=Paper=
{{Paper
|id=Vol-1459/paper18
|storemode=property
|title=Preferential description logics meet sports entertainment: cardinality
restrictions and perfect extensions for a better royal rumble match
|pdfUrl=https://ceur-ws.org/Vol-1459/paper18.pdf
|volume=Vol-1459
|dblpUrl=https://dblp.org/rec/conf/cilc/Pozzato15
}}
==Preferential description logics meet sports entertainment: cardinality
restrictions and perfect extensions for a better royal rumble match==
https://ceur-ws.org/Vol-1459/paper18.pdf
Preferential Description Logics meet Sports
Entertainment: Cardinality Restrictions and Perfect
Extensions for a Better Royal Rumble Match
Gian Luca Pozzato
Dipartimento di Informatica - Università di Torino - Italy
gianluca.pozzato@unito.it
Abstract. In this work we include cardinality restrictions and degrees of expect-
edness of inclusions in preferential Description Logics. We enrich the language
of the nonmonotonic Description Logic DL-Litec T, obtained by adding a typ-
icality operator T to standard DL-Litecore , by allowing inclusions of the form
T(C) vd D, where d is a degree of expectedness. We then propose a syntactic
notion of extension of an ABox, in order to assume typicality assertions about
individuals satisfying cardinality restrictions on concepts. Moreover, we define
an order relation among such extended ABoxes, that allows to define a notion of
perfect extension as the minimal one with respect to such an order relation. We
apply this machinery to a problem coming from sports entertainment, namely the
problem of maximizing the approval rating by the people attending to the Royal
Rumble match, an annual wrestling event involving thirty athletes.
1 Introduction
The term sports entertainment has been coined by the World Wrestling Federation (now
World Wrestling Entertainment, WWE, http://www.wwe.com) to describe professional
wrestling, a combat sport combining athletics with theatrical performance. As a differ-
ence with typical athletics and games, which are conducted for competition, the main
objective of sports entertainment, and especially of professional wrestling, is to enter-
tain an audience. The owner of WWE, Vincent Kennedy McMahon, often mentions
that his company has to “Give the People What They Want”. Wrestling matches are
driven by storylines provided by a creative team, and their outcomes are generally pre-
determined: duration, sequence of athletic moves, external interferences and, obviously,
winners of the contests. Each athlete plays a specific role and follows a script, whereas
injuries (and deaths) are only due to accidents, for instance because of a wrongly exe-
cuted maneuver.
One of the most attractive events in professional wrestling is the WWE Royal Rum-
ble match: thirty athletes are involved in this competition, and the winner receives a title
shot in the main annual event of the company. The objective of each participant is to
eliminate all the other competitors by tossing them over the top rope of the ring; an ath-
lete is eliminated if both his feet touch the floor outside the ring. The match starts with
the two participants who have drawn entry numbers one and two, with the remaining
competitors entering the ring at regular timed intervals, usually 90 seconds, according
to their entrance number assigned by means of a lottery. On the contrary, the assignment
of entrance numbers to the participants, the sequence of eliminations (who eliminates
who), the last man being eliminated, as well as the winner himself are determined by
the choices of the creative team and scheduled in all the details.
In the last two years, the Royal Rumble match has been marked by an extremely
negative audience reaction: the trivial sequence of eliminations, as well as the fact that
both the winners have been predicted before the match by professional wrestling web
sites, lead the people in the arena to “boo” every single action of the show.
In this work we move a first step in order to tackle the problem of defining the
script of the perfect Royal Rumble match. The idea it to support (not to replace) the
creative team in the activities of selecting 1. the entrance number of the participants
2. the group of finalists, i.e. the last two or three athletes remaining in the ring after
all other competitors have been eliminated 3. the winner of the match. To this aim, we
exploit preferential Description Logics recently introduced in [13, 17, 19, 14].
Nonmonotonic extensions of Description Logics (DLs) have been actively investi-
gated since the early 90s [5, 3, 6, 12, 13, 17, 10]. A simple but powerful nonmonotonic
extension of DLs is proposed in [13, 17, 16, 14]: in this approach “typical” or “normal”
properties can be directly specified by means of a “typicality” operator T enriching the
underlying DL; the typicality operator T is essentially characterized by the core proper-
ties of nonmonotonic reasoning axiomatized by either preferential logic [20] or rational
logic [21]. In these logics one can express defeasible inclusions such as “normally, a top
player returning from an injury wins the Royal Rumble match”:
T(Returning u Top) v Winner
As a difference with standard DLs, in these extensions one can consistently express
exceptions and reason about defeasible inheritance as well. For instance, a knowledge
base can consistently express that “normally, a face wrestler is supported by the crowd”,
whereas “typically, a face wrestler who is supposed to win the Royal Rumble match is
not supported by the crowd” as follows:
PredictedFace v Face
T(Face) v Supported
T(PredictedFace) v ¬Supported
The approach based on the typicality operator has been first introduced for the basic DL
ALC [13]. In [14], the authors have extended this approach also to the logic DL-Litecore
of the DL-Lite family. This logic is specifically tailored for effective query answering
over DL knowledge bases containing a large amount of data, however, thanks to its
computational complexity, it is considered a lightweight Description Logic: indeed, the
problem of subsumption and the satisfiability of a knowledge base in DL-Litecore are
NL OG S PACE in the size of the TBox [8, 9]. In this work, we restrict our concerns to
the logic DL-Litec T, whose expressive power is sufficient for the application to sports
entertainment presented in Section 4.
