concepts results in more inferences and a larger , which may explain the maxima as the annotationbased filtering depends solely on the number of axioms in . The size of will also increase in these cases, leading to more exponential entailment checks for full filtering. The full filtering took an average of 27, 11 and 70 times longer than the approximate case for the DOID, ICF and SYN ontologies respectively. This indicates that the cost of the full entailment check increased with the size of the ontology, particularly the size of the TBox, and not the size of .
In 100% of cases for both experiments the hypotheses were represented without fixpoints, indicating that cyclic, semantically minimal hypotheses seem rare in practice.
7 Discussion
The use of forgetting for abduction has been suggested in classical logics [Doherty, Łukaszewicz, and Szałas2001, Gabbay, Schmidt, and Szałas2008, Wernhard2013], and a form of TBox abduction [Koopmann and Schmidt2015b]. Our work extends on these suggestions in several ways. As the focus is on large DL ontologies, and not small theories in classical logics, an interpretable hypothesis cannot be obtained by negating the forgetting result as most of it will be redundant (Table 2). Thus, we gain insight into the redundancies in in terms of abductive notions, such as [Konolige1992, Halland, Britz, and Klarman2014], resulting in Definition 1(iii) and (iv). Efficient redundancy removal is then achieved via the annotationbased filtering. The overall approach, including two options emphasising (a) practicality and (b) full redundancy removal, is then evaluated over a corpus of realworld ontologies. This is the first realisation and evaluation of a practical forgettingbased approach to ABox abduction in DL ontologies.
Restricting inferences in to axioms dependent on , rather than filtering the output, was considered. However, this would not circumvent the need to perform entailment checking, as illustrated in Example 3. Second, computing the full uniform interpolant has an interesting use case: iterative abduction. For example:
Example 4.
Let and . In Step (1), using results in . Steps (2)–(3) result in . Now, forgetting from of the previous iteration results in . Repeating Steps (2) and (3) gives , which is stronger than and is the same as the result of computing the uniform interpolant of using , but will be more efficient.
This iterative process enables hypothesis refinement, and has potential synergy with induction. Data could inform the selection of new forgetting signatures to find stronger hypotheses following from prior likely hypotheses: a cycle of abduction, deduction and induction.
Limitations include the lack of role assertions in the observations and hypotheses, due to the inability of to handle negated role assertions, and the incompleteness of role forgetting for abduction, as illustrated by the following:
Example 5.
Let and . Using the result of Step (1) is . This is due to the fact that no inferences are possible on the symbol , since resolution is restricted to . Thus, the hypothesis obtained is , while the expected result is .
With the use of nominals, this limitation can be overcome. Options include the use of other forgetting approaches [Zhao and Schmidt2015, Zhao and Schmidt2016] or the extension of .
It should be noted that methods such as [Klarman, Endriss, and Schlobach2011, Pukancová and Homola2017] can already handle role assertions. The former is a purely theoretical work, which restricts the abductive observations and solutions to : the fragment of without disjunctions of concepts and allowing only atomic negation. The method of [Pukancová and Homola2017] performs abductive reasoning up to , restricting observations and hypotheses to atomic and negated atomic concept and role assertions. This method considers syntactic, but not semantic, minimality, though the authors note the importance of semantic minimality in practical applications.
8 Conclusion and Future Work
In this paper, a practical method for ABox abduction in ontologies was presented. The method computes semantically minimal hypotheses with independent disjuncts to explain observations, where both may contain complex concepts but not role assertions, and the set of abducibles must contain all role symbols. The practicality of the method, including the proposed annotationbased filtering, was evaluated over a corpus of realworld ontologies. To the best of our knowledge, this is the first method that computes such hypotheses efficiently in large ontologies. The ability to produce a semantically minimal space of independent explanations will likely be beneficial in realworld applications. For example, this can provide engineers with multiple, nonredundant suggestions for fixing errors in an ontology or explaining negative query results, even over large knowledge bases. For scientific investigation using ontologies, the ability to produce independent avenues of explanation starting with the least assumptions necessary captures the essence of scientific hypothesis formation. The ability to refine these hypotheses via repeated forgetting also provides a goaloriented, potentially data driven, way to derive stronger insights from the hypotheses produced.
Future work will include removing the restriction on role assertions. Also, though forgetting in DLs can be applied to a form of TBox abduction [Koopmann and Schmidt2015b], the hypotheses take the form where each is an concept. Thus, the problem of determining interdisjunct redundancy and the proposed approach differ in several aspects. This will be investigated, as will the iterative abduction use case.
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9 Appendix
9.1 Examples
For the inferences shown in the following examples, the rules in are referred to as follows, where X and Y are arbitrary axioms denoted by a number:
(1) res(X, Y): resolution
(2) role_ prop(X, Y): role propagation
(3) exis_ elim(X, Y): existential role elimination
Here is an example demonstrating the full procedure for an ABox observation, including the normal form used by and all inferences:
Example 6.
