Connection-minimal Abduction in EL via Translation to FOL

. Abduction in description logics ﬁnds extensions of a knowledge base to make it entail an observation. As such, it can be used to explain why the observation does not follow, to repair incomplete knowledge bases, and to provide possible explanations for unexpected observations. We consider TBox abduction in the lightweight description logic EL , where the observation is a concept inclusion and the background knowledge is a TBox, i.e., a set of concept inclusions. To avoid useless answers, such problems usually come with further restrictions on the solution space and/or minimality criteria that help sort the chaﬀ from the grain. We argue that existing minimality notions are insuﬃcient, and introduce connection minimality. This criterion follows Occam’s razor by rejecting hypotheses that use concept inclusions unrelated to the problem at hand. We show how to compute a special class of connection-minimal hypotheses in a sound and complete way. Our technique is based on a translation to ﬁrst-order logic, and constructs hypotheses based on prime implicates. We evaluate a prototype implementation of our approach on ontologies from the medical domain.


Introduction
Ontologies are used in areas like biomedicine or the semantic web to represent and reason about terminological knowledge.They consist normally of a set of axioms formulated in a description logic (DL), giving definitions of concepts, or stating relations between them.In the lightweight description logic EL [4], particularly used in the biomedical domain, we find ontologies that contain around a hundred thousand axioms.For instance, SNOMED CT 5 contains over 350,000 axioms, and the Gene Ontology GO 6 defines over 50,000 concepts.A central reasoning task for ontologies is to determine whether one concept is subsumed by another, a question that can be answered in polynomial time [2], and rather efficiently in practice using highly optimized description logic reasoners [34].If the answer to this question is unexpected or hints at an error, a natural interest is in an explanation for that answer-especially if the ontology is complex.But whereas explaining entailments-i.e., explaining why a concept subsumption holds-is well-researched in the DL literature and integrated into standard ontology editors [25,26], the problem of explaining non-entailments has received less attention, and there is no standard tool support.Classical approaches involve counter-examples [7], or abduction.
In abduction a non-entailment T |= α, for a TBox T and an observation α, is explained by providing a "missing piece", the hypothesis, that, when added to the ontology, would entail α.Thus it provides possible fixes in case the entailment should hold.In the DL context, depending on the shape of the observation, one distinguishes between concept abduction [8], ABox abduction [11-14, 17, 23, 28, 29, 35, 36], TBox abduction [15,38] or knowledge base abduction [19,30].We are focusing here on TBox abduction, where the ontology and hypothesis are TBoxes and the observation is a concept inclusion (CI), i.e., a single TBox axiom.
To illustrate this problem, consider the following TBox, about academia, • "Being employed in a research position and having a qualifying diploma implies being a researcher."• "Writing a research paper implies being a researcher." • "Being a doctor implies holding a PhD qualification." • "Being a professor is being a doctor employed at a (university) chair." • "Being a funds provider implies writing grant applications." The observation α a = Professor Researcher, "Being a professor implies being a researcher", does not follow from T a although it should.We can use TBox abduction to find different ways of recovering this entailment.
Commonly, to avoid trivial answers, the user provides syntactic restrictions on hypotheses, such as a set of abducible axioms to pick from [12,35], a set of abducible predicates [29,30], or patterns on the shape of the solution [16].But even with those restrictions in place, there may be many possible solutions and, to find the ones with the best explanatory potential, syntactic criteria are usually combined with minimality criteria such as subset minimality, size minimality, or semantic minimality [11].Even combined, these minimality criteria still retain a major flaw.They allow for explanations that go against the principle of parsimony, also known as Occam's razor, in that they may contain concepts that are completely unrelated to the problem at hands.As an illustration, let us return to our academia example.The TBoxes H a1 = { Chair ResearchPosition, PhD Diploma} and H a2 = { Professor FundsProvider, GrantApplication ResearchPaper} are two hypotheses solving the TBox abduction problem involving T a and α a .Both of them are subset-minimal, have the same size, and are incomparable w.r.t. the entailment relation, so that traditional minimality criteria cannot distinguish them.However, intuitively, the second hypothesis feels more arbitrary than the first.Looking at H a1 , Chair and ResearchPosition occur in T a in concept inclusions where the concepts in α a also occur, and both PhD and Diploma are similarly related to α a but via the role qualification.In contrast, H a2 involves the concepts FundsProvider and GrantApplication that are not related to α a in any way in T a .In fact, any random concept inclusion A ∃writes.B in T a would lead to a hypothesis similar to H a2 where A replaces FundsProvider and B replaces GrantApplication.Such explanations are not parsimonious.
We introduce a new minimality criterion called connection minimality that is parsimonious (Sect.3), defined for the lightweight description logic EL.This criterion characterizes hypotheses for T and α that connect the left-and right-hand sides of the observation α without introducing spurious connections.To achieve this, every left-hand side of a CI in the hypothesis must follow from the left-hand side of α in T , and, taken together, all the right-hand sides of the CIs in the hypothesis must imply the right-hand side of α in T , as is the case for H a1 .To compute connection-minimal hypotheses in practice, we present a technique based on first-order reasoning that proceeds in three steps (Sect.4).First, we translate the abduction problem into a first-order formula Φ.We then compute the prime implicates of Φ, that is, a set of minimal logical consequences of Φ that subsume all other consequences of Φ.In the final step, we construct, based on those prime implicates, solutions to the original problem.We prove that all hypotheses generated in this way satisfy the connection minimality criterion, and that the method is complete for a relevant subclass of connection-minimal hypotheses.We use the SPASS theorem prover [39] as a restricted SOS-resolution [22,40] engine for the computation of prime implicates in a prototype implementation (Sect.5), and we present an experimental analysis of its performances on a set of bio-medical ontologies.(Sect.6).Our results indicate that our method can in many cases be applied in practice to compute connection-minimal hypotheses.
There are not many techniques that can handle TBox abduction in EL or more expressive DLs [15,30,38].In [15], instead of a set of abducibles, a set of justification patterns is given, in which the solutions have to fit.An arbitrary oracle function is used to decide whether a solution is admissible or not (which may use abducibles, justification patterns, or something else), and it is shown that deciding the existence of hypotheses is tractable.However, different to our approach, they only consider atomic CIs in hypotheses, while we also allow for hypotheses involving conjunction.The setting from [38] also considers EL, and abduction under various minimality notions such as subset minimality and size minimality.It presents practical algorithms, and an evaluation of an implementation for an always-true informativeness oracle (i.e., limited to subset minimality).Different to our approach, it uses an external DL reasoner to decide entailment relationships.In contrast, we present an approach that directly exploits first-order reasoning, and thus has the potential to be generalisable to more expressive DLs.
While dedicated resolution calculi have been used before to solve abduction in DLs [13,30], to the best of our knowledge, the only work that relies on firstorder reasoning for DL abduction is [28].Similar to our approach, it uses SOSresolution, but to perform ABox adbuction for the more expressive DL ALC.Apart from the different problem solved, in contrast to [28] we also provide a semantic characterization of the hypotheses generated by our method.We believe this characterization to be a major contribution of our paper.It provides an intuition of what parsimony is for this problem, independently of one's ease with first-order logic calculi, which should facilitate the adoption of this minimality criterion by the DL community.Thanks to this characterization, our technique is calculus agnostic.Any method to compute prime implicates in first-order logic can be a basis for our abduction technique, without additional theoretical work, which is not the case for [28].Thus, abduction in EL can benefit from the latest advances in prime implicates generation in first-order logic.

Preliminaries
We first recall the descripton logic EL and its translation to first-order logic [4], as well as TBox abduction in this logic.
Let N C and N R be pair-wise disjoint, countably infinite sets of unary predicates called atomic concepts and of binary predicates called roles, respectively.Generally, we use letters A, B, E, F ,... for atomic concepts, and r for roles, possibly annotated.Letters C, D, possibly annotated, denote EL concepts, built according to the syntax rule We implicitly represent EL conjunctions as sets, that is, without order, nested conjunctions, and multiple occurrences of a conjunct.We use {C 1 , . . ., C m } to abbreviate C 1 . . .C m , and identify the empty conjunction (m = 0) with .
An EL TBox T is a finite set of concept inclusions (CIs) of the form C D.
EL is a syntactic variant of a fragment of first-order logic that uses N C and N R as predicates.Specifically, TBoxes T and CIs α correspond to closed first-order formulas π(T ) and π(α) resp., while concepts C correspond to open formulas π(C, x) with a free variable x.In particular, we have As common, we often omit the in conjunctions Φ, that is, we identify sets of formulas with the conjunction over those.The notions of a term t; an atom P ( t) where t is a sequence of terms; a positive literal P ( t); a negative literal ¬P ( t); and a clause, Horn, definite, positive or negative, are defined as usual for first-order logic, and so are entailment and satisfaction of first-order formulas.
We identify CIs and TBoxes with their translation into first-order logic, and can thus speak of the entailment between formulas, CIs and TBoxes.When T |= C D for some T , we call C a subsumee of D and D a subsumer of C. We adhere here to the definition of the word "subsume": "to include or contain something else", although the terminology is reversed in first-order logic.We say two TBoxes T 1 , T 2 are equivalent, denoted It is well known that, due to the absence of concept negation, every EL TBox is consistent.
The abduction problem we are concerned with in this paper is the following: where T is a TBox called the background knowledge, Σ is a set of atomic concepts called the abducible signature, and where m > 0, n ≥ 0 and such that T ∪ H |= C 1 C 2 and, for all CIs α ∈ H, T |= α.A solution to an abduction problem is called a hypothesis.
For example, H a1 and H a2 are solutions for T a , Σ, α a , as long as Σ contains all the atomic concepts that occur in them.Note that in our setting, as in [8,38], concept inclusions in a hypothesis are flat, i.e., they contain no existential role restrictions.While this restricts the solution space for a given problem, it is possible to bypass this limitation in a targeted way, by introducing fresh atomic concepts equivalent to a concept of interest.We exclude the consistency requirement T ∪ H |= ⊥, that is given in other definitions of DL abduction problem [29], since EL TBoxes are always consistent.We also allow m > 1 instead of the usual m = 1.This produces the same hypotheses modulo equivalence.
For simplicity, we assume in the following that the concepts C 1 and C 2 in the abduction problem are atomic.We can always introduce fresh atomic concepts A 1 and A 2 with A 1 C 1 and C 2 A 2 to solve the problem for complex concepts.