The logic DL-Litec T results to be too weak in several application domains. Indeed,
although the operator T is nonmonotonic (T(C) v E does not imply T(C u D) v E),
the logic DL-Litec T is monotonic, in the sense that if the fact F follows from a given
knowledge base KB, then F also follows from any KB’ ⊇ KB. As a consequence, unless
a KB contains explicit assumptions about typicality of individuals, there is no way of
inferring defeasible properties about them: in the above example, if KB contains the
fact that Daniel is a face wrestler, i.e. Face(daniel ) belongs to KB, it is not possible to
infer that he is supported by the crowd (Supported (daniel )). This would be possible
only if the KB contained the stronger information that Daniel is a typical face wrestler,
namely that T(Face)(daniel ) belongs to KB.
In order to overwhelm this limit and perform useful inferences, in [17, 14] the au-
thors have introduced a nonmonotonic extension of the logic DL-Litec T based on a
minimal model semantics. Intuitively, the idea is to restrict our consideration to models
that maximize typical instances of a concept when consistent with the knowledge base.
The resulting logic, called DL-Litec Tmin , supports typicality assumptions, so that if
one knows that Daniel is a face wrestler, one can nonmonotonically assume that he is
also a typical face wrestler and therefore that he is supported by the crowd.
From a semantic point of view, the logic DL-Litec Tmin is based on a preference
relation among DL-Litec T models and a subsequent notion of minimal entailment re-
stricted to models that are minimal with respect to such preference relation.
In several applications the assumptions of typicality in DL-Litec Tmin seem to be
too strong, for instance when the need arises of bounding the cardinality of the extension
of a given concept, that is to say the number of domain elements being members of such
a concept, as introduced in [4]. As an example, consider the following KB:
T(Face) v Winner
T(Returning) v Winner
T(Predicted ) v Winner
If the assertional part of the KB contains the facts that:
Face(daniel ),
Returning(dave),
Predicted (roman)
whose meaning is that Daniel is a face athlete, Dave is returning from an injury, and
that Roman has been predicted to win the Royal Rumble match, respectively, then in
DL-Litec Tmin we conclude that
T(Face)(daniel )
T(Returning)(daniel )
T(Predicted )(roman)
and then that Dave, Daniel and Roman are all winners. This happens in DL-Litec Tmin
because it is consistent to make the three assumptions above, that hold in all minimal
models, however one should be interested in three distinct, but related aspects that can-
not be captured by DL-Litec Tmin as it is:
– first, one would like to restrict his attention to models/situations satisfying cardi-
nality restrictions (in the example, there is only one winner, therefore the three
assumptions above must be mutually exclusive);
– second, one could need to express different degrees of expectedness of typicality
inclusions: for instance, normally a top face wrestler wins the Royal Rumble match,
however this is in general more surprising with respect to the fact that, typically,
a returning top wrestler wins. In other words, both the two inclusions represent
typical properties, but the latter one seems to be more predictable;
– third, making all the consistent assumptions about prototypical properties should
be in contrast with the need of taking into account a reasonable but “surprising
enough” (or not obvious) scenario: in sports entertainment, a quite unpredictable
script should help to obtain a positive reaction from the crowd.
In this work, we propose a new extension of the standard Description Logic DL-Litecore
for reasoning about typicality called DL-Litec Texp , whose aim is to restrict reasoning
in DL-Litec T to “non trivial” scenarios respecting restrictions on the cardinality of con-
cepts, in order to match the needs of proposing memorable scripts for events in sports
entertainment. The original contribution of this work can be summarized as follows:
– we introduce a new Description Logic of typicality, called DL-Litec Texp , allowing
to express a degree of expectedness of typicality assumptions, that is to say TBoxes
are extended by (i) inclusions of the form T(C) vd D where d is a positive integer,
such that an inclusion with degree d is more “trivial” (or “obvious”) with respect to
another one with degree d0 ≤ d, as well as by (ii) restrictions on the cardinality of
concepts;
– we introduce a notion of extension of an ABox for the logic DL-Litec Texp , corre-
sponding to a set of typicality assumptions that can be performed in DL-Litec Tmin
for individual constants, then we introduce an order relation among extensions
whose basic idea is to prefer extensions representing more surprising scenarios;
– we define notions of entailment in DL-Litec Texp , relying on existing reasoners for
DL-Litec T, but allowing to restrict our concern to “non trivial” scenarios, corre-
sponding to minimal extensions with respect to the order relation among extensions
of the previous point.
The plan of the paper is as follows. In Section 2 we briefly recall preferential DLs
DL-Litec T and DL-Litec Tmin . In Section 3 we introduce the logic DL-Litec Texp , al-
lowing to express degrees of expectedness of typicality inclusions as well as to deal
with cardinality restrictions: we introduce notions of eligible and perfect extensions of
an ABox for the logic DL-Litec Texp , allowing to describe a plausible, but unexpected
scenario. In Section 4 we apply DL-Litec Texp in the context of sports entertainment
to find a script for a better Royal Rumble match. Issues that will be object of future
research are described in the concluding Section 5.
2 Preferential Description Logics DL-Litec T and DL-Litec Tmin
The logic DL-Litec T is obtained by adding to standard DL-Litecore the typicality op-
erator T [14]. The intuitive idea is that T(C) selects the typical instances of a concept
C. We can therefore distinguish between the properties that hold for all instances of
concept C (C v D), and those that only hold for the normal or typical instances of C
(T(C) v D).
The language of DL-Litec T is defined as follows.
Definition 1. We consider an alphabet of concept names C, of role names R, and of
individual constants O. Given A ∈ C and S ∈ R, we define
R := S | S −
CL := A | ∃R.> | T(A)
CR := A | ¬A | ∃R.> | ¬∃R.>
A DL-Litec T KB is a pair (TBox, ABox). TBox contains a finite set of concept inclusions
of the form CL v CR . ABox contains assertions of the form C(a) and R(a, b), where
C is a concept CL or CR , R ∈ R, and a, b ∈ O.
