In this work we include cardinality restrictions and degrees of expectedness of inclusions in preferential Description Logics. We enrich the language of the nonmonotonic Description Logic DL-Litec T, obtained by adding a typicality 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.
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 difference with typical athletics and games, which are conducted for competition, the main objective of sports entertainment, and especially of professional wrestling, is to entertain 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 predetermined: 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 executed maneuver.
One of the most attractive events in professional wrestling is the WWE Royal Rumble 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 athlete 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 [
Nonmonotonic extensions of Description Logics (DLs) have been actively
investigated since the early 90s [
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:
The approach based on the typicality operator has been first introduced for the basic DL
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
NLOGSPACE in the size of the TBox [
The logic DL-LitecT 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-LitecT 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 [
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 restricted 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 [
Face (daniel ); Returning (dave );
Predicted (roman ) If the assertional part of the KB contains the facts that: 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 cannot be captured by DL-Litec Tmin as it is: – first, one would like to restrict his attention to models/situations satisfying cardinality 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 concepts, 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, corresponding 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, corresponding 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, allowing 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
operator T [
The language of DL-LitecT 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 2 C and S 2 R, we define
R := S j S CL := A j 9R:> j T(A)
CR := A j :A j 9R:> j :9R:> A DL-LitecT 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 2 R, and a; b 2 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 CI , 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 2 is a typical
instance of some concept C if x 2 CI 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 [
the “core” properties of nonmonotonic reasoning.
Definition 2 (Well-foundedness). Given an irreflexive and transitive relation < over and S , we define M in<(S) = fx : x 2 S and @y 2 S s.t. y < xg. We say that < is well-founded if and only if, for all S , for all x 2 S, either x 2 M in<(S) or 9y 2 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 2 , if u < z and v < z, then either u = v or u < v or v < u.
Definition 4. A model of DL-LitecT is any structure h ; <; Ii, where: is the domain; I is the extension function that maps each extended concept C to CI , and each role R to a RI ; < is an irreflexive, transitive, well-founded (Definition 2) and multilinear (Definition 3) relation over . I is defined for atomic concepts A 2 C end extended to complex concepts in the usual way (as for DL-Litecore ): (:A)I = nAI ; (9S:>)I = fx 2 j 9y 2 and (x; y) 2 SI g; (9S :>)I = fx 2 j 9y 2 and (y; x) 2 SI g; in addition, (T(C))I = M in<(CI ).
a of O a distinct element aI of the domain (unique name assumption). We say that M satisfies an inclusion C v D if CI DI , and that M satisfies C(a) if aI 2 CI , S(a; b) if (aI ; bI ) 2 SI , and S (a; b) if (bI ; aI ) 2 SI . 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-LitecT. 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-LitecT if F holds in all DL-LitecT models satisfying KB, and we write KB j=DL-LitecT F .
The semantics of the typicality operator can be specified by modal logic. The interpretation of T can be split into two parts: for any x of the domain , x 2 (T(C))I just in case (i) x 2 CI , and (ii) there is no y 2 CI 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 = fx 2 j for every y 2 , if y < x then y 2 CI g. We immediately get that x 2 (T(C))I if and only if x 2 (C u :C)I .
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 [
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 2 CI is typical for C if x 2 ( :C)I . Minimizing the untypical instances of
MLT = f(x; : :C) j x 62 ( :C)I ; with x 2 ; C 2 LTg: Definition 5 (Preferred and minimal models). Given two models M = h , <; Ii and M0 = h 0; <0; I0i of a knowledge base KB, we say that M is preferred to M0 w.r.t. LT, and we write M <LT M0, if (i) = 0, (ii) MLT M0LT , (iii) aI = aI0 for all a 2 O. M is a minimal model for KB (w.r.t. LT) if it is a model of KB and there is no other model M0 of KB such that M0 <LT M.
