An ASP approach for reasoning in
a concept-aware multipreferential lightweight DL

The need to reason about exceptions in ontologies has led to the development of many non-monotonic extensions of Description Logics (DLs), incorporating features from NMR formalisms in the literature [Straccia (1993), Baader and Hollunder (1995), Donini et al. (2002), Giordano et al. (2007), Britz et al. (2008), Bonatti et al. (2009), Casini and Straccia (2010), Motik and Rosati (2010)], and notably including extensions of rule-based languages [Eiter et al. (2008), Eiter et al. (2011), Knorr et al. (2012), Gottlob et al. (2014), Giordano and Theseider Dupré (2016), Bozzato et al. (2018)], as well as new constructions and semantics [Casini and Straccia (2013), Bonatti et al. (2015), Bonatti (2019)]. Preferential approaches [Kraus et al. (1990), Lehmann and Magidor (1992)] have been extended to description logics, to deal with inheritance with exceptions in ontologies, allowing for non-strict forms of inclusions, called typicality or defeasible inclusions, with different preferential semantics [Giordano et al. (2007), Britz et al. (2008)] and closure constructions [Casini and Straccia (2010), Casini et al. (2013), Giordano et al. (2013), Pensel and Turhan (2018)].

In this paper, we propose a “concept-aware multipreference semantics” for reasoning about exceptions in ontologies taking into account preferences with respect to different concepts and integrating them into a preferential semantics which allows a standard interpretation of defeasible inclusions. The intuitive idea is that the relative typicality of two domain individuals usually depends on the aspects we are considering for comparison: Bob may be a more typical as sport lover than Jim, but Jim may be a more typical swimmer than Bob. This leads to consider a multipreference semantics in which there is a preference relation $\leq_{C}$ among individuals for each aspect (concept) $C$. In the previous case, we would have $bob\leq_{SportLover}jim$ and $jim\leq_{Swimmer}bob$. Considering different preference relations associated with concepts, and then combining them into a global preference, provides a simple solution to the blocking inheritance problem, which affects rational closure, while still allowing to deal with specificity and irrelevance.

Our approach is strongly related with Gerard Brewka’s proposal of preferred subtheories [Brewka (1989)], later generalized within the framework of Basic Preference Descriptions for ranked knowledge bases [Brewka (2004)]. We extend to DLs the idea of having ranked or stratified knowledge bases (ranked TBoxes here) and to define preorders (preferences) on worlds (here, preferences among domain elements in a DL interpretation). Furthermore, we associate ranked TBoxes with concepts. The ranked TBox for concept $C$ describes the prototypical properties of $C$-elements. For instance, the ranked TBox for concept Horse describes the typical properties of horses, of running fast, having a long mane, being tall, having a tail and a saddle. These properties are defeasible and horses should not necessarily satisfy all of them.

The ranked TBox for $C_{h}$ determines a preference relation $\leq_{C_{h}}$ on the domain, defining the relative typicality of domain elements with respect to aspect $C_{h}$. We then combine such preferences into a global preference relation $<$ to define a concept-wise multipreference semantics, in which all conditional queries can be evaluated as usual in preferential semantics. For instance, we may want to check whether typical Italian employees have a boss, given the preference relation $\leq_{\mathit{Employee}}$, but no preference relation for concept $\mathit{Italian}$; or to check whether employed students are normally young or have a boss, given the preference relations $\leq_{\mathit{Employee}}$ and $\leq_{\mathit{Student}}$, resp., for employees and for students.

We introduce a notion of multipreference entailment and prove that it satisfies the KLM properties of preferential consequence relations. This notion of entailment deals properly with irrelevance and specificity, is not subject to the “blockage of property inheritance” problem, which affects rational closure [Pearl (1990)], i.e., if a subclass is exceptional with respect to a superclass for a given property, it does not inherit from that superclass any other property.

To prove the feasibility of our approach, we develop a proof method for reasoning under the proposed multipreference semantics for the description logic ${\mathcal{EL}}^{+}_{\bot}$ [Kazakov et al. (2014)], the fragment of OWL2 EL supported by ELK. We reformulate multipreference entailment as a problem of computing preferred answer sets and, as a natural choice, we develop an encoding of the multipreferential extension of ${\mathcal{EL}}^{+}_{\bot}$ in asprin [Brewka et al. (2015)], exploiting a fragment of Krötzsch’s Datalog materialization calculus \shortciteKrotzschJelia2010.