Consider the ontology containing the axioms:
Pogona livesIn.(Woodland Arid)
Sloth Mammal
Woodland Habitat
PineWoods Woodland
and an observation livesIn.Woodland(Gary). Using a forgetting signature Woodland, the first two steps in computing a hypothesis are as follows. Step (1): Negate and annotate it to obtain livesIn.Woodland)(Gary) and add this to . Convert to the normal form required by . This results in the clause set:
1. Pogona livesIn.
2. Woodland
3. Arid
4. Sloth Mammal
5. Woodland Habitat
6. PineWoods Woodland
7. livesIn.(Gary)
8. Woodland
where and are definer symbols. Step (2): Apply to this clause set to compute the uniform interpolant . The inferences are as follows:
9. Habitat  res(2,5) 
10. PineWoods  res(6,8) 
11. PineWoods Habitat  res(5,6) 
12. Pogona livesIn.(Gary)  role_ prop(1,7) 
13.  
14.  
15. Woodland  res(2,13) 
16. PineWoods  res(10,14) 
17. Woodland  res(8,14) 
18.  res(15,17) 
19. Pogona)(Gary)  exis_ elim(12,18) 
Here, all inferences possible under have been made. Now, definers are eliminated by the reverse of the Ackermann rules [Koopmann and Schmidt2015a], following which all clauses containing symbols in Woodland or any remaining definer symbols are deleted. The resulting uniform interpolant is
{ PogonalivesIn.(Habitat Arid),
Sloth Mammal,
PineWoods Habitat,
Pogona)(Gary),
livesIn.PineWoods)(Gary) }
Step (2): Remove all axioms with a signature that does not contain the annotation concept . The first, second and third axioms are all discarded and it is clear that these follow from the original ontology . The annotation concept can then be eliminated by setting , leaving the approximate reduced uniform interpolant:
Pogona(Gary), livesIn.PineWoods(Gary).
For the full filtering procedure, the remaining two axioms in are then subject to the entailment check in Step (2): for each , if then is redundant and is removed from . In this case, neither of the two axioms is redundant since:
Pogona(Gary)livesIn.PineWoods(Gary) and
livesIn.PineWoods(Gary)Pogona(Gary)
Thus, and the result of Step (2) is:
Pogona(Gary), livesIn.PineWoods(Gary).
Step (3): negate the set . This results in the hypothesis:
Pogona livesIn.PineWoods(Gary)
9.1.1 Cyclic Example
For the experimental corpus, no hypotheses containing fixpoints were observed. This means that no cyclic, semantically minimal hypotheses were obtained for each combination of observation and random signature of abducibles. This is consistent with the results obtained by [Koopmann and Schmidt2013], given that here the cycles would need to occur specifically over axioms dependent on .
However, despite the rarity of these hypotheses it is still important to consider the meaning of such hypotheses and how they might occur. Below is a small example of such a case using the method described in this paper:
Example 7.
Consider the following ontology :
Mammal hasParent.Mammal
and an observation Mammal with a forgetting signature Mammal. In Step (1), the observation is negated, annotated to obtain Mammal , and is added to . is applied as follows:
1. Mammal hasParent.  
2. Mammal  
3. Mammal  
4. hasParent.  res(1,2) 
5. hasParent.  res(1,3) 
At this point, all inferences have been made. Now definer symbols are eliminated and clauses containing symbols in Mammal are removed. The elimination of results in the introduction of a greatest fixpoint operator, representing a potentially infinite chain under the hasParent relation in axioms 4 and 5. The resulting uniform interpolant is
{ hasParent.hasParent
where represents a greatest fixpoint. In Step (2), the reduced uniform interpolant is simply:
hasParent.hasParent
In Step (3), the annotation is discarded by setting before negating to obtain the following hypothesis:
hasParent.hasParent.
where represents a least fixpoint.
The need to introduce the fixpoint operator in this example can be seen by seen by comparing the elimination of noncylic and cyclic definers under Ackermann’s lemma [Koopmann and Schmidt2013]:
Noncyclic Definer Elimination  Cyclic Definer Elimination 
where is a set of axioms, is an concept, is a definer symbol and represents a greatest fixpoint, where is concept variable. In the noncyclic case, it is assumed that does not appear in , while the opposite is true in the cyclic case. Thus, the introduction of a greatest fixpoint operator is due to the presence of axiom 4 in the above example.