Connection-minimal Abduction
To address the lack of parsimony of common minimality criteria, illustrated in the academia example, we introduce connection minimality, Intuitively, connection minimality only accepts those hypotheses that ensure that every CI in the hypothesis is connected to both C 1 and C 2 in T , as is the case for H a1 in the academia example.The definition of connection minimality is based on the following ideas: 1) Hypotheses for the abduction problem should create a connection between C 1 and C 2 , which can be seen as a concept D that satisfies 2) To ensure parsimony, we want this connection to be based on concepts D 1 and D 2 for which we already have This prevents the introduction of unrelated concepts in the hypothesis.Note however that D 1 and D 2 can be complex, thus the connection from C 1 to D 1 (resp.D 2 to C 2 ) can be established by arbitrarily long chains of concept inclusions.3) We additionally want to make sure that the connecting concepts are not more complex than necessary, and that H only contains CIs that directly connect parts of D 2 to parts of D 1 by closely following their structure.
To address point 1), we simply introduce connecting concepts formally.If there was only one concept, C 1 and C 2 would already be connected, and as soon as there are more than two concepts, hypotheses start becoming more arbitrary: for a very simple example with unrelated concepts, assume given a TBox that entails Lion Felidae, Mammal Animal and House Building.A possible hypothesis to explain Lion Animal is {Felidae House, Building Mammal} but this explanation is more arbitrary than {Felidae Mammal}-as is the case when comparing H a2 with H a1 in the academia example-because of the lack of connection of House Building with both Lion and Animal.Clearly this CI could be replaced by any other CI entailed by T , which is what we want to avoid.
We can represent the structure of D 1 and D 2 in graphs by using EL description trees, originally from Baader et al. [5].Definition 3.An EL description tree is a finite labeled tree T = (V, E, v 0 , l) where V is a set of nodes with root v 0 ∈ V , the nodes v ∈ V are labeled with l(v) ⊆ N C , and the (directed) edges vrw ∈ E are such that v, w ∈ V and are labeled with r ∈ N R .
Given a tree T = (V, E, v 0 , l) and v ∈ V , we denote by T(v) the subtree of T that is rooted in v.If l(v 0 ) = {A 1 , . . ., A k } and v 1 , . .., v n are all the children of v 0 , we can define the concept represented by T recursively using If T = ∅, then subsumption between EL concepts is characterized by the existence of a homomorphism between the corresponding description trees [5].We generalise this notion to also take the TBox into account.
and only if the following conditions are satisfied: If only 1 and 2 are satisfied, then φ is called a weak homomorphism.
T -homomorphisms for a given TBox T capture subsumption w.r.We can finally capture our ideas on connection minimality formally.
Definition 7 (Connection-Minimal Abduction).Given an abduction problem T , Σ, C 1 C 2 , a hypothesis H is connection-minimal if there exist concepts D 1 and D 2 built over Σ ∪ N R and a mapping φ satisfying each of the following conditions: φ is a weak homomorphism from the tree T D2 = (V 2 , E 2 , w 0 , l 2 ) to the tree H is additionally called packed if the left-hand sides of the CIs in H cannot hold more conjuncts than they do, which is formally stated as: for H, there is no H' defined from the same D 2 and a D 1 and φ s.t.there is a node w ∈ V 2 for which l 1 (φ(w)) l 1 (φ (w)) and l 1 (φ(w )) = l 1 (φ (w )) for w = w.

Computing Connection-minimal Hypotheses using Prime Implicates
To compute connection-minimal hypotheses in practice, we propose a method based on first-order prime implicates, that can be derived by resolution.We assume the reader is familiar with the basics of first-order resolution, and do not reintroduce notions of clauses, Skolemization and resolution inferences here (for details, see [6]).In our context, every term is built on variables, denoted x, y, a single constant sk 0 and unary Skolem functions usually denoted sk, possibly annotated.Prime implicates are defined as follows.Let Σ ⊆ N C be a set of unary predicates.Then PI g+ Σ (Φ) denotes the set of all positive ground prime implicates of Φ that only use predicate symbols from Σ ∪ N R , while PI g− Σ (Φ) denotes the set of all negative ground prime implicates of Φ that only use predicates symbols from Σ ∪ N R .
Example 9. Given a set of clauses Φ = {A 1 (sk 0 ), ¬B 1 (sk 0 ), ¬A 1 (x)∨r(x, sk(x)), ¬A 1 (x) ∨ A 2 (sk(x)), ¬B 2 (x) ∨ ¬r(x, y) ∨ ¬B 3 (y) ∨ B 1 (x)}, the ground prime implicates of Φ for Σ = N C are, on the positive side, PI g+ Σ (Φ) = {A 1 (sk 0 ), A 2 (sk(sk 0 )), r(sk 0 , sk(sk 0 ))} and, on the negative side, PI g− Σ (Φ) = {¬B 1 (sk 0 ), ¬B 2 (sk 0 ) ∨ ¬B 3 (sk(sk 0 ))}.They are implicates because all of them are entailed by Φ.For a ground implicate ϕ, another ground implicate ϕ such that ϕ |= ϕ and ϕ |= ϕ can only be obtained from ϕ by dropping literals.Such an operation does not produce another implicate for any of the clauses presented above as belonging to PI g+ Σ (Φ)and PI g− Σ (Φ), thus they really are all prime.To generate hypotheses, we translate the abduction problem into a set of firstorder clauses, from which we can infer prime implicates that we then combine to obtain the result as illustrated in Fig. 2. In more details: We first translate the problem into a set Φ of Horn clauses.Prime implicates can be computed using an off-the-shelf tool [18,33] or, in our case, a slight extension of the resolutionbased version of the SPASS theorem prover [39] using the set-of-support strategy and some added features described in Sect. 5. Since Φ is Horn, PI g+ Σ (Φ) contains only unit clauses.A final recombination step looks at the clauses in PI g− Σ (Φ) one after the other.These correspond to candidates for the connecting concepts D 2 of Def. 7. Recombination attempts to match each literal in one such clause with unit clauses from PI g+ Σ (Φ).If such a match is possible, it produces a suitable D 1 to match D 2 , and allows the creation of a solution to the abduction problem.The set S contains all the hypotheses thus obtained.
In what follows, we present our translation of abduction problems into firstorder logic and formalize the construction of hypotheses from the prime implicates of this translation.We then show how to obtain termination for the prime implicate generation process with soundness and completeness guarantees on the solutions computed.
Abduction Method.We assume the EL TBox in the input is in normal form as defined, e.g., by Baader et al. [4].Thus every CI is of one of the following forms: The use of normalization is justified by the following lemma (see App A.3 for its proof).After the normalisation, we eliminate occurrences of , replacing this concept everywhere by the fresh atomic concept A .We furthermore add ∃r.A A and B A in T for every role r and atomic concept B occurring in T .This simulates the semantics of for A , namely the implicit property that C holds for any C no matter what the TBox is.In particular, this ensures that whenever there is a positive prime implicate B(t) or r(t, t ), A (t) also becomes a prime implicate.Note that normalisation and elimination extend the signature, and thus potentially the solution space of the abduction problem.This is remedied by intersecting the set of abducible predicates Σ with the signature of the original input ontology.We assume that T is in normal form and without in the rest of the paper.
We denote by T − the result of renaming all atomic concepts A in T using fresh duplicate symbols A − .This renaming is done only on concepts but not on roles, and on C 2 but not on C 1 in the observation.This ensures that the literals in a clause of PI g− Σ (Φ) all relate to the conjuncts of a -minimal subsumee of C 2 .Without it, some of these conjuncts would not appear in the negative implicates due to the presence of their positive counterparts as atoms in PI g+ Σ (Φ).The translation of the abduction problem T , Σ, C 1 C 2 is defined as the Skolemization of where sk 0 is used as the unique fresh Skolem constant such that the Skolemization of ¬π(C 1 C − 2 ) results in {C 1 (sk 0 ), ¬C − 2 (sk 0 )}.This translation is usually denoted Φ and always considered in clausal normal form.
Theorem 11.Let T , Σ, C 1 C 2 be an abduction problem and Φ be its firstorder translation.Then, a TBox H is a packed connection-minimal solution to the problem if and only if an equivalent hypothesis H can be constructed from non-empty sets A and B of atoms verifying: We call the hypotheses that are constructed as in Th. 11 constructible.This theorem states that every packed connection-minimal hypothesis is equivalent to a constructible hypothesis and vice versa.A constructible hypothesis is built from the concepts in one negative prime implicate in PI g− Σ (Φ) and all matching concepts from prime implicates in PI g+ Σ (Φ).The matching itself is determined by the Skolem terms that occur in all these clauses.The subterm relation between the terms of the clauses in PI g+ Σ (Φ) and PI g− Σ (Φ) is the same as the ancestor relation in the description trees of subsumers of C 1 and subsumees of C 2 respectively.The terms matching in positive and negative prime implicates allow us to identify where the missing entailments between a subsumer D 1 of C 1 and a subsumee D 2 of C 2 are.These missing entailments become the constructible H.The condition C B,t C A,t is a way to write that C A,t C B,t is not a tautology, which can be tested by subset inclusion.
The formal proof of this result is detailed in App C. 7 We sketch it briefly here.To start, we link the subsumers of C 1 with PI g+ Σ (Φ).This is done at the semantics level: We show that all Herbrand models of Φ, i.e., models built on the symbols in Φ, are also models of PI g+ Σ (Φ), that is itself such a model.Then we show that C 1 (sk 0 ) as well as the formulas corresponding to the subsumers of C 1 in our translation are satisfied by all Herbrand models.This follows from the fact that Φ is in fact a set of Horn clauses.Next, we show, using a similar technique, how duplicate negative ground implicates, not necessarily prime, relate to subsumees of C 2 , with the restriction that there must exist a weak homomorphism from a description tree of a subsumer of C 1 to a description tree of the considered subsumee of C 2 .Thus, H provides the missing CIs that will turn the weak homomorphism into a (T ∪ H)-homomorphism.Then, we establish an equivalence between the -minimality of the subsumee of C 2 and the primality of the corresponding negative implicate.Packability is the last aspect we deal with, whose use is purely limited to the reconstruction.It holds because A contains all A(t) ∈ PI g+ Σ (Φ) for all terms t occurring in B.
Example 12. Consider the abduction problem T a , Σ, α a where Σ contains all concepts from T a .For the translation Φ of this problem, we have where sk 1 is the Skolem function introduced for Professor ∃employment.Chair and sk 2 is introduced for Doctor ∃qualification.PhD.This leads to two constructible solutions: {Professor Doctor Researcher} and H a1 , that are both packed connection-minimal hypotheses if Σ = N C .Another example is presented in full details in App B.
Termination.If T contains cycles, there can be infinitely many prime implicates.For example, for T = {C 1 A, A ∃r.A, ∃r.B B, B C 2 } both the positive and negative ground prime implicates of Φ are unbounded even though the set of constructible hypotheses is finite (as it is for any abduction problem): To find all constructible hypotheses of an abduction problem, an approach that simply computes all prime implicates of Φ, e.g., using the standard resolution calculus, will never terminate on cyclic problems.However, if we look only for subset-minimal constructible hypotheses, termination can be achieved for cyclic and non-cyclic problems alike, because it is possible to construct all such hypotheses from prime implicates that have a polynomially bounded term depth, as shown below.To obtain this bound, we consider resolution derivations of the ground prime implicates and we show that they can be done under some restrictions that imply this bound.Before performing resolution, we compute the presaturation Φ p of the set of clauses Φ, defined as where A and B are either both original or both duplicate atomic concepts.The presaturation can be efficiently computed before the translation, using a modern EL reasoner such as Elk [27], which is highly optimized towards the computation of all entailments of the form A B. While the presaturation computes nothing a resolution procedure could not derive, it is what allows us to bind the maximal depth of terms in inferences to that in prime implicates.If Φ p is presaturated, we do not need to perform inferences that produce Skolem terms of a higher nesting depth than what is needed for the prime implicates.
Starting from the presaturated set Φ p , we can show that all the relevant prime implicates can be computed if we restrict all inferences to those where R1 at least one premise contains a ground term, R2 the resolvent contains at most one variable, and R3 every literal in the resolvent contains Skolem terms of nesting depth at most n × m, where n is the number of atomic concepts in Φ, and m is the number of occurrences of existential role restrictions in T .
The first restriction turns the derivation of PI g+ Σ (Φ) and PI g− Σ (Φ) into an SOSresolution derivation [22] with set of support {C 1 (sk 0 ), C − 2 (sk 0 )}, i.e., the only two clauses with ground terms in Φ.This restriction is a straightforward consequence of our interest in computing only ground implicates, and of the fact that the non-ground clauses in Φ cannot entail the empty clause since every EL TBox is consistent.The other restrictions are consequences of the following theorems, whose proofs are available in App.D.
Theorem 13.Given an abduction problem and its translation Φ, every constructible hypothesis can be built from prime implicates that are inferred under restriction R2.
In fact, for PI g+ Σ (Φ) it is even possible to restrict inferences to generating only ground resolvents, as can be seen in the proof of Th. 13, that directly looks at the kinds of clauses that are derivable by resolution from Φ. Theorem 14.Given an abduction problem and its translation Φ, every subsetminimal constructible hypothesis can be built from prime implicates that have a nesting depth of at most n × m, where n is the number of atomic concepts in Φ, and m is the number of occurrences of existential role restrictions in T .
The proof of Th. 14 is based on a structure called a solution tree, which resembles a description tree, but with multiple labeling functions.It assigns to each node a Skolem term, a set of atomic concepts called positive label, and a single atomic concept called negative label.The nodes correspond to matching partners in a constructible hypothesis: The Skolem term is the term on which we match literals.The positive label collects the atomic concepts in the positive prime implicates containing that term.The maximal anti-chains of the tree, i.e., the maximal subsets of nodes s.t.no node is the ancestor of another are such that their negative labels correspond to the literals in a derivable negative implicate.For every solution tree, the Skolem labels and negative labels of the leaves determine a negative prime implicate, and by combining the positive and negative labels of these leaves, we obtain a constructible hypothesis, called the solution of the tree.We show that from every solution tree with solution H we can obtain a solution tree with solution H ⊆ H s.t. on no path, there are two nodes that agree both on the head of their Skolem labeling and on the negative label.Furthermore the number of head functions of Skolem labels is bounded by the total number n of Skolem functions, while the number of distinct negative labels is bounded by the number m of atomic concepts, bounding the depth of the solution tree for H at n × m.This justifies the bound in Th 14.This bound is rather loose.For the academia example, it is equal to 22 × 6 = 132.