As a consequence of the soundness and completeness of this reformulation of multipreference entailment, we prove that concept-wise multipreference entailment is $\Pi^{p}_{2}$-complete for ${\mathcal{EL}}^{+}_{\bot}$ ranked knowledge bases.

We consider the description logic ${\mathcal{EL}}^{+}_{\bot}$ [Kazakov et al. (2014)] of the ${\cal EL}$ family [Baader et al. (2005)]. Let ${N_{C}}$ be a set of concept names, ${N_{R}}$ a set of role names and ${N_{I}}$ a set of individual names. The set of ${\mathcal{EL}}^{+}_{\bot}$ concepts can be defined as follows: $C\ \ :=A\mid\top\mid\bot\mid C\sqcap C\mid\exists r.C$, where $a\in N_{I}$, $A\in N_{C}$ and $r\in N_{R}$. Observe that union, complement and universal restriction are not ${\mathcal{EL}}^{+}_{\bot}$ constructs. A knowledge base (KB) $K$ is a pair $({\cal T},{\cal A})$, where ${\cal T}$ is a TBox and ${\cal A}$ is an ABox. The TBox ${\cal T}$ is a set of concept inclusions (or subsumptions) of the form $C\sqsubseteq D$, where $C,D$ are concepts, and of role inclusions of the form $r_{1}\circ\ldots\circ r_{n}\sqsubseteq r$, where $r_{1},\ldots,r_{n},r\in N_{R}$. The ABox ${\cal A}$ is a set of assertions of the form $C(a)$ and $r(a,b)$ where $C$ is a concept, $r\in N_{R}$, and $a,b\in N_{I}$.

An interpretation for ${\mathcal{EL}}^{+}_{\bot}$ is a pair $I=\langle\Delta,\cdot^{I}\rangle$ where: $\Delta$ is a non-empty domain—a set whose elements are denoted by $x,y,z,\dots$—and $\cdot^{I}$ is an extension function that maps each concept name $C\in N_{C}$ to a set $C^{I}\subseteq\Delta$, each role name $r\in N_{R}$ to a binary relation $r^{I}\subseteq\Delta\times\Delta$, and each individual name $a\in N_{I}$ to an element $a^{I}\in\Delta$. It is extended to complex concepts as follows: $\top^{I}=\Delta$, $\bot^{I}=\emptyset$, $(C\sqcap D)^{I}=C^{I}\cap D^{I}$ and $(\exists r.C)^{I}=\{x\in\Delta\mid\exists y.(x,y)\in r^{I}\ \mbox{and}\ y\in C^{I}\}.$ The notions of satisfiability of a KB in an interpretation and of entailment are defined as usual:

To define a multipreferential semantics for ${\mathcal{EL}}^{+}_{\bot}$ we extend the language with a typicality operator ${\bf T}$, as done for $\mathcal{EL}^{\bot}$ [Giordano et al. (2011)]. In the language extended with the typicality operator, an additional concept ${\bf T}(C)$ is allowed (where $C$ is an ${\mathcal{EL}}^{+}_{\bot}$ concept), whose instances are intended to be the prototypical instances of concept $C$. Here, we assume that ${\bf T}(C)$ can only occur on the left hand side of concept inclusion, to allow typicality inclusions of the form ${\bf T}(C)\sqsubseteq D$, meaning that “typical C’s are D’s” or “normally C’s are D’s”. Such inclusions are defeasible, i.e., admit exceptions, while standard inclusions are called strict, and must be satisfied by all domain elements.

Let ${\cal C}$ be a (finite) set of distinguished concepts $\{C_{1},\ldots,C_{k}\}$, where $C_{1},\ldots,C_{k}$ are possibly complex ${\mathcal{EL}}^{+}_{\bot}$ concepts. Inspired to Brewka’s framework of basic preference descriptions [Brewka (2004)], we introduce a ranked TBox ${\cal T}_{C_{i}}$ for each concept $C_{i}\in{\cal C}$, describing the typical properties ${\bf T}(C_{i})\sqsubseteq D$ of $C_{i}$-elements. Ranks (non-negative integers) are assigned to such inclusions; the ones with higher rank are considered more important than the ones with lower rank.