In order to provide a semantics to the operator T, the definition of a model M = h∆, Ii
used in “standard” terminological logic DL-Litecore , where ∆ is the domain and I is
a function mapping each concept C to its extension C I ⊆ ∆, is extended by a global
preference relation among individuals of ∆: in this respect, x < y means that x is “more
normal” than y, and that the typical members of a concept C are the minimal elements
of C with respect to this relation. In this framework, an element x ∈ ∆ is a typical
instance of some concept C if x ∈ C I and there is no C-element in ∆ more typical
than x. The typicality preference relation is partial. The basic idea is that the operator T
is characterized by a set of postulates that are essentially a reformulation of the Kraus,
Lehmann and Magidor’s axioms of preferential logic P [20]. Intuitively, the assertion
T(C) v D corresponds to the conditional assertion C |∼ D of P. T has therefore all
the “core” properties of nonmonotonic reasoning.
Definition 2 (Well-foundedness). Given an irreflexive and transitive relation < over
∆ and S ⊆ ∆, we define M in< (S) = {x : x ∈ S and @y ∈ S s.t. y < x}. We say that
< is well-founded if and only if, for all S ⊆ ∆, for all x ∈ S, either x ∈ M in< (S) or
∃y ∈ M in< (S) such that y < x.
Definition 3 (Multilinearity). Given a preference relation < over a domain ∆, we say
that < is multilinear if, for all u, v, z ∈ ∆, if u < z and v < z, then either u = v or
u < v or v < u.
Definition 4. A model of DL-Litec T is any structure h∆, <, Ii, where: ∆ is the do-
main; I is the extension function that maps each extended concept C to C I ⊆ ∆, and
each role R to a RI ⊆ ∆ × ∆; < is an irreflexive, transitive, well-founded (Defi-
nition 2) and multilinear (Definition 3) relation over ∆. I is defined for atomic con-
cepts A ∈ C end extended to complex concepts in the usual way (as for DL-Litecore ):
(¬A)I = ∆\AI , (∃S.>)I = {x ∈ ∆ | ∃y ∈ ∆ and (x, y) ∈ S I }, (∃S − .>)I = {x ∈
∆ | ∃y ∈ ∆ and (y, x) ∈ S I }; in addition, (T(C))I = M in< (C I ).
Given a model M of Definition 4, I can be extended so that it assigns to each individual
a of O a distinct element aI of the domain ∆ (unique name assumption). We say that
M satisfies an inclusion C v D if C I ⊆ DI , and that M satisfies C(a) if aI ∈ C I ,
S(a, b) if (aI , bI ) ∈ S I , and S − (a, b) if (bI , aI ) ∈ S I . Moreover, M satisfies TBox
if it satisfies all its inclusions, and M satisfies ABox if it satisfies all its formulas. M
satisfies a KB (TBox,ABox), if it satisfies both TBox and ABox.
We can also define a notion of entailment in DL-Litec T. Given a query F (either an
inclusion C v D or an assertion of the form C(a) or an assertion of the form R(a, b)),
we say that F is entailed from a KB in DL-Litec T if F holds in all DL-Litec T models
satisfying KB, and we write KB |=DL-Litec T F .
The semantics of the typicality operator can be specified by modal logic. The in-
terpretation of T can be split into two parts: for any x of the domain ∆, x ∈ (T(C))I
just in case (i) x ∈ C I , and (ii) there is no y ∈ C I such that y < x. Condition (ii) can
be represented by means of an additional modality , whose semantics is given by the
preference relation < interpreted as an accessibility relation. The interpretation of in
M is as follows: (C)I = {x ∈ ∆ | for every y ∈ ∆, if y < x then y ∈ C I }. We
immediately get that x ∈ (T(C))I if and only if x ∈ (C u ¬C)I .
Even if the typicality operator T itself is nonmonotonic (i.e. T(C) v E does not
imply T(C u D) v E), what is inferred from a KB can still be inferred from any KB’
with KB ⊆ KB’. In order to perform nonmonotonic inferences, in [17] the authors have
strengthened the above semantics by restricting entailment to a class of minimal (or
preferred) models. Intuitively, the idea is to restrict entailment to models that minimize
the untypical instances of a concept. The resulting logic is called DL-Litec Tmin .
Given a KB, we consider a finite set LT of concepts: these are the concepts whose
untypical instances we want to minimize. We assume that the set LT contains at least
all concepts C such that T(C) occurs in the KB or in the query F . As we have already
said, x ∈ C I is typical for C if x ∈ (¬C)I . Minimizing the untypical instances of
C therefore means to minimize the objects falsifying ¬C for C ∈ LT . Hence, for a
given model M = h∆, <, Ii, we can define:
−
I
M
LT = {(x, ¬¬C) | x 6∈ (¬C) , with x ∈ ∆, C ∈ LT }.
Definition 5 (Preferred and minimal models). Given two models M = h∆, <, Ii
and M0 = h∆0 , <0 , I 0 i of a knowledge base KB, we say that M is preferred to M0
−
0− I0
w.r.t. LT , and we write M | ¬∃R.>
A DL-Litec Texp KB is a pair (TBox, ABox). TBox contains axioms of the form:
– CR v CR ;
– T(A) vd CR , where A ∈ C and d ∈ N+ is called the degree of expectedness;
– (≥ n CR ), where n ∈ N+ ;
– (≤ n CR ), where n ∈ N+ ;
– (= n CR ), where n ∈ N+ .
ABox contains assertions of the form C(a) and R(a, b), where C is a concept of CR ,
R ∈ R, and a, b ∈ O.