Implementation
We implemented our method to compute all subset-minimal constructible hypotheses in the tool CAPI. 8To compute the prime implicates, we used SPASS [39], a first-order theorem prover that includes resolution among other calculi.We implemented everything before and after the prime implicate computation in Java, including the parsing of ontologies, preprocessing (detailed below), clausification of the abduction problems, translation to SPASS input, as well as the parsing and processing of the output of SPASS to build the constructible hypotheses and filter out the non-subset-minimal ones.On the Java side, we used the OWL API for all DL-related functionalities [24], and the EL reasoner Elk for computing the presaturations [27].
Preprocessing.Since realistic TBoxes can be too large to be processed by SPASS, we replace the background knowledge in the abduction problem by a subset of axioms relevant to the abduction problem.Specifically, we replace the abduction problem (T , Σ, , where M ⊥ C1 is the ⊥-module of T for the signature of C 1 , and M C2 is themodule of T for the signature of C 2 [20].Those notions are explained in App E. Their relevant properties are that For the presaturation, we compute with Elk all CIs of the form A B s.t.
Prime implicates generation.We rely on a slightly modified version of SPASS v3.9 to compute all ground prime implicates.In particular, we added the possibility to limit the number of variables allowed in the resolvents to enforce R2.
For each of the restrictions R1 -R3 there is a corresponding flag (or set of flags) that is passed to SPASS as an argument.
Recombination.The construction of hypotheses from the prime implicates found in the previous stage starts with a straightforward process of matching negative prime implicates with a set of positive ones based on their Skolem terms.It is followed by subset minimality tests to discard non-subset-minimal hypotheses, since, with the bound we enforce, there is no guarantee that these are valid constructible hypotheses because the negative ground implicates they are built upon may not be prime.If SPASS terminates due to a timeout instead of reaching the bound, then it is possible that some subset-minimal constructible hypotheses are not found, and thus, some non-constructible hypotheses may be kept.Note that these are in any case solutions to the abduction problem.

Experiments
There is no benchmark suite dedicated to TBox abduction in EL, so we created our own, using realistic ontologies from the bio-medical domain.For this, we used ontologies from the 2017 snapshot of Bioportal [32].We restricted each ontology to its EL fragment by filtering out unsupported axioms, where we replaced domain axioms and n-ary equivalence axioms in the usual way [4].Note that, even if the ontology contains more expressive axioms, an EL hypothesis is still useful if found.From the resulting set of TBoxes, we selected those containing at least 1 and at most 50,000 axioms, resulting in a set of 387 EL TBoxes.Precisely, they contained between 2 and 46,429 axioms, for an average of 3,039 and a median of 569.Towards obtaining realistic benchmarks, we created three different categories of abduction problems for each ontology T , where in each case, we used the signature of the entire ontology for Σ.
• Problems in ORIGIN use T as background knowledge, and as observation a randomly chosen A B s.t.A and B are in the signature of T and T |= A B. This covers the basic requirements of an abduction problem, but has the disadvantage that A and B can be completely unrelated in T .
• Problems in JUSTIF contain as observation a randomly selected CI α s.t., for the original TBox, T |= α and α ∈ T .The background knowledge used is a justification for α in T [37], that is, a minimal subset I ⊆ T s.t.I |= α, from which a randomly selected axiom is removed.The TBox is thus a smaller set of axioms extracted from a real ontology for which we know there is a way of producing the required entailment without adding it explicitly.Justifications were computed using functionalities of the OWL API and Elk.• Problems in REPAIR contain as observation a randomly selected CI α s.t.
T |= α, and as background knowledge a repair for α in T , which is a maximal subset R ⊆ T s.t.R |= α.Repairs were computed using a justificationbased algorithm [37] with justifications computed as for JUSTIF.This usually resulted in much larger TBoxes, where more axioms would be needed to establish the entailment.
All experiments were run on Debian Linux (Intel Core i5-4590, 3.30 GHz, 23 GB Java heap size).The code and scripts used in the experiments are available online [21].The three phases of the method (see Fig. 2) were each assigned a hard time limit of 90 seconds.
For each ontology, we attempted to create and translate 5 abduction problems of each category.This failed on some ontologies because either there was no corresponding entailment (25/28/25 failures out of the 387 ontologies for ORIGIN/JUSTIF/REPAIR), there was a timeout during the translation (5/5/5 failures for ORIGIN/JUSTIF/REPAIR), or because the computation of justifications caused an exception (-/2/0 failures for ORIGIN/JUSTIF/REPAIR).The final number of abduction problems for each category is in the first column of Table 1.
We then attempted to compute prime implicates for these benchmarks using SPASS.In addition to the hard time limit, we gave a soft time limit of 30 seconds to SPASS, after which it should stop exploring the search space and return the implicates already found.In Table 1 we show, for each category, the percentage of problems on which SPASS succeeded in computing a non-empty set of clauses (Success) and the percentage of problems on which SPASS terminated within the time limit, where all solutions are computed (Compl.).The high number of CIs in the background knowledge explains most of the cases where SPASS reached the soft time limit.In a lot of these cases, the bound on the term depth goes into the billion, rendering it useless in practice.However, the "Compl."column shows that the bound is reached before the soft time limit in most cases.
The reconstruction never reached the hard time limit.We measured the median, average and maximal number of solutions found (#H), size of solutions in number of CIs (|H|), size of CIs from solutions in number of atomic concepts (|α|), and SPASS runtime (time, in seconds), all reported in Table 1.Except for the simple JUSTIF problems, the number of solutions may become very large.At the same time, solutions always contain very few axioms (never more than 3), though the axioms become large too.We also noticed that highly nested Skolem terms rarely lead to more hypotheses being found: 8/1/15 for ORIGIN/JUSTIF/REPAIR, and the largest nesting depth used was: 3/1/2 for ORIGIN/JUSTIF/REPAIR.This hints at the fact that longer time limits would not have produced more solutions, and motivates future research into redundancy criteria to stop derivations (much) earlier.

Conclusion
We have introduced connection-minimal TBox abduction for EL which finds parsimonious hypotheses, ruling out the ones that entail the observation in an arbitrary fashion.We have established a formal link between the generation of connection-minimal hypotheses in EL and the generation of prime implicates of a translation Φ of the problem to first-order logic.In addition to obtaining these theoretical results, we developed a prototype for the computation of subsetminimal constructible hypotheses, a subclass of connection-minimal hypotheses that is easy to construct from the prime implicates of Φ.Our prototype uses the SPASS theorem prover as an SOS-resolution engine to generate the needed implicates.We tested this tool on a set of realistic medical ontologies, and the results indicate that the cost of computing connection-minimal hypotheses is high but not prohibitive.We see several ways to improve our technique.The bound we computed to ensure termination could be advantageously replaced by a redundancy criterion discarding irrelevant implicates long before it is reached, thus greatly speeding computation in SPASS.We believe it should also be possible to further constrain inferences, e.g., to have them produce ground clauses only, or to generate the prime implicates with terms of increasing depth in a controlled incremental way instead of enforcing the soft time limit, but these two ideas remain to be proved feasible.As an alternative to using prime implicates, one may investigate direct method for computing connection-minimal hypotheses in EL.
The theoretical worst-case complexity of connection-minimal abduction is another open question.Our method only gives a very high upper bound: by bounding only the nesting dept of Skolem terms polynomially as we did with Th. 13, we may still permit clauses with exponentially many literals, and thus double exponentially many clauses in the worst case, which would give us an 2ExpTime upper bound to the problem of computing all subset-minimal con-structible hypotheses.Using structure-sharing and guessing, it is likely possible to get a lower bound.We have not looked yet at lower bounds for the complexity either.
While this work focuses on abduction problems where the observation is a CI, we believe that our technique can be generalised to knowledge that also contains ground facts (ABoxes), and to observations that are of the form of conjunctive queries on the ABoxes in such knowledge bases.The motivation for such an extension is to understand why a particular query does not return any results, and to compute a set of TBox axioms that fix this problem.Since our translation already transforms the observation into ground facts, it should be possible to extend it to this setting.We would also like to generalize TBox abduction by finding a reasonable way to allow role restrictions in the hypotheses, and to extend connection-minimality to more expressive DLs such as ALC.