A ranked ${\mathcal{EL}}^{+}_{\bot}$ knowledge base $K$ over ${\cal C}$ is a tuple $\langle{\cal T}_{strict},{\cal T}_{C_{1}},\ldots,{\cal T}_{C_{k}},{\cal A}\rangle$, where ${\cal T}_{strict}$ is a set of standard concept and role inclusions, ${\cal A}$ is an ABox and, for each $C_{j}\in{\cal C}$, ${\cal T}_{C_{j}}$ is a ranked TBox of defeasible inclusions, $\{(d^{j}_{i},r^{j}_{i})\}$, where each $d^{j}_{i}$ is a typicality inclusion of the form ${\bf T}(C_{j})\sqsubseteq D^{j}_{i}$, having rank $r^{j}_{i}$, a non-negative integer.

The ranked Tbox ${\cal T}_{Horse}$ can be used to define an ordering among domain elements comparing their typicality as horses. For instance, given two horses Spirit and Buddy, if Spirit has long mane, no saddle, has a tail and runs fast, it is intended to be more typical than Buddy, a horse running fast, with saddle and long mane, but without tail, as having a tail (rank 2) is a more important property for horses wrt having a saddle (rank 0).

In order to define an ordering for each $C_{i}\in{\cal C}$, where $x\leq_{C_{i}}y$ means that $x$ is at least as typical as $y$ wrt $C_{i}$ (in the example, $\mathit{Spirit\leq_{Horse}Buddy}$ and, actually, $\mathit{Spirit<_{Horse}Buddy}$), among the preference strategies considered by Brewka, we adopt strategy $\#$, which considers the number of formulas satisfied by a domain element for each rank.

Given a ranked knowledge base $K=\langle{\cal T}_{strict},{\cal T}_{C_{1}},\ldots,{\cal T}_{C_{k}},{\cal A}\rangle$, where ${\cal T}_{C_{j}}=\{(d^{j}_{i},r^{j}_{i})\}$ for all $j=1,\ldots,k$, let us consider an ${\mathcal{EL}}^{+}_{\bot}$ interpretation $I=\langle\Delta,\cdot^{I}\rangle$ satisfying all the strict inclusions in ${\cal T}_{strict}$ and assertions in ${\cal A}$. For each $j$, to define a preference ordering $\leq_{C_{j}}$ on $\Delta$, we first need to determine when a domain element $x\in\Delta$ satisfies/violates a typicality inclusion for $C_{j}$. We say that $x\in\Delta$ satisfies ${\bf T}(C_{j})\sqsubseteq D$ in $I$, if $x\not\in C_{j}^{I}$ or $x\in D^{I}$, while $x$ violates ${\bf T}(C_{j})\sqsubseteq D$ in $I$, if $x\in C_{j}^{I}$ and $x\not\in D^{I}$. Note that any element which is not an instance of $C_{j}$ trivially satisfies all conditionals ${\bf T}(C_{j})\sqsubseteq D^{j}_{i}$. For a domain element $x\in\Delta$, let ${\cal T}_{C_{j}}^{l}(x)$ be the set of typicality inclusions in ${\cal T}_{C_{j}}$ with rank $l$ which are satisfied by $x$: ${\cal T}_{C_{j}}^{l}(x)=\{d\mid(d,l)\in{\cal T}_{C_{j}}\mbox{ and }x\mbox{ satisfies }d\mbox{ in }I\}$.

Informally, $\leq_{C_{j}}$ gives higher preference to domain individuals violating less typicality inclusions with higher rank. Definition 2 exploits Brewka’s $\#$ strategy in DL context. In particular, all $x,y\not\in C_{j}^{I}$, $x\sim_{C_{j}}y$, i.e., all elements not belonging to $C_{j}^{I}$ are assigned the same rank, the least one, as they trivially satisfy all the typical properties of $C_{j}$’s. As, for a ranked knowledge base, the $\#$ strategy defines a total preorder [Brewka (2004)] and, for each ${\cal T}_{C_{j}}$, we have applied this strategy to the materializations $C_{j}\sqsubseteq D$ of the typicality inclusions ${\bf T}(C_{j})\sqsubseteq D$ in the ranked TBox ${\cal T}_{C_{j}}$, the relation $\leq_{C_{j}}$ is a total preorder on the domain $\Delta$. Then, the strict preference relation $<_{C_{j}}$ is a strict modular partial order, i.e., an irreflexive, transitive and modular relation (where modularity means that: for all $x,y,z\in\Delta$, if $x<_{C_{j}}y$ then $x<_{C_{j}}z$ or $z<_{C_{j}}y$); $\sim_{C_{j}}$ is an equivalence relation.