To interpret the intuition and meaning behind the fixpoint hypothesis hasParent.hasParent. from the above example, consider the nonfinite form of without fixpoints:
hasParent.hasParent.(hasParent
Effectively this means that, if is not a Mammal as in , then it must “not have a parent” or ”must have a parent who does not have a parent…” and so on. In this limited ontology, this is the semantically minimal hypothesis not involving the concept Mammal.
9.2 Proofs of Theorems
9.2.1 Theorem 1
First the following connection between uniform interpolants and strongest necessary conditions or entailments is proven:
Theorem 1: is a uniform interpolant of ontology for iff is the strongest necessary entailment of in .
Proof: for any . Since , then by the reverse direction of Definition 2(ii). To show that is the strongest necessary entailment of , let be any uniform interpolant of such that . Since and , it follows by Definition 2(ii) that . Thus, for any . Now let be the strongest necessary entailment of in the signature . Trivially, this means that satisfies condition (i). Let be an axiom such that . Since is a set of entailments of , it follows that if then . As is also the strongest set of entailments of , it follows that for any other axiom such that then . Thus both directions of condition (ii) are satisfied and is a uniform interpolant.
9.2.2 Soundness and Completeness
Here, the soundness and completeness of the method with respect to the abduction problem outlined in Definition 1 are proved. Note, this is first proved for the general case without the annotationbased preprocessing step in Step (2), i.e., by applying the entailment check directly to and not . Following this, the soundness of the annotationbased filtering is proved, which ensures that the overall soundness of the method is retained when the annotationbased preprocessing is used.
Let and be defined as in Section 5. First, Lemmas 1–5 cover key properties of the uniform interpolant and the reduced uniform interpolant which are useful in proving the soundness and completeness of the abduction method.
Lemma 1:
Proof: The soundness of for computing uniform interpolants has been proven in [Koopmann and
Schmidt2013, Koopmann and Schmidt2015a]. Thus, the set is a uniform interpolant of the input and satisfies the conditions in Definition 2. The property in Lemma 1 then follows from Theorem 1: if is the strongest necessary entailment of in a signature , then trivially .
Lemmas 2, 3, 4 and 5 follow from the definition of Step (2) of the method: the reduction of the uniform interpolant to only the set of axioms such that for each , .
Lemma 2:
Proof: Given that and , it then follows that .
Lemma 3: for every
Proof: Since Step (2) of the method omits all axioms such that via the annotationbased filtering followed by entailment checking on any remaining axioms, it follows that for every .
Lemma 3 implies that .
Lemma 4:
Proof: The extraction of from is performed sequentially. Thus, we can define a sequence:
where , and for each with :
(1)
and
(2)
where is the redundant axiom removed at step . Now we can prove Lemma 4 by induction. Consider an axiom , the base case is:
There are two possible cases to consider. (i) , in which case and so the base case trivially holds. (ii) , in which case must be redundant according to the dual of Definition 1(iii) and thus statement (1) holds at step 0. We now define the induction hypothesis:
and the induction step:
There are three possible cases for the induction step. (i) , i.e., the axiom has not yet been discarded as of step . Then the induction step holds since . (ii) , i.e., the axiom is removed at step . Then statement (1) holds at step under the definition of redundancy in 1(iii), and thus the induction step holds. (iii) , i.e., was checked and discarded prior to step . Then from (1):
and we can also write:
.
Which simplifies to . By substituting statement (2) into this, we obtain:
From the induction hypothesis, . Thus, the following holds:
meaning that the induction step holds for all . As a result, we have that:
for all and finally, by setting we have:
by substituting (2):
for all . Since :
as required.
Lemma 5: For any in the signature such that for every and , if , then .
Proof: We have that . We also have that , since is the strongest necessary condition of in the signature . We can write as:
and thus:
We can then write:
(1)
From (1) and Lemma 4, we then derive:
as required.
Using Lemmas 1–5, it is possible to prove the soundness of the method with respect to the abduction problem in Definition 1.
Theorem 2.Let be an ontology, an observation as a set of ABox axioms, excluding role assertions, and a set of abducible symbols such that it includes all role symbols in and .
(i) Soundness: The hypothesis returned by the method is a disjunction of ABox axioms such that sig( and satisfies Definition 1(i)–(iv).
Proof: We obtain the hypothesis by negating under contrapositive reasoning, which is then added to . Thus, condition (i) follows from Lemma 3 since for all and thus for every disjunct . Condition (ii) follows from Lemma 2: since , under contraposition where . Condition (iii) is guaranteed via the strict check performed in Step (2) of the method, which is the dual of condition (iii). Thus, since is obtained by applying contraposition to , and all axioms in satisfy the check in Step (2), will satisfy condition (iii). Condition (iv) follows from Lemma 5, which shows that if there exists a set of axioms in the signature such that , satisfies the dual of condition (iii) and then . Since the hypothesis is obtained by negating , the dual of Lemma 5 holds for