A Various Minor Results
A.1 T -homomorphism and Entailment Lemma 15.
Proof.We prove this result by induction on the structure of T 2 . If In the general case, let us consider any child w i of w 0 in T 2 since there must be at least one.Then there is a corresponding child . And in particular, for the r i such that w 0 r i w ∈ E 2 , we have T |= ∃r i .C T1(v) ∃r i .C T2(w) .This applies to all the children w 1 , . . ., w n of w 0 , and since

B Detailed Example
Consider the abduction problem T , Σ, C 1 C 2 where and Σ = {A, B, D, E, F, G, H, L}.Consider the concepts Indeed, the concepts D 1 and D 2 are such that There is also a weak homomorphism from T D2 to T D1 , as illustrated in Fig. 3. Thus, is a connection-minimal hypothesis.Note that the tautology M M , that is one of the entailments that must hold in T ∪ H, as is visible in Fig. 3, is not included in H since it is a tautology and thus T |= M M .The hypothesis H is even packed.In contrast, that are both connection-minimal but lack either L or H on the left-hand side of their first CI when compared with H, are not packed.
We apply our technique to compute the hypotheses of this abduction problem.Since T is not in normal form, we must normalize it before the translation.The CIs to normalize are ∃r 1 .X E C 2 and ∃r 2 .M ∃r 2 .Z Y for which we introduce the fresh concepts U , V and W and corresponding CIs ∃r 2 .M U , ∃r 2 .Z V and ∃r 1 .X W , along with the normalized form of the two initial CIs, i.e., W E C 2 and U V Y .Since T does not contain , there is no need to introduce A .We write the CIs in the normalization of T and their first-order translation after Skolemization side by side.
The translation of T − is identical to that of T up to the replacement of every unary predicate with its duplicate and the introduction of fresh Skolem functions distinct from the ones used for T .Let Φ denote the full translation of the problem.The ground prime implicates for Σ = {A, B, E, F, G, H, L, M } are as follows: where sk 1 is the Skolem function corresponding to the existential quantifier introduced by the translation of ∃r 1 .X to first-order logic, and where sk 2 and sk 2 correspond respectively to ∃r 2 .M and ∃r 2 .Z.The only constructible hypothesis out of this configuration is H, the packed connection-minimal hypothesis already introduced.Finally, H 3 is the smallest TBox that fixes all entailments missing between D 1 and D 2 , ensuring the connection minimality of H 3 and it is packed, contrarily to H 1 and H 2 that lack either L or H on the left-hand side of their first CI.Note that the signature restriction has been made to capture only these solutions, but there would be many more if we considered the whole signature after normalization for Σ.In particular, including A to Σ would produce all solutions where replaces the left-hand side of some CIs in another solution, so it should generally be avoided.

C Construction
We recall the statement of the main theorem that we prove in this appendix: Theorem 11.Let T , Σ, C 1 C 2 be an abduction problem and Φ be its firstorder translation.Then, a TBox H is a packed connection-minimal solution to the problem if and only if an equivalent hypothesis H can be constructed from non-empty sets A and B of atoms verifying: Until the end of this appendix, we assume Σ = N C .For Th. 11, the case where Σ N C trivially follows, but that is not the case for the intermediate results.
To prove Th. 11, we first establish the link between the positive prime implicates in PI g+ Σ (Φ) and the subsumers of C 1 , then we do the same for the negative side.First, we adapt the notion of a canonical model by Baader et al. [5], to construct a minimal Herbrand Interpretation M ensuring that, for a given concept C, at least one Skolem term t is such that t ∈ C M .We show how to extend a canonical model so that it also satisfies T and we link the existence of such a model built for some C T and sk 0 to the existence of the entailment T |= C 1 C T , while showing that this model is in fact a subset of PI g+ Σ (Φ).Second, we show how renamed negative ground implicates, not necessarily prime, relate to the subsumees of C 2 .To that aim, we again rely on a canonical model, but this time for the renamed version of some C T subsumee of C 2 , with the restriction that there must exist a weak homomorphism from a subsumer of C 1 to this C T , the idea being that H is built to provide the missing CIs that will turn the weak homomorphism into a (T ∪ H)-homomorphism.
Finally, we establish an equivalence between the -minimality of C T and the fact that the corresponding renamed negative implicate is prime.
Before diving into the proofs, remember that we work under the assumptions that T is in normal form and without , and as a consequence π(T ) contains only axioms of the following shapes: After Skolemization, the clauses are all Horn.
We provide a direct specification of EL semantics, that we work with in the proofs.It uses first-order structures or interpretations, which are tuples I = (∆ I , • I ) made of a domain ∆ I and an interpretation function • I that maps atomic concepts A ∈ N C to sets A I ⊆ ∆ I and roles r ∈ N R to relations r I ⊆ ∆ I × ∆ I .The interpretation function • I is extended to complex concepts as follows: Let N S denote the set of all monadic Skolem functions that are used to Skolemize the translation of an abduction problem to first-order logic.We call an interpretation I with ∆ I = T sk0 (N S ), where T sk0 (N S ) is the set of terms built on the constant sk 0 and functions from N S , a Herbrand intepretation, which for convenience, we identify with the set of ground atoms that are satisfied in it.Specifically, for a Herbrand interpretation I, we write A(t) ∈ I if t ∈ A I , and r(t, t ) ∈ I if (t, t ) ∈ r I .

C.1 Subsumers of C 1 and Positive Prime Implicates
We derive a relation between the subsumers of C 1 in T and the prime implicates of Φ = Π(T , C 1 C 2 ).This relation is established at the semantics level, by constructing a Herbrand model of T and showing it necessarily contains the prime implicates of Φ.
For this purpose, we adapt the definition of a canonical model from [5] by using T sk0 (N S ) for the domain of the EL-description tree T corresponding to a subsumer of C 1 .9 Definition 16 (Canonical Model).Given a description tree T = (V, E, v 0 , l), a Skolem labeling sl T : V → T sk0 (N S ) of T maps the vertices of T to ground Skolem terms.A canonical model M(sl T ) of T is a Herbrand interpretation consisting of the following atoms: We denote by M A (sl T ) the subset of M(sl T ) made of all atoms built over unary predicates, and by M r (sl T ), the rest of M(sl T ), that contains all atoms built over binary predicates.
It is always possible to find a canonical model of an EL-description tree T as a subset in any Herbrand interpretation I for which (C T ) I is not empty.This is formally stated, and proven, in the following lemma.
Lemma 17.Given an EL-description tree T = (V, E, v 0 , l) and a Herbrand interpretation I, if t ∈ (C T ) I then there exists a Skolem labeling sl T s.t.sl T (v 0 ) = t and M(sl T ) ⊆ I Proof.Given an EL-description tree T = (V, E, v 0 , l), a Herbrand interpretation I and a Skolem term t, such that t ∈ (C T ) I , let us construct the suitable Skolem labeling sl T .We proceed inductively on the depth of T.
where T(w) is the subtree of T rooted in w, V T(w) is the subset of V that occurs in T(w) and sl T(w) is defined as sl T but on t instead of t, for a t such that r(t, t ) ∈ I and t ∈ (C T(w) ) I .Such a t must exist because ∃r.C T(w) is a conjunct in C T and t ∈ (C T ) I .Hence sl T(w) is well-defined.This construction terminates because the depth of all T(w) is strictly smaller than that of T.
If T is of depth 0, then sl T is simply defined on v 0 such that sl T (v 0 ) = t, and C T is a conjunction of atomic concepts A ∈ l(v 0 ).Thus, t ∈ (C T ) I is equivalent to A(t) ∈ I for all A ∈ l(v 0 ).Hence, any atom A(sl If T is of depth i > 0, for any v ∈ V \ {v 0 }, there exists a w ∈ V such that v 0 rw ∈ E and v ∈ V T(w) , i.e., v must belong to a subtree rooted in one of the children w of the root of T. Then sl T (v) = sl T(w) (v).By induction, M(sl T(w) ) ⊆ I. Moreover A(t) ∈ I for all A ∈ l(v 0 ) as in the base case; and r(t, sl T(w) (w)) ∈ I and sl T(w) (w) ∈ (C T(w) ) I for all w children of v 0 by construction of sl T(w) .Thus I(sl T ) ⊆ I. Now, we show that PI g+ Σ (Φ) holds the role of universal Herbrand model for Φ = Π(T , C 1 C 2 ).The proof adapts a result by Bienvenu et al. [9] to the case with only one constant but a possibly infinite domain.
Lemma 18 (PI g+ Σ (Φ) as a universal model).Given the translation Φ of an abduction problem, the set PI g+ Σ (Φ) considered as a Herbrand interpretation is a model of Φ and for any other Herbrand model I of Φ, PI g+ Σ (Φ) ⊆ I.
Proof.By the definition of a prime implicate, any model of Φ must be a model of any ϕ ∈ PI g+ Σ (Φ).Moreover, a positive prime implicate can only be an atom since Φ contains only Horn clauses.Thus all Herbrand models must contain PI g+ Σ (Φ).
To show that PI g+ Σ (Φ) is itself a Herbrand model, we construct the Herbrand Interpretation I = i I i for i ∈ N where: • given I j , We show that I is a model of Φ.We know that I |= C 1 (sk 0 ) by construction of I 0 , and that all other clauses containing only non-duplicated literals are also satisfied by I, again by construction.Note that there are cases where no I j alone is enough to satisfies a clause, but they are all satisfied at the limit by I (e.g., if T includes a concept inclusion A ∃r.A, possibly leading to the presence of infinitely many atoms of the form A(sk n (t)) ∈ I).Regarding the remaining clauses in Φ, they all contain at least one literal of the form ¬A − (x) and since I includes no atom A − (t) at all, I also satisfies that part, thus I is a model of Φ.
It remains only to show that I ⊆ PI g+ Σ (Φ) to have PI g+ Σ (Φ) = I, thus showing that PI g+ Σ (Φ) is a model of Φ.This is done by induction.Clearly I 0 ⊆ PI g+ Σ (Φ), and, assuming I j ⊆ PI g+ Σ (Φ) for some j ≥ 0, then any atom in I j can be derived by resolution from Φ, thus, by construction, any atom in I j+1 \ I j can be derived from Φ by one additional resolution step, making them implicates of Φ.Because they are atoms, they must be prime implicates, thus I j+1 ⊆ PI g+ Σ (Φ), completing the induction.
Note that if Φ was a set of definite Horn clauses, the above result would be immediate because it is well-known in logic programming [31].The presence of the negative clause ¬C − 2 (sk 0 ) ∈ Φ is what justifies the existence of the current proof.
Now that PI g+ Σ (Φ) has been established as the universal Herbrand model of Φ, the atoms it contains can be used to reconstruct concepts subsuming C 1 by means of a canonical model.