As ${\mathcal{EL}}^{+}_{\bot}$ has the finite model property [Baader et al. (2005)], we can restrict our consideration to interpretations $I$ with a finite domain. In principle, we would like to consider, for each concept $C_{j}\in{\cal C}$, all possible domain elements compatible with the inclusions in ${\cal T}_{strict}$, and compare them according to $\leq_{C_{i}}$ relation. This leads us to restrict the consideration to models of ${\cal T}_{strict}$ that we call canonical, in analogy with the canonical models of rational closure [Giordano et al. (2013)]. For each concept $C$ occurring in $K$, let us consider a new concept name $\overline{C}$, (representing the negation of $C$) such that $C\sqcap\overline{C}\sqsubseteq\bot$. Let $\mathcal{S}_{K}$ be the set of all such $C$ and $\overline{C}$, and let ${\cal T}_{Constr}$ the set of all subsumptions $C\sqcap\overline{C}\sqsubseteq\bot$. A set $\{D_{1},\ldots,D_{m}\}$ of concepts in $\mathcal{S}_{K}$ is consistent with $K$ if ${\cal T}_{Strict}\cup{\cal T}_{Constr}\not\models_{{\mathcal{EL}}^{+}_{\bot}}D_{1}\sqcap\dots\sqcap D_{m}\sqsubseteq\bot$.

The idea is that, in a canonical model for $K$, any conjunction of concepts occurring in $K$, or their complements, when consistent with $K$, must have an instance in the domain. Existence of canonical interpretations is guaranteed for knowledge bases which are consistent under the preferential (or ranked) semantics for typicality. ${\mathcal{EL}}^{+}_{\bot}$ with typicality is indeed a fragment of the description logic $\mathcal{SHIQ}$ with typicality, for which existence of canonical models of consistent knowledge bases was proved [Giordano et al. (2018)].

In agreement with the preferential interpretations of typicality logics, we further require that, if there is some $C_{h}$-element in a model, then there is at least one $C_{h}$-element satisfying all typicality inclusions for $C_{h}$ (i.e., a prototypical $C_{h}$-element).

In a canonical and ${\bf T}$-compliant interpretation for $K$, for each $C_{j}\in{\cal C}$, the relation $\leq_{C_{j}}$ on the domain $\Delta$ provides a preferential interpretation for the typicality concept ${\bf T}(C_{j})$ as $min_{<_{C_{j}}}(C_{j}^{I})$, in which all typical $C_{j}$ satisfy all typicality inclusions in ${\cal T}_{C_{h}}$.

Existence of a ${\bf T}$-compliant canonical interpretation is not guaranteed for an arbitrary knowledge base. For instance, a knowledge base whose typicality inclusions conflict with strict ones (e.g, ${\bf T}(C_{j})\sqsubseteq D$ and $C_{j}\sqcap D\sqsubseteq\bot$) has no ${\bf T}$-compliant interpretation. However, existence of ${\bf T}$-compliant interpretations is guaranteed for knowledge bases which are consistent under the preferential (or ranked) semantics for typicality (see Appendix A, Proposition A.8), and consistency can be tested in polynomial time in Datalog [Giordano and Theseider Dupré (2018)].

For a ranked knowledge base $K=\langle{\cal T}_{strict},{\cal T}_{C_{1}},\ldots,{\cal T}_{C_{k}},{\cal A}\rangle$, and a given ${\mathcal{EL}}^{+}_{\bot}$ interpretation $I=\langle\Delta,\cdot^{I}\rangle$, the strict modular partial order relations $<_{C_{1}},\ldots,<_{C_{k}}$ over $\Delta$, defined according to Definition 2 above, determine the relative typicality of domain elements w.r.t. each concept $C_{j}$. Clearly, the different preference relations $<_{C_{j}}$ do not need to agree, as seen in the introduction.