3.1 Extensions of the ABox and order among extensions
Given an inclusion T(C) vd D, the more the degree of expectedness is high, the
more the inclusion is, in some sense, “obvious”, not surprising. Given another inclusion
T(C 0 ) vd0 D0 , with d0 < d, we assume that this inclusion is less “obvious”, more
surprising with respect to the other one. As an example, let KB contain T(Student) v4
SocialNetworkUser and T(Student) v2 PartyParticipant, representing that typical
students make use of social networks, and that normally they go to parties; however, the
second inclusion is less obvious with respect to the first one.
Given a KB, we define a finite set C of concepts for the evaluation of typical prop-
erties. We assume that, for all T(C) vd D ∈ KB, then C ∈ C.
Given an individual a explicitly named in the ABox, we define the set of “plausible”
typicality assumptions T(C)(a) that can be minimally entailed from KB in the logic
DL-Litec Tmin , with C ∈ C. We then consider an ordered set of pairs (a, C) of all
possible assumptions T(C)(a), for all concepts C ∈ C and all individual constants a
occurring in ABox. This is formally stated in the next definition:
Definition 8 (Assumptions in DL-Litec Texp ). Given a KB=(TBox,ABox) and the set
of concepts C, we define, for each individual name a occurring in ABox:
Ca = {C ∈ C | KB |=DL-Litec Tmin T(C)(a)}
We also define CABox = {(a, C) | C ∈ Ca and a occurs in ABox} and we impose an
order on the elements of CABox :
CABox =< (a1 , C1 ), (a2 , C2 ), . . . , (an , Cn ) > .
Furthermore, we define the ordered multiset:
dABox =< d1 , d2 , . . . , dn >
respecting the order imposed on CABox , where di = avg({d ∈ N+ | T(Ci ) vd D ∈
TBox}).
Intuitively, the ordered multiset dABox contains tuples of the form < d1 , d2 , . . . , dn >,
where di is the degree of expectedness of the assumption T(C)(a), such that (a, C) ∈
CABox at position i. di corresponds to the average of all the degrees d of typicality
inclusions T(C) vd D in the TBox.
In order to define alternative scenarios, where not all plausible assumptions are
taken into account, we consider different extensions of the ABox and we introduce an
order among them, allowing to range from unpredictable to trivial ones. Starting from
tuples < d1 , d2 , . . . , dn > in dABox , the first step is to build all alternative tuples where
0 is used in place of some di to represent that the corresponding typicality assertion
T(C)(a) is no longer assumed (Definition 9). Furthermore, we define the extension of
the ABox corresponding to a string so obtained (Definition 10). To give an intuitive
idea, before introducing the formal definitions, let us consider the following example:
Example 1. Given a KB, let the only typicality inclusions in TBox be T(C) v1 D
and T(E) v2 F . Let a and b be the only individual constants occurring in the ABox.
Suppose also that (i) KB |=DL-Litec Tmin T(C)(a), (ii) KB |=DL-Litec Tmin T(C)(b),
and (iii) KB |=DL-Litec Tmin T(E)(b). We have that:
CABox = {(a, C), (b, C), (b, E)}
dABox =< 1, 1, 2 >
Other possible tuples are: < 0, 0, 2 >, corresponding to extending the ABox with the
only assumption T(E)(b); < 0, 1, 0 >, corresponding to extending the ABox with
the only assumption T(C)(b); < 1, 0, 0 >, corresponding to extending the ABox
with T(C)(a); < 0, 1, 2 >, corresponding to extending the ABox with the assump-
tions T(C)(b) and T(E)(b); < 1, 0, 2 >, corresponding to extending the ABox with
T(C)(a) and T(E)(b); < 1, 1, 0 >, corresponding to extending the ABox with T(C)(a)
and T(C)(b); < 0, 0, 0 >, corresponding to not extending the ABox (the set of typical-
ity assumptions is empty).
Let us now introduce formal definitions for the above mentioned notions of string of
plausible assumptions and of extension of an ABox corresponding to a string.
Definition 9 (Strings of plausible assumptions S). Given a KB=(TBox,ABox) and the
set CABox , let dABox =< d1 , d2 , . . . , dn > be the ordered multiset of Definition 8. We
define the set S of all the strings of plausible assumptions with respect to KB as
S = {< s1 , s2 , . . . , sn >| ∀i = 1, 2, . . . , n either si = di or si = 0}
Definition 10 (Extension of the ABox). Let KB=(TBox,ABox) and let CABox =<
(a1 , C1 ), (a2 , C2 ), . . . , (an , Cn ) > as in Definition 8. Given a string of plausible as-
sumptions < s1 , s2 , . . . , sn >∈ S of Definition 9, we define the extension ABox [ of the
ABox corresponding to the string as
[ = {T(Ci )(ai ) | (ai , Ci ) ∈ C
ABox ABox and si 6= 0}
It is easy to observe that, in DL-Litec Tmin , the set of typicality assumptions that can
be inferred from a KB corresponds to the extension of the ABox corresponding to the
string dABox , that is to say no element is set to 0: all the typicality assertions of in-
dividuals occurring in the ABox, that are consistent with the KB, are assumed. This
corresponds to the “most obvious” situation. On the contrary, in DL-Litec T, no typical-
ity assumptions can be derived from a KB, and this corresponds to extending the ABox
by the assertions corresponding to the string < 0, 0, . . . , 0 >, i.e. by the empty set. This
corresponds to the most surprising situation. Between them, all the other strings of S
(Definition 9), corresponding to alternative extensions of the ABox, that we propose to
order as follows:
Definition 11 (Order between extensions). Given a KB=(TBox,ABox) and the set S
of strings of plausible assumptions (Definition 9), let s =< s1 , s2 , . . . , sn > and r =<
r1 , r2 , . . . , rn >, with s, r ∈ S. Furthermore, let ABox
\s and ABox
\r be the extensions
of the ABox corresponding to s and r (Definition 10), respectively. We say that s ≤ r if
there exists a bijection δ between s and r such that, for each (si , rj ) ∈ δ, it holds that
si ≤ rj , and there is at least one (si , rj ) ∈ δ such that si < rj . We say that ABox
\s is
more surprising (or less trivial) than ABoxr if s ≤ r.