Lemma 19 (Canonical Model and PI g+
Σ (Φ)).Given an abduction problem T , H, C 1 C 2 , its first-order translation Φ and an EL-description tree T = (V, E, v 0 , l), the entailment T |= C 1 C T holds if and only if there exists a Skolem labeling sl T such that sl T (v 0 ) = sk 0 and M(sl T ) ⊆ PI g+ Σ (Φ).To prove the opposite implication, we assume given a Skolem labeling verifying sl T (v 0 ) = sk 0 and M(sl T ) ⊆ PI g+ Σ (Φ) and show that T |= C 1 C T by contradiction.Then sk 0 ∈ (C T ) M(sl T ) because sl T (v 0 ) = sk 0 .Towards contradiction, we assume T |= C 1 C T .Then π(T ) |= π(C 1 C T ), since the standard translation from EL to first-order logic preserves entailment [4].Thus π(T ) ∧ ¬π(C 1 C T ) is satisfiable and hence, the Skolemizations of are also satisfiable.Let us consider the particular Skolemization ϕ of π(T ) ∧ ¬π(C 1 C T ) that coincides with Φ on the Skolemization of T and uses sk 0 to Skolemize the existential variable in ¬π(C 1 C T ).Let I be a minimal Herbrand model of ϕ.It verifies sk 0 ∈ (C 1 ) I and sk 0 ∈ (C T ) I .We show that I is a model of Φ, which will allow us to raise a contradiction on that last statement.Since, by design, ϕ contains all the non-renamed clauses in Φ, it follows that I satisfies these non-renamed clauses also for Φ.Since ϕ does not include renamed atoms, the minimality of I ensures that it does not include any renamed atoms.This ensures that I also models the renamed part of Φ: for any renamed C − D − , it holds that (C − ) I = (D − ) I = ∅, and ¬C − 2 (sk 0 ) is also true in I'.Thus, I is a model of Φ.However, since M(sl T ) ⊆ PI g+ Σ (Φ), and PI g+ Σ (Φ) ⊆ I by Lemma 18, it follows that M(sl T ) ⊆ I must hold.In addition, since C T (sk 0 ) ∈ M(sl T ) because sl T (v 0 ) = sk 0 , it follows that sk 0 ∈ (C T ) I , a contradiction.Proof.Given an abduction problem T , H, C 1 C 2 , its first-order translation Φ and a set A = {A 1 (t 1 ), . . ., A n (t n )} ⊆ PI g+ Σ (Φ) where n > 0, we notice that every singleton set {A i (t i )} ⊆ A also verifies that {A i (t i )} ⊆ PI g+ Σ (Φ).Thus to prove the property for any A, we first show it for singletons and then we show how to construct a description tree for any A given the description trees for each singleton containing an element of A.
Let A = {A(t)} be a singleton.In practice, we need a slightly stronger property: we show the existence of a T = (V, E, v 0 , l) and sl T such that As shown in the proof of Lemma 18, we can write PI g+ Σ (Φ) as i∈N I i , where: • given I j , Since A(t) ∈ PI g+ Σ (Φ), there exists an i ∈ N that is the smallest such that A(t) ∈ I i .We construct T inductively, depending on the value of i.If i = 0, then A(t) = C 1 (sk 0 ) and thus defining T and sl T as T = ({v 0 }, ∅, v 0 , {v 0 → {C 1 }}) and sl T = {v 0 → sk 0 } ensures additionally that Assuming we know how to construct suitable description trees and Skolem labelings up to a given j ∈ N, when i = j + 1, the construction of T depends on the reason for which A(t) ∈ I j+1 \ I j .
where D is in fact an atomic concept B then B(t) ∈ I j and thus B(t) ∈ PI g+ Σ (Φ).By induction, let T = (V , E , v 0 , l ) and sl T be a description tree and Skolem labeling such that sl T (v 0 ) = sk 0 , M(sl T ) ⊆ PI g+ Σ (Φ) and {B(t)} = M A (sl T ).Let v ∈ V be the node such that l T (v) = {B} and sl T (v) = t.Then the Skolem labeling sl T is defined as identical to sl T and we define T as (V , E , v 0 , l [v → {A}]), where l [v → {A}] denotes the function l except on v for which the value returned is {A} so that {A(t)} = M A (sl T ) as wanted.Since M(sl T ) ⊆ {A(t)} ∪ M(sl T ) and A(t) ∈ PI g+ Σ (Φ), it follows that M(sl T ) ⊆ PI g+ Σ (Φ).Ij , ¬π(D, x) ∨ B(x) ∈ Φ}, where D is in fact the conjunction of two atomic concepts B 1 and B 2 , then both B 1 (t) and B 2 (t) belong to I j and thus to PI g+ Σ (Φ).We adapt exactly as in the last case any of the description trees T 1 or T 2 and associated Skolem labeling sl T1 or sl T2 , that respectively correspond to B 1 and B 2 and verify the properties by induction.
¬π(A , x) ∨ r(x, sk(x)) ∈ Φ} then t = sk(t ) for some sk and t such that A (t ) ∈ I j , ¬A (x) ∨ A(sk(x)), ¬A (x) ∨ r(x, sk(x)) ∈ Φ for some r and A .Since A (t ) ∈ PI g+ Σ (Φ), there exists a description tree T = (V , E , v 0 , l ) such that sl T (v 0 ) = sk 0 , M A (sl T ) = {A (t )}, and M(sl T ) PI g+ Σ (Φ).Let v be the leaf node such that l (v) = {A } and sl T (v) = t .We introduce a fresh node v to define T as (V ∪{v}, ∈ Φ} then there exist A , r and sk such that A (sk(t)) ∈ I j , r(t, sk(t)) ∈ I j , and ¬r(x, y) ∨ ¬A (y) ∨ A(x) ∈ Φ.By induction, we consider a description tree T = (V , E , v 0 , l ) and associated Skolem labeling sl T for which sl T (v 0 ) = sk 0 , M A (sl T ) = {A (t )}, and M(sl T ) ⊆ PI g+ Σ (Φ).Let v be the leaf in V such that l (v) = {A } and w be its parent in the tree, such that wr v ∈ E for some r .We define T as (V \ {v}, E \ {wr v}, v 0 , l [w → {A}] \ {v → {A }}) and sl T = sl T \ {v → sk(t)}.Thus, sl Let us now consider the case of non-singleton A = {A 1 (t 1 ), . . ., A n (t n )} (n > 1).We have just seen how to obtain description trees T i = (V i , E i , v 0 , l i ) and Skolem labelings sl Ti for i ∈ {1, . . ., n} such that {A i (t i )} = M A (sl Ti ), sl Ti (v 0 ) = sk 0 and M(sl Ti ) ⊆ PI g+ Σ (Φ).We define T = (V, E, v 0 , l) and sl T by introducing a node v ∈ V for each t ∈ n i=1 {sl Ti (v ) | v ∈ V i } and setting sl T (v) = t in each case.For t = sk 0 , the introduced node v ∈ V is named v 0 and declared as the root of T. It remains to define E and l.For E we collect all edges from the description trees T i to obtain For l, we proceed similarly to collect labels, producing for each v ∈ V ,

C.2 Subsumees of C 2 and Negative Prime Implicates
Next, we show how negative ground implicates are related to the solutions of the abduction problem.
Let us now assume (FO) in order to prove (EL).We consider a Herbrand interpretation I = PI g+ Σ (Φ) ∪ i I i where I i for i ∈ N is defined inductively as: ) and, • given I j , The I i s are built to collect all the elements necessary to make the renamed part of Φ true, one step at a time, starting from an interpretation that satisfies ), and is thus incompatible with Φ under the (FO) assumption.Indeed since T , and by extension T − , is in normal form, it contains only concept inclusions of the form D B, A ∃r.B and ∃r.A B, where D is either an atomic concept or a conjunction of two atomic concepts.These correspond in Φ respectively to the clauses ¬π(D − , x)∨B − (x), to the pair of clauses {¬π(D − , x) ∨ B − (sk(x)), ¬π(D − , x) ∨ r(x, sk(x))} and to the clause ¬r(x, sk(x)) ∨ ¬A − (sk(x)) ∨ B − (x).Hence, if t ∈ (D − ) I (resp.sk(t) ∈ (A − ) I and (t, sk(t)) ∈ r I ) then there exists some i ∈ N such that t ∈ (D − ) Ii (resp.sk(t) ∈ (A − ) Ii and (t, sk(t)) ∈ r Ii ) and then all concept inclusions where D − occurs on the right-hand side (resp.where ∃r.A − occurs on the right-hand side for some r) are satisfied in I i+1 .
Since I includes PI g+ Σ (Φ), the satisfiability of C 1 (sk 0 ) and the Skolemization of π(T ) can be shown as in Lemma 20.Thus, given that 2 (sk 0 ).To make use of that fact, we must first prove the following statement: ( * ) For any set B = {B − 1 (t 1 ), . . ., B − k (t k )} ⊆ I, there exists T = (V, E, v 0 , l) and Without loss of generality, we can consider the biggest such B, that is the set of all atoms of the form A − (t) in I.The smaller Bs simply correspond to concepts C T with fewer conjuncts.
Since PI g+ Σ (Φ) only contains non-renamed concepts, we prove ( * ) by induction on the I j for j ∈ N. When j = 0, I 0 = M(sl T2 ), thus C T = C T2 and the result directly follows.Assuming the result holds for a given I j , let The induction hypothesis applies to B * and we conclude that there exists an EL-description tree T * = (V * , E * , v 0 , l * ) and a Skolem labeling Let us now consider the literals in B \ B * .They are all of the form B − (t) and belong to I j+1 .We define T and sl T by extending T * and sl T * .The extension for each B − (t) depends of which set it originates from.
the node in T * such that l * (v) contains all atomic concepts from D and sl T * (v) = t.We add B to l * (v) and the rest of T * and sl T * is unchanged.Note that, in that case, D B ∈ T by construction of Φ.
, and A ∈ l * (w).As in the previous case, we simply add B to l * (v).In that case, ∃r.A B ∈ T for the corresponding r.
Then we add a fresh node w to V * as well as an edge vrw to E * .We also extend sl T * so that w is mapped to t.In that case, A ∃r.B ∈ T for the corresponding r.
Note that in all cases, T |= C T * C T because the conjunct(s) added from C T * to C T is(/are) justified by the concept inclusion from T that is ultimately to blame for the existence of B − (t) in I j+1 \ I j .Since, by the induction hypothesis, * ) holds for that case and thus also for I. Because

Lemma 22 (EL-Description Tree and
).Given two EL-description trees Proof.If C T1 ≡ C T2 , we are done because then T 1 and T 2 obviously have the same shape.Otherwise, the missing conjuncts in C 1 would correspond to either: • some missing atomic concepts in some l 2 (w) from T 2 or • a subtree of T 2 that is not in the image of the ∅-homomorphism from T 1 .
Last, we show how prime implicates are related to the connection-minimal solutions of the abduction problem.
For any EL-description tree T 2 = (V 2 , E 2 , w 0 , l 2 ) with a weak homomorphism φ from T 2 to T 1 , the following equivalence holds:

and
Proof.Thanks to Lemma 21, we know that, in the conditions of the lemma, the existence of a C T2 such that T |= C T2 C 2 is equivalent to the existence of a Skolem labeling sl T2 for T 2 such that sl T2 (v) = sl T1 (φ(v)) for all v ∈ V T2 , and Φ |= v∈V2,B∈l2(v) ¬B − (sl T2 (v)).It remains to show the equivalence between the -minimality of C T2 and the fact that v∈V2,B∈l2(v) ¬B − (sl T2 (v)) ∈ PI g− Σ (Φ).By Lemma 22, for any two trees T 2 and T 2 and corresponding Skolem labelings sl T 2 and sl T 2 for which there are respective weak homomorphisms φ 1 and φ is not prime and vice-versa.
Theorem 11, that shows how to construct solutions for an abduction problem from prime implicates of its translation to first-order logic, is a consequence of Lemma 23.
Theorem 11.Let T , Σ, C 1 C 2 be an abduction problem and Φ be its firstorder translation.Then, a TBox H is a packed connection-minimal solution to the problem if and only if an equivalent hypothesis H can be constructed from non-empty sets A and B of atoms verifying: Proof.Let T , Σ, C 1 C 2 be an abduction problem and Φ be its first-order translation.
We begin by assuming given a packed connection-minimal hypothesis H. Then there exist concepts D 1 and D 2 , and weak homomorphism φ verifying points 1-3 of Def. 7 while H verifies point 4 of the same definition for these D 1 , D 2 and φ.W.l.o.g., we consider that D 1 is such that every node in T D1 is in the range of φ.Such a D 1 can always be obtained from a D 1 that has too much nodes by pruning the extra nodes, since they cannot have children that are in the range of φ.Since, by Def. 7 point 1, T |= C 1 D 1 , by Lemma 19 there exists a Skolem labeling sl 1 for ) cannot be empty and that there are no two nodes in T D1 with the same Skolem label, otherwise M(sl 1 ) ⊆ PI g+ Σ (Φ) would not hold since this would imply that two occurrences of Skolem terms in Φ share the same Skolem function, which is forbidden in the standard Skolemization procedure.From point 3 of Def. 7, we know that φ is a weak homomorphism from T D2 to T D1 and from point 2 that D 2 is a -minimal concept s.t.T |= D 2 C 2 .Hence, by Lemma 23, there also exists a Skolem labeling sl 2 for Our choice of D 1 allows us to define A as M A (sl 1 ) since it holds that sl 2 (v) = sl 1 (φ(v)) and there are no nodes in V 1 outside the range of φ.Thus A and B verify the first two points of Th. 11.Let us now consider any concept inclusion in H.It is of the form l 1 (φ(w)) l 2 (w) for some w ∈ V 2 and s.t.
C B,t for t = sl 1 (v).This means in particular that this CI is not a tautology, ensuring that C B,t C A,t .Thus H is equivalent to the constructible hypothesis built for A and B as just defined.Now, let us consider that H is a constructible hypothesis obtained from a given A and B verifying the constraints from Th. 11.Then A is a subset of PI g+ Σ (Φ), thus, by Lemma 20, there is a description tree T 1 = (V, E, v 0 , l 1 ) and associated Skolem labeling sl s.t.A = M A (sl) and T |= C 1 C T1 .We define , where for all v ∈ V , l 2 (v) = {B | B(sl(v)) ∈ B} and φ as the identity over V .Then φ is a weak homomorphism from T 2 to T 1 and sl can also be associated to T 2 and it is such that v∈V,B∈l2(v) ¬B − (sl(v)) ∈ PI g− Σ (Φ).Thus, by Lemma 23, C T2 is a -minimal concept s.t.T |= C T2 C 2 .As seen in the first part of this proof, sl must be injective on V due to its association with T 1 , and thus, for all v ∈ V , C A,sl(v) = l 1 (v) and by construction of C A,sl(v) as wanted.Hence H is a connection-minimal hypothesis.It remains only to show that it is packed.Any tree T built from T 1 by extending the label of some v ∈ V must be such that M(sl T ) ⊆ PI g+ Σ (Φ), where sl T is identical to sl but associated to T , since the labels of T 1 are already maximal in that regard.Thus, by Lemma 19, T |= C 1 C T , hence any such C T cannot be used to create constructible hypotheses, proving H packed.