Definition 6 (Minimal Entailment in DL-LitecTmin). A query F is minimally entailed in DL-LitecTmin by KB with respect to LT if F is satisfied in all models of KB that are minimal with respect to LT. We write KB j=DL-LitecTmin F . 3
The Logic DL-Litec Texp: between DL-Litec T and DL-Litec Tmin
In this section we define an alternative semantics that allows us to express a degree of
expectedness for the typicality inclusions and to limit the number of typicality
assumptions in the ABox in order to obtain less predictable scenarios. The basic idea is similar
to the one proposed in [
First of all, let us define the language L of the logic DL-LitecTexp: Definition 7. We consider an alphabet of concept names C, of role names R, and of individual constants O. Given A 2 C and S 2 R, we define
R := S j S
CR := A j :A j 9R:> j :9R:> A DL-LitecTexp KB is a pair (TBox, ABox). TBox contains axioms of the form: – CR v CR; – T(A) vd CR, where A 2 C and d 2 N+ is called the degree of expectedness; – ( n CR), where n 2 N+; – ( n CR), where n 2 N+; – (= n CR), where n 2 N+.
ABox contains assertions of the form C(a) and R(a; b), where C is a concept of CR, R 2 R, and a; b 2 O. 3.1
Extensions of the ABox and order among extensions
more the inclusion is, in some sense, “obvious”, not surprising. Given another inclusion
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 properties. We assume that, for all T(C) vd D 2 KB, then C 2 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
possible assumptions T(C)(a), for all concepts C 2 C and all individual constants a occurring in ABox. This is formally stated in the next definition: Definition 8 (Assumptions in DL-LitecTexp). Given a KB=(TBox,ABox) and the set of concepts C, we define, for each individual name a occurring in ABox:
Ca = fC 2 C j KB j=DL-LitecTmin T(C)(a)g We also define CABox = f(a; C) j C 2 Ca and a occurs in ABoxg 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(fd 2 N+ j T(Ci) vd D 2 TBoxg).
where di is the degree of expectedness of the assumption T(C)(a), such that (a; C) 2
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:
CABox = f(a; C); (b; C); (b; E)g 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 assumptions 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 typicality 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 = f< s1; s2; : : : ; sn >j 8i = 1; 2; : : : ; n either si = di or si = 0g 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 assumptions < s1; s2; : : : ; sn >2 S of Definition 9, we define the extension A[Box of the ABox corresponding to the string as
A[Box = fT(Ci)(ai) j (ai; Ci) 2 CABox and si 6= 0g 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 individuals 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 typicality 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 2 S . Furthermore, let A\Boxs and A\Boxr be the extensions of the ABox corresponding to s and r (Definition 10), respectively. We say that s r if there exists a bijection between s and r such that, for each (si; rj ) 2 , it holds that si rj , and there is at least one (si; rj ) 2 such that si < rj . We say that A\Boxs is more surprising (or less trivial) than A\Boxr 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 f(1; 1); (0; 0); (1; 2)g 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 A\Boxs and A\Boxr, we assume that A\Boxs is weakly less trivial than A\Boxr if A\Boxr is strictly included in another extension A\Boxu more trivial than A\Boxs. Definition 12 (Weak preference). Given a KB=(TBox,ABox), let A\Boxs and A\Boxr be two extensions of the ABox such that neither A\Boxs is more surprising than A\Boxr nor A\Boxr is more surprising than A\Boxs. We say that A\Boxs is (weakly) more surprising (or (weakly) less trivial) than A\Boxr if there exists an extension A\Boxu of ABox such that (i) A\Boxs is more surprising than A\Boxu (Definition 11) and (ii) A\Boxr A\Boxu. As an example, let
A\Boxs = fT(C)(a)g, A\Boxr = fT(D)(b)g,
A\Boxu = fT(D)(b); T(E)(b)g 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 . However, we have that (i) s u and that (ii) A\Boxr A\Boxu, therefore we conclude that A\Boxs is (weakly) more surprising (or (weakly) less trivial) than A\Boxr. 3.2
Cardinality restrictions on concepts and perfect extensions
In general, it could be useful to restrict logical entailment to models in which the
cardinality 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
elements 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” [
In the logic DL-LitecTexp, 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-LitecT 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 CI DI ; a typicality inclusion T(C) vd D if Min<(CI ) DI ; a cardinality restriction of the form ( n C) if ]CI n a cardinality restriction of the form ( n C) if ]CI n a cardinality restriction of the form (= n C) if ]CI = n – (elements of an ABox) an assertion of the form C(a) if aI 2 CI an assertion of the form R(a; b) if (aI ; bI ) 2 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-LitecT model as in Definition 13: Definition 14 (Eligible extension A\Box). Given a DL-Litec T KB=(TBox,ABox) and an extension A[Box of ABox as in Definition 10, we say that A[Box is eligible if there exists a DL-Litec T model M that satisfies KB’=(TBox, ABox [ A[Box). 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 A\Boxs is minimal if there is no other eligible extension A\Boxr which is (weakly) more surprising (or (weakly) less trivial) than it.