\
Intuitively, a string s whose elements are “lower” than the ones of another string r
corresponds to a less trivial ABox. For instance, recalling Example 1, let us consider
the strings s =< 1, 1, 0 > and r =< 1, 0, 2 >, we have that s ≤ r, because there
exists a bijection {(1, 1), (0, 0), (1, 2)} whose pairs (si , ri ) are such that si ≤ ri . The
assumptions T(C)(a) and T(C)(b) corresponding to s are then considered less trivial
than T(C)(a) and T(E)(b) corresponding to r. It is worth noticing that the order of
Definition 11 is partial: as an example, the strings < 1, 1 > and < 0, 2 > are not
comparable, in the sense that neither < 1, 1 > ≤ < 0, 2 > nor < 0, 2 > ≤ < 1, 1 >. In
order to choose between two incomparable situations, we introduce the following notion
of weak order. Intuitively, the idea is as follows: given two incomparable extensions
ABox
\s and ABox \r , we assume that ABox \s is weakly less trivial than ABox \r if ABox\r
is strictly included in another extension ABox
\u more trivial than ABox \s .
Definition 12 (Weak preference). Given a KB=(TBox,ABox), let ABox
\s and ABox
\r be
\s is more surprising than ABox
two extensions of the ABox such that neither ABox \r nor
\r is more surprising than ABox
ABox \s is (weakly) more surprising
\s . We say that ABox
(or (weakly) less trivial) than ABox
\r if there exists an extension ABox
\u of ABox such
\s is more surprising than ABox
that (i) ABox \r ⊂ ABox
\u (Definition 11) and (ii) ABox \u .
As an example, let
\s = {T(C)(a)},
ABox
\r = {T(D)(b)},
ABox
\u = {T(D)(b), T(E)(b)}
ABox
be three extensions of the ABox of a given KB=(TBox,ABox), corresponding to s =<
1, 0, 0 >, r =< 0, 1, 0 >, and u =< 0, 1, 2 >, respectively. We have that s =<
1, 0, 0 > and r =< 0, 1, 0 > are not comparable with respect to the relation ≤. How-
ever, we have that (i) s ≤ u and that (ii) ABox
\r ⊂ ABox
\u , therefore we conclude that
ABox
\s is (weakly) more surprising (or (weakly) less trivial) than ABox
\r .
3.2 Cardinality restrictions on concepts and perfect extensions
In general, it could be useful to restrict logical entailment to models in which the car-
dinality of the extensions of some concepts is bounded. More expressive DLs allow to
specify (un)qualified number restrictions, in order to specify the number of possible el-
ements filling a given role R. As an example, number restrictions allow to express that
a student attends to 3 courses. Number restrictions are therefore “localized to the fillers
of one particular role” [4], for instance we can have Student v≥ 3Attends.Course as
a restriction on the number of role fillers of the role Attends. However one could need
to express global restrictions on the number of domain elements belonging to a given
concept, for instance to express that in the whole domain there are exactly 3 courses. In
DLs not allowing cardinality restrictions one can only express that every student must
attend to three courses, but not that all must attend to the same ones.
In the logic DL-Litec Texp , cardinality restrictions on concepts are added to the
TBox as in Definition 7. They are expressions of the form either (≥ n C) or (≤ n C)
or (= n C), where n is a positive integer and C is an extended concept.
Definition 13. Given a DL-Litec T model M = h∆, <, Ii, where I is extended so that
it assigns to each individual a of O a distinct element aI of the domain ∆ (unique name
assumption), we say that M satisfies:
– (elements of a TBox)
• an inclusion C v D if C I ⊆ DI ;
• a typicality inclusion T(C) vd D if Min < (C I ) ⊆ DI ;
• a cardinality restriction of the form (≥ n C) if ]C I ≥ n
• a cardinality restriction of the form (≤ n C) if ]C I ≤ n
• a cardinality restriction of the form (= n C) if ]C I = n
– (elements of an ABox)
• an assertion of the form C(a) if aI ∈ C I
• an assertion of the form R(a, b) if (aI , bI ) ∈ RI .
Given a KB=(T ∪ C,ABox), where T is a set of inclusions and C is a set of axioms
of cardinality restrictions, we say that a model M satisfies KB if it satisfies all the
inclusions in T , all the axioms of cardinality restrictions in C and all the assertions in
ABox.
Given a KB=(TBox,ABox), we say that an extension of ABox is an eligible extension
if it admits a DL-Litec T model as in Definition 13:
\ Given a DL-Litec T KB=(TBox,ABox) and
Definition 14 (Eligible extension ABox).
[ is eligible if there
[ of ABox as in Definition 10, we say that ABox
an extension ABox
exists a DL-Litec T model M that satisfies KB’=(TBox, ABox ∪ ABox).
[
Definition 15 (Minimal (perfect) extensions). Given a KB=(TBox,ABox) and the set
S of strings of plausible assumptions (Definition 9), we say that an eligible extension
\r which is (weakly) more
\s is minimal if there is no other eligible extension ABox
ABox
surprising (or (weakly) less trivial) than it.