D Termination
The proofs of Theorem 13 and 14 are detailed in this appendix.
We first recall the notions used to describe the resolution calculus.A substitution is a function mapping variables to terms.The result of applying a substitution σ on a clause ϕ is denoted by ϕσ, and is the clause obtained by replacing every variable x in ϕ by σ(x).A most general unifier (mgu) of the atoms P (t) and P (t ), is a substitution s.t.P (t)σ = P (t )σ, and for any other such substitution σ , there exists a substitution σ so that σ = σ • σ .The resolution calculus is made of two rules: resolution and factorization.The resolution rule infers from two premises of the form ϕ ∨ P (t) and ϕ ∨ ¬P (t ) the resolvent (ϕ ∨ ϕ )σ, given that an mgu σ exists for P (t) and P (t ).The factorization rule infers from a premise of the form ϕ ∨ P (t) ∨ P (t ) the resolvent (ϕ ∨ P (t))σ and from ϕ ∨ ¬P (t) ∨ ¬P (t ) the resolvent (ϕ ∨ ¬P (t))σ, provided σ is the mgu of P (t) and P (t ).For our purpose, a derivation of a clause ϕ from a set of clauses Φ is a sequence of inferences where all premises are either in Φ or the resolvent of an inference occurring earlier in the sequence, and where the last resolvent is ϕ itself.A derivation is linear when the resolvent of one inference is always a premise of the next inference.
A general observation regarding the clauses that are relevant to this work is that, due to the shape of clauses in π(T ), the sets Φ and Φ p only contain clauses of the following shapes: where A 1 , A 2 and A 3 are either all original literals or all duplicate literals.We abbreviate a "clause of the form Ix" as an "Ix-clause" for x ∈ {1, .., 7}.Observe that there is exactly one I1-clause and one I2-clause, both for the same constant sk 0 .Moreover, for every Skolem function sk occurring in Φ p , there is exactly one pair of clauses where one is an I6-clause and the other an I7-clause where a given sk ∈ N S occurs.We call them the clauses introducing sk.To every Skolem function sk, we associate the atomic concept A sk that occurs positively in the I7-clause introducing sk.
Relying on Φ p allows to derive all ground implicates by increasing term depth, which is possible thanks to the following result.
Lemma 24.It is not necessary to use I5-clauses to derive PI g+ Σ (Φ) from Φ p by resolution.
Proof.Since Φ and Φ p are equivalent, they have the same prime implicates, that can be derived by resolution from any of them.We construct a Herbrand model for Φ from all clauses in Φ p except the I5-clauses.Then, by Lemma 18, all clauses from PI g+ Σ (Φ) will be included in this model and thus derivable by resolution from the restriction of Φ p to non-I5-clauses.
Let I = i I i for i ∈ N, such that: • I 0 = {C 1 (sk 0 )} and, • given I j , This interpretation is similar to the one used in the proof of Lemma 18, but uses the clauses in Φ p , I5-clauses excepted, instead of the clauses in Φ.Thus every atom in I can be derived by resolution from the clauses of Φ p that are not I5-clauses.
We show that I is a model of Φ.By construction, I |= C 1 (sk 0 ) and all I1-, I3-, I4-, I6and I7-clauses with original predicates in Φ since they also occur in Φ p .The clauses with duplicate predicates are also satisfied since I contains no duplicates at all.It remains only to show that the I5-clauses in Φ are true in I.By contradiction, consider that the clause ϕ = ¬r(x, y) ∨ ¬A 1 (y) ∨ A 2 (x) is not satisfied by I. Then there must exist terms t, t such that r(t, t ) ∈ I, A 1 (t ) ∈ I but A 2 (t) / ∈ I.By construction, t = sk(t) for some Skolem function sk.The only clauses with sk in Φ p are the clauses introducing sk, that we denote by ϕ 1 = ¬A 3 (x) ∨ r(x, sk(x)) and ϕ 2 = ¬A 3 (x) ∨ A 4 (sk(x)) for some original atomic concept A 4 .These clauses are the only possible cause for the presence of r(t, sk(t)) and A 1 (sk(t)) in I j for some j ≥ 1, and thus there must be an i < j s.t.A 3 (t) ∈ I i .The presence of ϕ 1 , ϕ 2 and ϕ in Φ ensures that Φ |= ¬A 3 (x) ∨ A 2 (x) and thus that ¬A 3 (x) ∨ A 2 (x) ∈ Φ p .Combined with the fact that A 3 (t) ∈ I i , it means that A 2 (t) ∈ I i+1 ⊆ I, a contradiction.Thus I is also a model of all clauses of the form I5 in Φ, so it is a model of Φ and it is possible to derive all clauses in PI g+ Σ (Φ) from Φ p without using the clauses of the form I5.
A direct consequence of this lemma is that, regarding derivations of PI g+ Σ (Φ), we only need to consider those where every inference preserves or increases the depth of terms from premises to conclusion, because the only way to decrease this depth is by using an I5-clause.This allows us to prove Th. 13 Theorem 13.Given an abduction problem and its translation Φ, every constructible hypothesis can be built from prime implicates that are inferred under restriction R2.
Proof.By Th. 11, it suffices to show that all clauses in PI g+ Σ (Φ) ∪ PI g− Σ (Φ) that contain no binary predicate can be derived using only inferences of clauses with at most one variable.Since I1 is the only clause containing no negative literals, any clause ϕ ∈ PI g+ Σ (Φ) must be derived using the I1-clause.Moreover I5-clauses are the only ones in the input that would introduce a variable when resolved with a ground clause.By Lemma 24, we can ignore these clauses to infer ϕ ∈ PI g+ Σ (Φ).Thus, any clause in PI g+ Σ (Φ) can be derived by inferring only ground clauses from Φ p , which is even more than what R2 requires.
For ϕ ∈ PI g− Σ (Φ), Lemma 24 does not apply.In general, any derivation from Φ p of a clause that contains a constant involves C 1 (sk 0 ) or ¬C − 2 (sk 0 ) or both, and only I5-clauses would introduce a variable into such a derivation.Let ϕ be the first clause with a variable that occurs as a resolvent in the derivation of ϕ from Φ p , and let ϕ be without binary predicates, since it must be usable to build a constructible hypothesis following Th.11.The premises of the inference producing ϕ are a ground clause and an I5-clause, ¬r(x, y) ∨ ¬A 1 (y) ∨ A 2 (x).We show that any occurrence of a variable in ϕ can be immediately eliminated by another inference, creating a new derivation for ϕ .Depending on the literal resolved upon in the I5-clause to obtain ϕ several cases occur.
• This literal cannot be ¬r(x, y), or ϕ would be ground because both x and y would be unified with ground terms.• If the literal resolved upon is A 2 (x), then y occurs in ϕ as its only variable, in the literals ¬r(t, y) for some ground t and ¬A 1 (y).The literal ¬r(t, y) is eliminated later in the derivation since ϕ contains no binary predicate.All positive occurrences of r that can be derived are of the form r(t , sk(t )) for some sk, because in the Φ p , the only positive occurrences of r are found in I6-clauses.Thus, to obtain a clause in PI g− Σ (Φ) without roles, we need to eventually unify the variable y with a ground term of the form sk(t) for some sk.Since Φ p is Horn, we can rearrange any derivation from ϕ to ϕ so that we first resolve upon ¬r(t, y) in ϕ with the suitable I6-clause, i.e., the one introducing the appropriate sk.As a result, we obtain another ground clause with no variables, before any further inference is performed if needed.
• If the literal resolved upon is ¬A 1 (y), then x occurs in ϕ in the literals A 2 (x) and ¬r(x, t), where t is ground.The argument unfolds as in the previous case, with the nuance that the considered sk function is the one s.t.t = sk(t ) for some t .
It follows that a derivation of ϕ that does not respect R2 can always be rearranged to eliminate occurrences of variables (and binary literals) as soon as they occur, before the next variable is introduced.The rearranged derivation respects R2.
The proof of Th. 14 is based on a structure called a solution tree, that resembles a description trees, but instead collects in its (multiple) labels information on all the clauses that helped derive the prime implicates needed to build a constructible hypothesis.A solution tree for a hypothesis H is defined as tuple (S, l + , l − ), which is a tree-shaped labeled graph S = (V, E, s) together with two negative duplicate clauses in Φ p if I5and I6-clauses are not used, because Φ p is Horn.This leaves C 1 (sk 0 ) and the I3-, I4and I7-clauses as the ones that are used to derive A(sk(t)) from Φ p .Moreover, since Φ p is Horn, linear resolution can be used to derive A(sk(t)), thus inferences between two resolvents are not necessary [1].Inferences in such a derivation can only preserve the Skolem term occurring in the ground premise when resolving with an I3or an I4-clause, and increase the depth of the term in the resolvent when resolving with an I7-clause.Thus A(sk(t)) can only be derived after A sk (sk(t)) has been introduced by the I7-clause introducing sk, and from A sk (sk(t)) only I3or I4-clauses can be used to derive A(sk(t)).Let us consider the (linear) derivation of A(sk(t)) and remove from it all the inferences on I3and I4-clauses upon literals where sk(t) occurs.The only literals that remain in the derived clause are copies of A sk (sk(t)) because it is the only literal with the term sk(t) that can be derived, and because all other literals are resolved upon in parts of the derivation that have not been removed.Thus, it is enough to append a few factorization inferences at the end of this derivation, if at all needed, to derive A sk (sk(t)).Hence A sk (sk(t)) ∈ PI g+ Σ (Φ).The inferences from the removed parts of the derivation of A(sk(t)), introducing only I3and I4-clauses can be used together with A sk (sk(t)) to construct a derivation of A(sk(t)).
To prove that ancestors of v 2 have a non-empty positive label if v 2 has a non-empty positive label, let us consider the case when v 1 is the direct parent of v 2 (the case when v 1 = v 2 is trivial).Let us consider once again the part of the linear derivation of A(sk(t)) from which we created the derivation of A sk (sk(t)).If we remove from it the inference(s) introducing sk, the clause that is derived must contain only A (t) literals, where ¬A (t) ∨ A sk (sk(t)) is the I7 clause that introduced sk.These A (t) literals can be factorized following this derivation to obtain a derivation of A (t) from Φ p .Thus v 1 has a non-empty positive label and this result also holds for any ancestor of v 2 by induction.
Finally, we have proven that for v 1 s.t.s(v 1 ) = sk (t ), there is a derivation of any A (sk (t )) ∈ l + (v 1 ) from A sk (sk (t )) and the I3and I4-clauses in Φ p .Moreover, in the previous paragraph, assuming v 1 is the parent of v 2 , we have seen that there is at least some A (sk (t )) ∈ l + (v 1 ) from which A sk (sk(sk (t ))) can be inferred by using the I7-clause introducing sk where sk(sk (t )) = s(v 2 ).Thus A(s(v 2 )) can be derived from A sk (s(v 1 )) and the I3-, I4and I7-clauses in Φ p , and this result can be extended to any ancestor of v 2 (except the root v 0 , for which C 1 (sk 0 ) could be used instead of A sk (s(v 0 )), that does not exist).
The following lemma is central to the proof of the theorem.It bounds the range of the positive labeling by ensuring that each path from the root to a leaf of the tree that is longer than the size of N S contains two nodes with the same positive labeling.