We are interested in defining a notion of typical $C$-element, and defining an interpretation of ${\bf T}(C)$, which works for all concepts $C$, not only for the distinguished concepts in ${\cal C}$. This can be used to evaluate subsumptions of the form ${\bf T}(C)\sqsubseteq D$ when $C$ does not belong to ${\cal C}$. We address this problem by introducing a notion of multipreference concept-wise interpretation, which generalizes the notion of preferential interpretation [Kraus et al. (1990)] by allowing multiple preference relations and, then, combining them in a single (global) preference. Let us consider the following example:

We might be interested to check whether typical Italian students are young or whether typical employed students are young. This would require the typicality inclusions $\mathit{{\bf T}(Student\sqcap Italian)\sqsubseteq Young}$ and $\mathit{{\bf T}(Employee\sqcap Student)\sqsubseteq Young}$ to be evaluated. Nothing should prevent Italian students from being young (irrelevance). Also, we expect not to conclude that typical employed students are young nor that they are not, as typical students and typical employees have conflicting properties concerning age. However, we would like to conclude that typical employed students have a boss, have classes and have no scholarship, as they should inherit the properties of typical students and of typical employees which are not overridden (i.e., there is no blocking of inheritance). As PhD students are students, they should inherit all the typical properties of Students, except having no scholarship, which is overridden by ($d_{6}$).

To evaluate conditionals ${\bf T}(C)\sqsubseteq D$ for any concept $C$ we introduce a concept-wise multipreference interpretation, that combines the preference relations $<_{C_{1}},\ldots,<_{C_{k}}$ into a single (global) preference relation $<$ and interpreting $\mathit{{\bf T}(C)}$ as $\mathit{({\bf T}(C))^{I}}=$ $min_{<}(C^{I})$. The relation $<$ should be defined starting from the preference relations $<_{C_{1}},\ldots,<_{C_{k}}$ also considering specificity.

Let us consider the simplest notion of specificity among concepts, based on the subsumption hierarchy (one of the notions considered for ${\cal DL}^{N}$ [Bonatti et al. (2015)]).

Relation $\succ$ is irreflexive and transitive [Bonatti et al. (2015)]. Alternative notions of specificity can be used, based, for instance, on the rational closure ranking.

We are ready to define a notion of multipreference interpretation. Let a relation $<_{C_{i}}$ be well-founded when there is no infinitely-descending chain of domain elements $x_{1}<_{C_{i}}x_{0},\;x_{2}<_{C_{i}}x_{1},\;x_{3}<_{C_{i}}x_{2},\ldots$.

Notice that the relation $<$ is defined from $<_{C_{1}},\ldots,<_{C_{k}}$ based on a modified Pareto condition: $x<y$ holds if there is at least a $C_{i}\in{\cal C}$ such that $x<_{C_{i}}y$ and, for all $C_{j}\in{\cal C}$, either $x\leq_{C_{j}}y$ holds or, in case it does not, there is some $C_{h}$ more specific than $C_{j}$ such that $x<_{C_{h}}y$ (preference $<_{C_{h}}$ in this case overrides $<_{C_{j}}$). For instance, in Example 2, for two domain elements $x,y$, both instances of $\mathit{PhDStudent,Student,}$ $\mathit{\exists has\_Classes.\top,Young}$, and such that $x$ is instance of $\mathit{has\_no\_Scholarship}$, while $\mathit{y}$ is not, we have that $\mathit{x<_{Student}y}$ and $\mathit{y<_{PhDStudent}x}$. As $\mathit{PhDStudent}$ is more specific than $\mathit{Student}$, globally we get $\mathit{y<x}$. We can prove the following result.

In a cw^m-interpretation we have assumed each $<_{C_{j}}$ to be any irreflexive, transitive, modular and well-founded relation. In a cw^m-model of $K$, the preference relations $<_{C_{j}}$’s will be defined from the ranked TBoxes ${\cal T}_{C_{j}}$’s according to Definition 2.

As the preferences $<_{C_{j}}$’s, defined according to Definition 2, are irreflexive, transitive, well-founded and modular relations over $\Delta$, a cw^m-model ${\mathcal{M}}$ is indeed a cw^m-interpretation. By definition ${\mathcal{M}}$ satisfies all strict inclusions and assertions in $K$, but is not required to satisfy all typicality inclusions ${\bf T}(C_{j})\sqsubseteq D$ in $K$, unlike in preferential typicality logics [Giordano et al. (2007)].