Given the above definitions, we can define a notion of entailment in DL-Litec Texp . Intu-
itively, given a query F , we check whether F follows in the monotonic logic DL-Litec T
from a given KB, whose ABox is augmented with extensions that are minimal (perfect)
as in Definition 15. We can reason either in a skeptical way, by allowing that F is en-
tailed if it follows in all KBs, obtained by considering each minimal extension of the
ABox, or in a credulous way, by assuming that F is entailed if there exists at least one
extension of the ABox allowing such inference. This is stated in a rigorous manner by
the following definition:
Definition 16 (Entailment in DL-Litec Texp ). Given a KB=(TBox,ABox) and given C
a set of concepts, let E the set of all extensions of ABox that are minimal as in Definition
15. Given a query F , we say that (i) F is skeptically entailed from KB in DL-Litec Texp ,
written KB |=skDL-Litec Texp F , if (TBox, ABox ∪ ABox) |=DL-Litec T F for all ABox ∈
[ [
exp cr
E; (ii) F is credulously entailed from KB in DL-Litec T , written KB |=DL-Lite Texp
c
[ ∈ E such that (TBox, ABox ∪ ABox)
F , if there exists ABox [ |=
DL-Litec T F .
Let us conclude this section with an example of how the proposed approach works.
Example 2. Let us recall and simplify the example of the Introduction. Consider a
KB=(TBox,ABox) where TBox is as follows:
T(Face) v1 Winner
T(Predicted ) v2 Winner
T(Returning) v3 Winner
expressing that, normally, a returning athlete wins the Royal Rumble match, and this
is more predictable with respect to the fact that an athlete whose victory has been pre-
dicted, typically wins the match. Furthermore, normally a face wrestler wins the Royal
Rumble match, but this inclusion is the most unexpected among the ones belonging to
the KB. ABox contains the following facts about Dean, Roman, and Dave:
Face(dean)
Face(roman)
Predicted (roman)
Returning(dave)
Moreover, the TBox is enriched by the cardinality restriction (= 1 Winner ), i.e. we
restrict our concern to models in which there is only one winner.
Let C = {Face, Predicted , Returning}. By Definition 8 above, we have that:
Cdean = {Face}
Croman = {Face, Predicted }
Cdave = {Returning}
and, obviously, CABox = Cdean ∪ Croman ∪ Cdave . Concerning the degrees of
expectedness, we have:
dABox =< 1, 1, 2, 3 >
The leftmost 1 is due to the fact that T(Face) v1 Winner belongs to the TBox, and we
have that in DL-Litec Tmin one can assume that Dean is a T(Face). Similarly, the other
1 is due to the fact the in DL-Litec Tmin we can assume T(Face)(roman). Similarly,
we have 2 in the multiset dABox , by the presence of T(Predicted ) v2 Winner in the
TBox and the fact that T(Predicted )(roman) is minimally entailed from the KB. Last,
3 is justified by the presence of T(Returning) v3 Winner and the fact that we can
assume T(Returning)(dave).
As mentioned above, in DL-Litec Tmin the minimal model semantics forces all the
consistent typicality assumptions, namely we are considering an ABox extended with
the following facts:
T(Face)(dean)
T(Face)(roman)
T(Predicted )(roman)
T(Returning)(dave)
corresponding (in the sense of Definition 10) to the multiset < 1, 1, 2, 3 >. However,
from the resulting KB, in DL-Litec T we obtain that Dean, Roman and Dave are all win-
ners, against the fact that we want to have only one winner: the extension corresponding
to < 1, 1, 2, 3 > is indeed not eligible in the sense of Definition 14.
In order to find only one winner and to obtain a non-trivial outcome of the match,
let us consider the set S of all plausible strings of typicality assumptions (Definition 9):
S = {< 1, 1, 2, 3 >, < 0, 1, 2, 3 >, < 1, 0, 2, 3 >, < 1, 1, 0, 3 >, < 1, 1, 2, 0 >,
< 0, 0, 2, 3 >, < 1, 0, 0, 3 >, < 0, 1, 0, 3 >, < 1, 0, 2, 0 >, < 0, 1, 2, 0 >, < 1, 1, 0, 0 >,
< 0, 0, 0, 3 >, < 0, 0, 2, 0 >, < 1, 0, 0, 0 >, < 0, 1, 0, 0 >, < 0, 0, 0, 0 >}
The only eligible extensions of ABox, corresponding to the above strings are:
\1 = {T(Returning)(dave)} , corresponding to < 0, 0, 0, 3 >
ABox
\2 = {T(Predicted )(roman)} , corresponding to < 0, 0, 2, 0 >
ABox
\3 = {T(Face)(dean)} , corresponding to < 1, 0, 0, 0 >
ABox
\4 = {T(Face)(roman)} , corresponding to < 0, 1, 0, 0 >
ABox
\5 = {T(Face)(roman), T(Predicted )(roman)}, corresponding to <
ABox
0, 1, 2, 0 >
We aim at choosing the less trivial scenario. To this aim, we observe that ABox
\3
and ABox4 are less trivial than ABox5 , because < 1, 0, 0, 0 > ≤ < 0, 1, 2, 0 > and
\ \
< 0, 1, 0, 0 > ≤ < 0, 1, 2, 0 >. Furthermore, ABox
\3 and ABox \4 are less trivial than
\1 (again, < 1, 0, 0, 0 > ≤ < 0, 0, 0, 3 > and < 0, 1, 0, 0 > ≤ < 0, 0, 0, 3 >).
ABox
T(Predicted ) v4 Winner T(MidCarder ) v4 FastExit Heel (wyatt)
T(Face) v1 Winner T(MidHeel ) v1 ¬EarlyEntrance Returning(bryan)
Face(bryan)
T(Returning) v4 Winner T(MidFace) v4 EarlyEntrance
Heel (kane)
T(BigMan) v4 Final T(Heel ) v2 EarlyEntrance
BigMan(kane)
T(Face) v3 Final T(Face) v2 EarlyEntrance Predicted (reigns)
T(Heel ) v2 Final T(BodyBuilder ) v3 FastExit Face(reigns)
MidFace v Face (= 2 EarlyEntrance) BigMan(bigshow )
MidHeel v Heel ( 2 Final ) Face(ryback )
MidHeel v MidCarder ( 3 Final ) BodyBuilder (ryback )
MidFace v MidCarder (= 1 Winner ) Face(ziggler )
Fig. 1. A portion of the KB in DL-Litec Texp adopted for the application to sports entertainment.