Lemma 28. For any two nodes
for the same Skolem function sk and some terms t 1 and t 2 , if l + (v 1 ) and l + (v 2 ) are not empty, then l + (v 1 ) = l + (v 2 ).
Proof.If r(t, sk(t)) ∈ PI g+ Σ (Φ), then r occurs in the I6-clause introducing sk, thus any (linear) derivation of r(t, sk(t)) can be transformed into a derivation of A sk (sk(t)) by replacing this I6-clause with the I7-clause introducing sk when resolving upon A 1 (t) (remember it is possible to ensure that the other premise is ground).Thus A sk (sk(t)) ∈ PI g+ Σ (Φ).In a Skolem tree with a node v 2 s.t.s(v 2 ) = sk(t) for the t and sk from the previous paragraph, there must be an ancestor v 1 of v 2 s.t.s(v 1 ) = sk(t ) since it is a subterm of sk(t).Thus by Lemma 28, A sk (sk(t )) ∈ PI g+ Σ (Φ).Moreover, a (linear) derivation of A sk (sk(t )) from Φ p can be turned into a derivation of r(t , sk(t )) by applying the reverse transformation as in the previous paragraph, thus r(t , sk(t )) ∈ PI g+ Σ (Φ).The last remaining ingredient to obtain the solution tree is the negative labeling.
Lemma 31.If a Skolem tree S has a negative labeling l − , then every maximal antichain {v 1 , . . ., v m } in S corresponds to a clause ¬B − 1 (t 1 ) ∨ . . .∨ ¬B − m (t m ) that can be derived from Φ p .This follows directly by induction starting from the root and it holds in particular for ϕ S,l − .Note that any such clause ¬B − 1 (t 1 )∨. ..∨¬B − m (t m ) is a ground negative implicate, not necessarily prime.Conversely, it is not hard to see that for any negative ground implicate of Φ p of the form ¬B 1 (t 1 ) − ∨ . . .∨ ¬B n (t n ) − , we can construct a Skolem tree and a negative labeling s.t.ϕ S,l − is the negative ground implicate, up to the repetition of literals.This formulation allows the same B i to be represented by several leaves, which is necessary because the tree captures resolution inferences in the derivation but not factorization inferences.This Skolem tree can be constructed together with the negative labeling l − by following the derivation from ¬C − 2 (sk 0 ) to ¬B − 1 (t 1 ) ∨ . . .∨ ¬B − n (t n ) in Φ p .Specifically, we note that: • Such a derivation must exist, because Φ p is Horn and ¬C − 2 (sk 0 ) is the only negative clause and thus every derivation of a negative ground clause must use this clause.
• Moreover, starting from Φ p ∪PI g+ Σ (Φ) all resolvents in a derivation of a negative ground clause can be negative clauses and only {r(t, sk(t)) ∈ PI g+ Σ (Φ) | sk ∈ N S , t ∈ T sk0 (N S )} is needed because the other positive prime implicates contain original literals that are not needed to derive a clause with only duplicate literals.Such derivations can be linear.
• Following the argument used in the proof for Th. 13, we can rearrange any linear derivation of a ground negative clause so that variables and binary literals are eliminated as soon as they are introduced.This step is done by resolving first a ground clause ϕ and an I5-clause ¬r(x, y) ∨ ¬A 1 (y) ∨ A 2 (x), and then resolving an I6-clause ¬A 1 (x ) ∨ r(x , sk(x )) and the resolvent of the previous step.These two steps amount to replacing the literal ¬A 2 (t) in ϕ by ¬A 1 (t) ∨ ¬A 1 (sk(t)).
We thus obtain the following lemma.
Lemma 32.For any B s.t.B(t)∈B ¬B − (t) ∈ PI g− Σ (Φ), we can construct a Skolem tree S that has a negative labeling l − s.t. for every B(t) ∈ B, s(v) = t and l − (v) = B for a leaf v of S, and for all leaves v in S, l − (v)(s(v)) ∈ B.
Combining all three labels, we obtain a solution tree.By reading off the three labels on the leaves of a solution tree, we obtain a connection-minimal hypothesis after Th. 11, when the positive labels of all leaves are not empty.Definition 33.A solution tree is a tuple (S, l + , l − ) where • S = (V, E, s) is a Skolem tree for the terms from which some constructible hypothesis H is defined, • l + is a positive labeling for S such that all nodes in S have a non-empty positive label, and • l − is a negative labeling for S s.t.ϕ S,l − ∈ PI g− Σ (Φ) modulo the repetition of literals.
The solution of the tree is a TBox equivalent to H, which is the set { l + (v) l − (v) | v is a leaf of S}.
Note that the CIs in the solution of the tree use only one atomic concept on the right-hand side, while the equivalent H may contain a conjunction.Moreover, removing all tautologies from this solution results in a packed connection-minimal hypothesis.
Lemma 34.Let (S, l + , l − ) be a solution tree, where S = (V, E, s) and v 1 , v 2 ∈ V be such that v 1 is an ancestor of v 2 , s(v 1 ) = sk(t) and s(v 2 ) = sk(t ) for some sk, t and t , and l − (v 1 ) = l − (v 2 ).Let S = (V , E , s ) the result of replacing in S the subtree under v 1 by the subtree under v 2 , adapting the Skolem labeling s to s appropriately, and let l + and l − be l + and l − restricted to V .Then, (S , l + , l − ) is also a solution tree.
Proof.We have to show that all primed labelings are valid labelings, that the positive one is not empty for any node in S' and that ϕ S ,l − ∈ PI g− Σ (Φ) modulo the repetition of literals.
The adaptation of s to create s consists in replacing the subterm sk(t ), in every term s(v) for v descending from v 2 in S by sk(t) in S'.That way, s is also a Skolem labeling.By Def. 26 and Lemma 28, l + is a positive labeling for S' and none of its labels are empty because none of the labels of l + are empty.By Def. 30, all properties needed to ensure l − is a negative labeling for S' are trivially verified for the nodes outside of the descendants of v 1 because they are the same as in S.
Let w 1 , . . ., w n be the children of v 1 in S'.We show that ϕ = ¬B − 1 (s(w 1 )) ∨ . . .∨ ¬B − n (s(w n )) where B i = l − (w i ) for i ∈ {1, . . ., n} can be derived from the non-ground clauses in Φ p , the set {r(t , sk(t )) ∈ PI g+ Σ (Φ) | sk ∈ N S , t ∈ T sk0 (N S )} and ¬B − (s(v 1 )), where B = l − (v 1 ).In S, the w i nodes are the children of v 2 , thus ϕ = ¬B − 1 (s(v 2 )) ∨ . . .∨ ¬B − n (s(v 2 )) can be derived from the non-ground clauses in Φ p , the set {r(t , sk(t )) ∈ PI g+ Σ (Φ) | sk ∈ N S , t ∈ T sk0 (N S )} and ¬B − (s(v 2 )).We can write s(v 2 ) as g(s(v 1 )), where g is a composition of Skolem functions, and every s(w i ) can be written as sk i (g(s(v 1 ))) for some sk i ∈ N S .The derivation of ϕ can thus be transformed into a derivation of ϕ by replacing B(t ) by B(t) everywhere it is used, and replacing any r(s(v 2 ), sk i (s(v 2 ))) ∈ PI g+ Σ (Φ) used by r(s(v 1 ), sk i (s(v 1 ))) because the latter also belongs to PI g+ Σ (Φ) by Lemma 29.The same argument applies to any other descendant v of v 1 in S' so that the second point of Def. 30 holds for l − .Thus l − is a negative labeling for S'.
If l − is such that ϕ S ,l − is not in PI g− Σ (Φ) modulo the repetition of literals, then there exists a solution tree S" for a strict subclause of ϕ S ,l − modulo the repetition of literals, i.e., where one literal of ϕ S ,l − does not appear at all, that is an implicate of Φ p and it is possible to apply the transformation from S to S' backward from S".This would produce a solution forest for a strict subclause of ϕ S,l − modulo the repetition of literals derivable from Φ p by Lemma 31, which is impossible since ϕ S,l − ∈ PI g− Σ (Φ) modulo the repetition of literals.Thus ϕ S ,l − ∈ PI g− Σ (Φ) modulo the repetition of literals.Definition 35.A solution tree S for a hypothesis H is minimal if there exists no solution tree S for a hypothesis H ⊆ H s.t.S uses strictly less nodes.Lemma 36.Let (S, l + , l − ) be a minimal solution tree where S = (V, E, s).Let v, v ∈ V be nodes such that v is an ancestor of v.Then, either s(v) and s(v ) are not headed by the same Skolem term or l − (v) = l − (v ).
Proof.Let (S, l + , l − ) be a minimal solution tree for the hypothesis H where S = (V, E, s) such that v is an ancestor of v, l − (v) = l − (v ) and s(v) = sk(t) and s(v ) = sk(t ) for some sk, t and t .By applying Lemma 34, we obtain a solution tree S with less nodes than S and for a solution H s.t.H ⊆ H. Consequently, S cannot be minimal.
Corollary 37. Let = (S, l + , l − ) be a minimal solution tree.Then, the depth of S is bounded by n × m, where n is the number of Skolem functions in Φ introduced for the transformation of T , and m is the number of atomic concepts in Φ.
Proof.Let S be a minimal solution tree.By Lemma 36, on every path in S, there are no two nodes v, v such that l − (v) = l − (v ), s(v) = sk(t) and s(v ) = sk(t ) for some sk, t and t .Thus in a path, for every Skolem function sk, there can be at most m nodes on a path, each with a different negative label.However, the Skolem functions introduced during the translation of T − are never needed since they do not occur in PI g+ Σ (Φ).The range of l − is bounded by the number of atomic concepts in Φ, that we denote m.We additionally denote by n the number of Skolem functions in Φ introduced by the translation of T , and thus, every path in S can have a length of at most n × m.Theorem 14 is a direct consequence of Corollary 37, because for a subsetminimal constructible hypothesis H, there is no constructible hypothesis H' s.t.