Consider, in fact, a situation in which typical birds are fliers and typical fliers are birds (${\bf T}(B)\sqsubseteq F$ and ${\bf T}(F)\sqsubseteq B$). In a cw^m-model two domain elements $x$ and $y$, which are both birds and fliers, might be incomparable wrt $<$, as $x$ is more typical than $y$ as a bird, while $y$ is more typical than $x$ as a flier, even if one of them is minimal wrt $<_{Bird}$ and the other is not. In this case, they will be both minimal wrt $<$. In preferential logics, we would conclude that ${\bf T}(B)\equiv{\bf T}(F)$, which is not the case under the cw^m-semantics. This implies that the notion of cw^m-entailment (defined below) is not stronger than preferential entailment. It is also not weaker as, for instance, in Example 2, cw^m-entailment allows to conclude that typical employed students have a boss, have classes and no scholarship (although defaults $(d_{1})$ and $(d_{4})$ are conflicting), while neither preferential entailment nor the rational closure would allow such conclusions; cw^m-entailment does not suffer from inheritance blocking, and is then incomparable with preferential entailment and with entailment under rational closure, being neither weaker nor stronger.

The notion of cw^m-entailment exploits canonical and ${\bf T}$-compliant cw^m-models of $K$. A cw^m-model ${\mathcal{M}}=\langle\Delta,<_{C_{1}},\ldots,<_{C_{k}},<,\cdot^{I}\rangle$ is canonical (${\bf T}$-compliant) for $K$ if the ${\mathcal{EL}}^{+}_{\bot}$ interpretation $\langle\Delta,\cdot^{I}\rangle$ is canonical (${\bf T}$-compliant) for $K$.

It can be proved that cw^m-entailment satisfies the KLM postulates of a preferential consequence relation (see Appendix A, Proposition A.10).

In this section we consider the problem of checking cw^m-entailment of a typicality subsumption ${\bf T}(C)\sqsubseteq D$ as a problem of determining preferred answer sets. Based on this formulation, that we prove to be sound and complete, we show that the problem is in $\Pi^{p}_{2}$. We exploit asprin [Brewka et al. (2015)] to compute preferred answer sets. The proofs for this section can be found in Appendix C.

In principle, for checking ${\bf T}(C)\sqsubseteq D$ we would need to consider all possible typical $C$-elements in all possible canonical and ${\bf T}$-compliant cw^m-model of $K$, and verify whether they are all instances of $D$. However, we will prove that it is sufficient to consider, among all the (finite) cw^m-models of $K$, the polynomial ${\mathcal{EL}}^{+}_{\bot}$ models that we can construct using the ${\mathcal{EL}}^{+}_{\bot}$ fragment of the materialization calculus for $\mathcal{SROEL}(\sqcap,\times)$ [Krötzsch (2010)], by considering all alternative interpretations for a distinguished element $aux_{C}$, representing a prototypical $C$-element. In preferred models, which minimize the violation of typicality inclusions by $aux_{C}$, it indeed represents a typical $C$-element. An interesting result is that neither we need to consider all the possible interpretations for constants in the model nor to minimize violation of typicalities for them. Essentially, when evaluating the properties of typical employed students we are not concerned with the typicality (or atypicality) of other constants in the model (e.g., with typical cars, with typical birds, and with typical named individuals). Unlike a previous semantics by Giordano and Theseider Dupré \shortciteTPLP2016, which generalizes rational closure by allowing typicality concepts on the rhs of inclusions, we are not required to consider all possible alternative interpretations and ranks of individuals in the model. We will see, however, that we do not loose solutions (models) in this way.

In the following we first describe how answer sets of a base program, corresponding to cw^m-models of $K$, are generated. Then we describe how preferred models can be selected, where $\mathit{aux_{C}}$ represent a typical $C$-element.

We will assume that assertions ($C(a)$ and $r(a,b)$) are represented using nominals as inclusions (resp., $\{a\}\sqsubseteq A$ and $\{a\}\sqsubseteq\exists R.\{b\}$), where a nominal $\{a\}$ is a concept containing a single element and $(\{a\})^{I}=\{a^{I}\}$. We also assume that the knowledge base $K$ is in normal form [Baader et al. (2005)], where a typicality inclusion ${\bf T}(B)\sqsubseteq C$ is in normal form when $B,C\in N_{C}$ [Giordano and Theseider Dupré (2016)]. Extending the results in [Krötzsch (2010)], it can be proved that, given a KB, a semantically equivalent KB in normal form (over an extended signature) can be computed in linear time. We refer to a previous paper [Giordano and Theseider Dupré (2018)] for the details on normalization.