Moreover, ABox
\3 and ABox \2 (again, < 1, 0, 0, 0 > ≤
\4 are less trivial than ABox
< 0, 0, 0, 2 > and < 0, 1, 0, 0 > ≤ < 0, 0, 0, 2 >). The strings < 1, 0, 0, 0 > and
< 0, 1, 0, 0 > are not comparable, however ABox
\3 is weakly less trivial than ABox\4 ,
since ABox4 ⊂ ABox5 and < 1, 0, 0, 0 > ≤ < 0, 1, 2, 0 >. This allows to conclude
\ \
that ABox
\3 is minimal (the perfect extension) and to suggest that Dean has to be chosen
as the winner of the Royal Rumble match.
4 DL-Litec Texp meets Sports Entertainment: a Better Royal
Rumble match
In Figure 1 we present a small portion of the simple ontology with exceptions we have
considered in order to suggest an alternative, possibly better script of the Royal Rumble
2015. We have considered athletes involved in the Royal Rumble 2015, that took place
on January 25, 2015 at the Wells Fargo Center in Philadelphia, Pennsylvania: Roman
Reigns, widely predicted, won the contest. One minimal/perfect extension suggests the
following alternative script: the match starts with a returning, face athlete, Daniel Bryan,
and another face superstar, Ryback. Dolph Ziggler is the winner, with Bray Wyatt being
the last athlete eliminated. We have then asked over 30 wrestling experts and fans about
the result, and all of them found it very interesting and significantly better than the
original one. Obviously, this is only a preliminary feedback, we aim at taking care
of a more precise evaluation of the quality (in terms of non triviality) of the scenario
proposed by adopting the machinery described in this work.
5 Conclusions and Future Issues
We have moved a first step in the direction of an alternative semantics for preferen-
tial Description Logics and its application in the context of sports entertainment. We
have introduced the Description Logic of typicality DL-Litec Texp , an extension of
DL-Litecore with a typicality operator T allowing to:
– express typicality inclusions of the form T(A) vd B, where d is a positive integer
representing a degree of expectedness;
– reason in presence of restrictions on the cardinality of concepts;
– perform plausible inferences in presence of alternative, non-trivial scenarios.
We are currently working on studying the complexity of standard reasoning tasks in
DL-Litec Texp . We have chosen the logic DL-Litec Tmin as the base logic of our ap-
proach also thanks to its computational properties: in [14] the authors have shown that
minimal entailment in DL-Litec Tmin is in Π2p . We strongly conjecture that adding car-
dinality restrictions and the machinery for finding the perfect extension of an ABox is
absorbed by the complexity of reasoning in DL-Litec Tmin , and is therefore inexpen-
sive. We also intend to develop and implement proof methods for reasoning in the logic
DL-Litec Texp of optimal complexity.
One limit of the proposed approach is that the computation of the extension of the
ABox only applies to individuals explicitly named in the knowledge base. This is one
of the limits of the completion of an ALC + T ABox in [13] as well. We aim at ex-
tending our work in order to also consider the individuals introduced by the existential
restrictions (e.g. (∃HasSon.>)(bob), the son of Bob). To this aim, we can define the
assumptions in DL-Litec Texp on domain elements rather than on individual constants.
In our approach, we first consider all possible typicality assumptions that are minimally
entailed in DL-Litec Tmin from a KB without cardinality restrictions, and then we re-
strict our concern to models satisfying cardinality restrictions and the ABox extended
with such assumptions. We aim at studying an alternative approach in which cardinality
restrictions are directly expressed in the initial KB, and the notion of preference among
extensions of the ABox is replaced by a preference relation of expectedness among
models, thus allowing to consider domain elements not explicitly named in the ABox.
The above mentioned alternative approach suggests the opportunity of studying ex-
tensions of Description Logics of typicality with restrictions on the cardinality of con-
cepts. This task is of its own interest. As far as we know, no other non-monotonic
extension of DLs (DLS+default rules [3], DLs+circumscription [5], DLs+ Lifschitz’s
nonmonotonic logic MKNF [12, 22], DLs+rational closure [11, 18, 19]) has been ex-
tended to reason in presence of cardinality constraints.
In this work we have tried to tackle a problem coming from sports entertainment,
however the logic DL-Litec Texp of typicality with degree of expectedness can find sev-
eral alternative applications. As an example, it could be applied in healthcare and med-
ical diagnosis, where ontologies with exceptions should be useful for reasoning about
defeasible inheritance (e.g. normally, the heart is positioned in the left-hand side of the
chest, however people with situs inversus have the heart positioned in the right-hand
side). Sometimes, the “obvious” diagnosis given a set of symptoms is not the right one:
the semantics of DL-Litec Texp could be used in order to formulate a “mystery” diagno-
sis, alternative to the standard one.
As a further direction, we aim at extending our approach to other Description Log-
ics. On the one hand, we want to take into account other lightweight DLs, for instance
the logics of the EL family, allowing for conjunction (u) and (qualified) existential re-
striction (∃R.C). Despite their relatively low expressivity, they are relevant for several
applications, in particular in the bio-medical domain; for instance, small extensions
of EL can be used to formalize medical terminologies, such as the GALEN Medical
Knowledge Base, the Systemized Nomenclature of Medicine, and the Gene Ontology
used in bioinformatics. In [1, 2, 7] it is shown that reasoning in EL and several of its
extensions remains tractable (i.e., polynomial-time decidable) in the presence of the
TBox, and even of general concept inclusions (GCIs). An extension EL⊥ Tmin with the
typicality operator has been introduced in [14]. On the other hand, we want to consider
more expressive Description Logics, in particular the logics underlying the standard
language for ontology engineering OWL. In this direction, it seems to be promising an
alternative approach to non-monotonic semantics for the typicality operator based on a
notion of rational closure in DLs [18]. As a difference with the semantics adopted in
this paper, the semantics in [15, 18] exploits an alternative notion of preference among
models, based on the idea of minimizing the rank of objects in the domain (that is, their
level of “untypicality”), rather than minimizing the ¬2¬C-elements in the models.