H
H.

E Locality-based Modules
Realistic ontologies easily get too large to be processed by SPASS in reasonable time for the abduction task.We therefore use module extraction to obtain a relevant subset of the ontology before translating the abduction problem.For a signature Σ and an ontology T , a module M of T for Σ is a subset of T that preserves all entailments of closed second-order formulas that only use predicates from Σ.However, for our particular reasoning task, the signature Σ from the abduction problem T , Σ, C 1 C 2 is not known in advance, so that we have to be a bit more careful when extracting the module.Specifically, we need to ensure that for the observation C 1 C 2 , the module M preserves all subsumers of C 1 and all subsumees of C 2 , as these are the backbone of connection minimality (see Def. 7).This can be done using special locality-based modules, as presented by Grau et al. [20].This means the axioms outside of the ∅-module M do not contribute to nontrivial entailments using the signature Σ ∪ Σ(M).
A fast approximation of ∅-modules are ⊥-modules.The exact definition of ⊥-modules is given in [20], but not needed for the following.It suffices to know that if M is the ⊥-module of T for Σ, and M a ∅-module of T for Σ, then M ⊆ M .In the same way, -modules approximate ∆-modules.The relevant property for us is the following.

Definition 6 .
Let C and D be arbitrary concepts.Then C D if either: • C = D, • D = D D , and C D , or • C = ∃r.C , D = ∃r.D and C D .

Lemma 10 .
For every EL TBox T , we can compute in polynomial time an EL TBox T in normal form such that for every other TBox H and every CI C D that use only names occurring in T , we have T ∪H |= C D iff T ∪H |= C D.
Straightforward Consequences of Definition 7 Point 4 of Def. 7 turns φ from a weak homomorphism to a T -homomorphism.The hypothesis H is made to add exactly the entailments that are missing in T to ensure that point 3 of Def. 4 is satisfied.Thanks to this T ∪ H |= D 1 D 2 and thus both D 1 and D 2 become connecting concepts from C 1 to C 2 in T ∪ H, because we have T ∪ H |= C 1 D 1 D 2 C 2 .A.3 Normalization Lemma 10.For every EL TBox T , we can compute in polynomial time an EL TBox T in normal form such that for every other TBox H and every CI C D that use only names occurring in T , we have T ∪H |= C D iff T ∪H |= C D. Proof.Most of the lemma is well-known.How normalization can be performed, and that it is possible in polynomial time, is shown in [4, Lemma 6.2].Furthermore, by [4, Proposition 6.5], the result of this transformation is a conservative extension T of the original TBox T in the sense that: 1. T |= T , and 2. for every model I of T , there exists a model I of T s.t. for every concept name A occurring in T , A I = A I , and for every role name r ∈ N R occurring in T , r I = r I .Now let H be a TBox and C D a CI such that both only use names occurring in T .If T ∪ H |= C D, we observe that by Item 1, we have T ∪ H |= T ∪ H, and thus by transitivity of entailment, T ∪ H |= C D. Assume T ∪ H |= C D. Then there exists a model I of T ∪ H s.t.I |= C D. Since H, C and D only use names occurring in T , by Item 2, we can find a model I of T s.t.I |= H and I |= C D, and consequently, T ∪ H |= C D. We obtain that T ∪ H |= C D iff T ∪ H |= C D.

Fig. 3 .
Fig. 3. Two description trees with a weak homomorphism between them.

(
∃r.C) I = {d ∈ ∆ I | ∃(d, e) ∈ r I s.t. e ∈ C I } The interpretation I satisfies a CI C D, in symbols I |= C D, if C I ⊆ D I .If I satisfies all axioms in a TBox T , we write I |= T and call I a model of T .If a CI C D is satisfied in every model of T , we write T |= C D and say that CD is entailed by T .In this case, we say that D subsumes C, or that C is subsumed by D and call C a subsumee of D and D a subsumer of C. One easily verifies that the above translation of EL axioms and TBoxes is consistent with their semantics, that is, that I |= π(α) iff I |= α for any CI α (or TBox)[3,10].

Proof.
Given the preconditions of the lemma, let us first assume T |= C 1 C 2 to show the existence of a Skolem labeling sl T such that sl T (v 0 ) = sk 0 and M(sl T ) ⊆ PI g+ Σ (Φ).Since C 1 (sk 0 ) ∈ Φ by definition, we have sk 0 ∈ (C 1 ) I for any Herbrand model I of Φ.Moreover, because T |= C 1 C T , it follows that sk 0 ∈ (C T ) I for any Herbrand model I of Φ.By Lemma 18, we know that PI g+ Σ (Φ) can be seen as a Herbrand model of Φ, thus sk 0 ∈ (C T ) PI g+ Σ (Φ) .The existence of a Skolem labeling with the desired properties follows by Lemma 17.
Lemma 19 establishes a relation between the FOL encoding Φ and the original EL problem, but we need a stronger result to know how to construct the C T such that T |= C 1 C T from PI g+ Σ (Φ).Lemma 20 does the job, by showing that it is only necessary to collect the atomic prime implicates about unary predicates (the ones from N C ) to construct all relevant C T .Lemma 20 (Construction of Subsumers of C 1 ).Given an abduction problem T , H, C 1 C 2 , its first-order translation Φ and a set A = {A 1 (t 1 ), . . ., A n (t n )} ⊆ PI g+ Σ (Φ) where n > 0, there exists T = (V, E, v 0 , l) and sl T s.t.A = M A (sl T ) and T |= C 1 C T .
sk 1 (sk 0 ) .For C T2 = ∃r 1 .C ∃r 1 .D, and C T1 = ∃r 1 .(AB) the set {A B C, A B D} is a packed connection-minimal hypothesis and the equivalent constructible hypothesis {A B C D} is the one found by applying Th. 11.

Definition 38 .
A CI α is ∅-local (reps.∆-local) for a signature Σ if every interpretation I s.t.X I = ∅ (resp.X I = ∆) for all X ∈ (N C ∪ N R ) \ Σ satisfies I |= α.Definition 39.The ∅-module (resp.∆-module) of T for Σ is the smallest subset M ⊆ T s.t.every axiom in T \ M is ∅-local (resp.∆-local) for Σ ∪ Σ(M),where Σ(M) denotes the restriction of the signature to the symbols occurring in the TBox M.

Lemma 40 .
For a concept C 1 and a TBox T , the ⊥-module M of T for Σ(C 1 ) satisfies T |= C 1 D iff M |= C 1 D for all concepts D. Proof sketch.Thanks to the relation between the ⊥-module and the ∅-module, it suffices to prove that the ∅-module M' for Σ(C 1 ) satisfies the property to obtain the same result for the ⊥-module M. We first observe that every nontautological axiom C C ∈ T s.t.Σ(C) ⊆ Σ(M) occurs in M. Otherwise, we would have Σ(C ) ⊆ (M), and C I = ∅ for an interpretation ∅-local for Σ(M), while C I = ∅, and thus I |= C D. Moreover, in EL, all subsumers of C 1 can be generated by unfolding, i.e., by iteratively replacing sub-concepts C in C 1 by concepts C s.t.C C ∈ T .By using our first observation, we obtain that any axiom C C ∈ T that could be involved by such an unfolding operation must be included in M. It follows then that M |= C 1 D iff T |= C 1 D for all concepts D. Lemma 41.For a concept C 2 , the -module M of T for Σ(C 2 ) satisfies T |= D C 2 iff M |= D C 2 for all concepts D. Proof sketch.Can be shown in the same way as Lemma 40.It follows from Lemma 40 and 41, as well as from the definition of connection minimality (Def.7), that by replacing T in the abduction problem by the union of the ⊥-module for Σ(C 1 ) and the -module for Σ(C 2 ), we do not loose solutions of the original problem.
Common minimality criteria include subset minimality, size minimality and semantic minimality, that respectively favor H over H if: H H ; the number of atomic concepts in H is smaller than in H ; and if H |= H but H |= H.
Note that if T |= C 1 C 2 then both C 1 and C 2 are connecting concepts from C 1 to C 2 , and if T |= C 1 C 2 , the case of interest, neither of them are.To address point 2), we must capture how a hypothesis creates the connection between the concepts C 1 and C 2 .As argued above, this is established via concepts D 1 and D 2 that satisfy T |= C 1 D 1 , D 2 C 2 .Note that having only two concepts D 1 and D 2 is exactly what makes the approach parsimonious.
Definition 2. Let C 1 and C 2 be concepts.A concept D connects C 1 to C 2 in T if and only if T |= C 1 D and T |= D C 2 .
The figure can also be used to illustrate what we mean by connection minimality: in order to create a connection between D 1 and D 2 , we should only add the CIs from H a1 ∪ { } unless they are already entailed by T a .In practice, this means the weak homomorphism from D 2 to D 1 becomes a (T a ∪ H a1 )-homomorphism.
t. T .If there exists a T -homomorphism φ from T 2 to T 1 , then T |= C T1 C T2 .This can be shown easily by structural induction using the definitions (see App A.1).The weak homomorphism is the structure on which a T -homomorphism can be built by adding some hypothesis H to T .It is used to reveal missing links between a subsumee D 2 of C 2 and a subsumer D 1 of C 1 , that can be added using H. Example 5. Consider the concepts D 1 = ∃employment.Chair ∃qualification.PhD D 2 = ∃employment.ResearchPosition ∃qualification.Diploma from the academia example.Figure 1 illustrates description trees for D 1 (left) and D 2 (right).The curved arrows show a weak homomorphism from T D2 to T D1 that can be strengthened into a T -homomorphism for some TBox T that corresponds to the set of CIs in H a1 ∪ { }.
Let Φ be a set of clauses.A clause ϕ is an implicate of Φ if Φ |= ϕ.Moreover ϕ is prime if for any other implicate ϕ of Φ s.t.ϕ |= ϕ, it also holds that ϕ |= ϕ .
is one of t 1 , . . ., t m }, and • H = {C A,t C B,t | t is one of t 1 , . . ., t m and C B,t C A,t }, where C A,t =
is one of t 1 , . . ., t m }, and • H = {C A,t C B,t | t is one of t 1 , . . ., t m and C B,t C A,t }, where C A,t = A(t)∈A A and C B,t = B(t)∈B B.
sl(v).We show that this implies C B,sl(v)C A,sl(v).Consider any t s.t.T |= C A,t C B,t .Then by translation, it means that Φ |= ¬π(C A,t ) ∨ π(C B,t ) and since both concepts do not contain role restrictions, it means in particular that Φ |= A∈C A,t ¬A(x) ∨ B(x) for all B ∈ C B,t .Since A(t) ∈ PI g+ Σ (Φ) for all A ∈ C A,t , Φ |= B(t) for all B ∈ C B,t and since those are atomic ground positive implicates, for all B ∈ C B,t , B(t) ∈ PI g+ Σ (Φ).Furthermore, by definition of A and C A,t , this leads to B ∈ C A,t for all B ∈ C B,t , hence C B,t C A,t .In the particular case that interests us, it means that C B,sl(v)