The base program $\Pi(K,C,D)$ for the (normalized) knowledge base $K$ and typicality subsumption ${\bf T}(C)\sqsubseteq D$ is composed of three parts, $\Pi(K,C,D)=\Pi_{K}\cup\Pi_{IR}\cup\Pi_{C,D}$.

$\Pi_{K}$ is the representation of $K$ in Datalog [Krötzsch (2010)], where, to keep a DL-like notation, we do not follow the convention where variable names start with uppercase; in particular, $A$, $C$, $D$ and $R$, are intended as ASP constants corresponding to the same class/role names in $K$. In this representation, $\mathit{nom(a)}$, $\mathit{cls(A)}$, $\mathit{rol(R)}$ are used for $\mathit{a\in N_{I}}$ , $\mathit{A\in N_{C}}$, $\mathit{R\in N_{R}}$, and, for example, $\mathit{subClass(a,C)}$, $\mathit{subClass(A,C)}$ are used for $C(a)$, $A\sqsubseteq C$. Additionally, $\mathit{subTyp(C,D,N)}$ is used for $\mathit{T(C)\sqsubseteq D}$ having rank $N$, and the following definitions for distinguished concepts, typical properties, and valid ranks, will be used in defining preferences:

For each distinguished concept $C_{i}$, $\mathit{auxtc(aux\_Ci,Ci)}$ is included, where $\mathit{aux\_Ci}$ is an auxiliary individual name. Other auxiliary constants (one for each inclusion $A\sqsubseteq\exists R.B$) are needed [Krötzsch (2010)] to deal with existential rules.

$\Pi_{IR}$ contains the subset of the inference rules (1-29) for instance checking [Krötzsch (2010)] that is relevant for ${\mathcal{EL}}^{+}_{\bot}$ (reported in Appendix B), for example $\mathit{inst(x,z)\leftarrow}$ $\mathit{subClass(y,z),inst(x,y)}$; for $\bot$, an additional rule is used: $\mathit{\leftarrow bot(z),inst(x,z)}$. Additionally, $\Pi_{IR}$ contains the version of the same rules for subclass checking (where $\mathit{inst\_sc(A,B,A)}$ represents $A\sqsubseteq B$ [Krötzsch (2010)]), and then the following rule encodes specificity $C_{h}\succ C_{j}$:

$\Pi_{IR}$ also contains the following rules:

Rule (a) generates alternative answer sets (corresponding to different interpretations) where $\mathit{aux_{C}}$ may have the typical properties of the concepts it belongs. The constant $aux\_Ci$, such that $auxtc(aux\_Ci,Ci)$ holds, represents a typical $Ci$ (a minimal element wrt. $\leq_{C_{i}}$) only in case it is an instance of $C_{i}$ (i.e., $\mathit{inst(aux\_Ci,Ci)}$ holds). Rule (b) establishes that, if there is an instance $x$ of concept $C_{i}$ in the interpretation, then $aux\_Ci$ must be an instance of $C_{i}$ (it models ${\bf T}$-compliance) and, by rule (c), $aux\_Ci$ is a typical instance of $C_{i}$, i.e., it is minimal wrt. $\leq_{C_{i}}$ among $C_{i}$-elements in the interpretation at hand. By rule (d), a typical instance of $C_{i}$ has all typical properties of $C_{i}$. The rules (b)-(d) only allow to derive conclusions involving $aux\_Ci$ constants.

$\Pi_{C,D}$ contains (if necessary) normalized axioms defining $C,D$ in ${\bf T}(C)\sqsubseteq D$ in terms of other concepts (e.g., replacing $\mathit{{\bf T}(Employee\sqcap Student)\sqsubseteq Young}$ with $\mathit{{\bf T}(A)\sqsubseteq Young}$ and $\mathit{A\sqsubseteq Employee}$, $\mathit{A\sqsubseteq Student}$ and $\mathit{Employee\sqcap Student\sqsubseteq A}$) plus the facts $\mathit{auxtc(aux_{C},C)}$, $\mathit{nom(aux_{C})}$, $\mathit{inst(aux_{C},C).}$