This alternative way of minimizing typicality has the nice property that corresponds
to a simple reformulation of rational closure for the Description Logic ALC [18]. A
similar correspondence has also been proved for the more expressive SHIQ in [19].
References
1. Baader, F.: Terminological cycles in a description logic with existential restrictions. In: Pro-
ceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI-
03). pp. 325–330 (2003)
2. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope. In: Kaelbling, L., Saffiotti, A.
(eds.) Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJ-
CAI 2005). pp. 364–369. Professional Book Center, Edinburgh, Scotland, UK (August 2005)
3. Baader, F., Hollunder, B.: Priorities on defaults with prerequisites, and their application in
treating specificity in terminological default logic. Journal of Automated Reasoning (JAR)
15(1), 41–68 (1995)
4. Baader, F., Buchheit, M., Hollunder, B.: Cardinality restrictions on concepts. Artificial Intel-
ligence 88(1-2), 195–213 (1996)
5. Bonatti, P.A., Lutz, C., Wolter, F.: DLs with Circumscription. In: KR. pp. 400–410 (2006)
6. Bonatti, P.A., Faella, M., Sauro, L.: Defeasible inclusions in low-complexity dls: Prelimi-
nary notes. In: Boutilier, C. (ed.) Proceedings of the 21st International Joint Conference on
Artificial Intelligence (IJCAI 2009). pp. 696–701 (2009)
7. Brandt, S.: Polynomial time reasoning in a description logic with existential restrictions,
gci axioms, andwhat else? In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of the 16th
European Conference on Artificial Intelligence (ECAI 2004). pp. 298–302. IOS Press (2004)
8. Calvanese, D., Giacomo, G.D., Lembo, D., Lenzerini, M., Rosati, R.: DL-Lite: Tractable
Description Logics for Ontologies. In: Veloso, M., Kambhampati, S. (eds.) Proceedings of
the 20th National Conference on Artificial Intelligence and the 17th Innovative Applications
of Artificial Intelligence Conference. pp. 602–607. AAAI Press / The MIT Press, Pittsburgh,
Pennsylvania, USA (July 2005)
9. Calvanese, D., Giacomo, G.D., Lembo, D., Lenzerini, M., Rosati, R.: Tractable Reasoning
and Efficient Query Answering in Description Logics: The DL-Lite Family. Journal of Au-
tomated Reasoning (JAR) 39(3), 385–429 (2007)
10. Casini, G., Straccia, U.: Rational closure for defeasible description logics. In: Janhunen,
T., Niemelä, I. (eds.) Logics in Artificial Intelligence - Proceedings of the12th European
Conference (JELIA 2010). Lecture Notes in Computer Science (LNCS), vol. 6341, pp. 77–
90. Springer (2010)
11. Casini, G., Straccia, U.: Defeasible Inheritance-Based Description Logics. Journal of Artifi-
cial Intelligence Research (JAIR) 48, 415–473 (2013)
12. Donini, F.M., Nardi, D., Rosati, R.: Description logics of minimal knowledge and negation
as failure. ACM Transactions on Computational Logics (ToCL) 3(2), 177–225 (2002)
13. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: ALC+T: a preferential extension of
description logics. Fundamenta Informaticae 96, 341–372 (2009)
14. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Reasoning about typicality in low com-
plexity DLs: the logics EL⊥ Tmin and DL-Litec Tmin . In: Walsh, T. (ed.) Proceedings of the
22nd International Joint Conference on Artificial Intelligence (IJCAI 2011). pp. 894–899.
IOS Press, Barcelona, Spain (2011)
15. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: A minimal model semantics for non-
monotonic reasoning. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) Logics in Artificial
Intelligence - Proceedings of the 13th European Conference (JELIA 2012). Lecture Notes
in Artificial Intelligence (LNAI), vol. 7519, pp. 228–241. Springer-Verlag, Toulouse, France
(September 2012)
16. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Preferential vs Rational Description
Logics: which one for Reasoning About Typicality? . In: Coelho, H., Studer, R., Wooldridge,
M. (eds.) Proceedings of the 19th European Conference on Artificial Intelligence (ECAI
2010). FAIA (Frontiers in Artificial Intelligence and Applications), vol. 215, pp. 1069 –
1070. IOS Press, Lisbon, Portugal (August 2010)
17. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: A NonMonotonic Description Logic
for Reasoning About Typicality. Artificial Intelligence 195, 165 – 202 (2013)
18. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Minimal model semantics and rational
closure in description logics. In: Eiter, T., Glimm, B., Kazakov, Y., Krötzsch, M. (eds.) Infor-
mal Proceedings of the 26th International Workshop on Description Logics. CEUR Work-
shop Proceedings, vol. 1014, pp. 168–180. CEUR-WS.org (2013)
19. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Rational closure in SHIQ. In: DL
2014, 27th International Workshop on Description Logics. CEUR Workshop Proceedings,
vol. 1193, pp. 543–555. CEUR-WS.org (2014)
20. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and
cumulative logics. Artificial Intelligence 44(1-2), 167–207 (1990)
21. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artificial Intel-
ligence 55(1), 1–60 (1992)
22. Motik, B., Rosati, R.: Reconciling Description Logics and rules. Journal of the ACM 57(5)
(2010)