Given the above definitions, we can define a notion of entailment in DL-Litec Texp. Intuitively, 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 entailed 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 j=sDkL-Litec Texp F , if (TBox, ABox [ A[Box) j=DL-LitecT F for all A[Box 2 E ; (ii) F is credulously entailed from KB in DL-Litec Texp, written KB j=cDrL-Litec Texp F , if there exists A[Box 2 E such that (TBox, ABox [ A[Box) j=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:
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 predicted, 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 = fFace ; Predicted ; Returning g. By Definition 8 above, we have that: Cdean = fFaceg Croman = fFace; Predicted g
Cdave = fReturning g and, obviously, CABox = Cdean [ Croman [ Cdave . Concerning the degrees of expectedness, we have:
dABox =< 1; 1; 2; 3 >
have that in DL-LitecTmin one can assume that Dean is a T(Face). Similarly, the other 1 is due to the fact the in DL-LitecTmin 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,
assume T(Returning )(dave).
As mentioned above, in DL-LitecTmin 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-LitecT we obtain that Dean, Roman and Dave are all winners, 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 = f< 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 >g The only eligible extensions of ABox, corresponding to the above strings are: A\Box1 = fT(Returning )(dave)g , corresponding to < 0; 0; 0; 3 > A\Box2 = fT(Predicted )(roman)g , corresponding to < 0; 0; 2; 0 > A\Box3 = fT(Face)(dean)g , corresponding to < 1; 0; 0; 0 > A\Box4 = fT(Face)(roman)g , corresponding to < 0; 1; 0; 0 > A\Box5 = fT(Face)(roman); T(Predicted )(roman)g, corresponding to < 0; 1; 2; 0 >
We aim at choosing the less trivial scenario. To this aim, we observe that A\Box3 and A\Box4 are less trivial than A\Box5, because < 1; 0; 0; 0 > < 0; 1; 2; 0 > and < 0; 1; 0; 0 > < 0; 1; 2; 0 >. Furthermore, A\Box3 and A\Box4 are less trivial than A\Box1 (again, < 1; 0; 0; 0 > < 0; 0; 0; 3 > and < 0; 1; 0; 0 > < 0; 0; 0; 3 >).
T(Predicted ) v4 Winner T(Face) v1 Winner T(Returning ) v4 Winner T(BigMan) v4 Final T(Face) v3 Final T(Heel ) v2 Final MidFace v Face MidHeel v Heel MidHeel v MidCarder MidFace v MidCarder
T(MidCarder ) v4 FastExit T(MidHeel ) v1 ¬EarlyEntrance T(MidFace) v4 EarlyEntrance T(Heel ) v2 EarlyEntrance T(Face) v2 EarlyEntrance T(BodyBuilder ) v3 FastExit (= 2 EarlyEntrance) (
2 Final ) ( 3 Final ) (= 1 Winner )
Heel (wyatt ) Returning (bryan) Face(bryan) Heel (kane) BigMan(kane) Predicted (reigns) Face(reigns) BigMan(bigshow ) Face(ryback ) BodyBuilder (ryback ) Face(ziggler ) Fig. 1. A portion of the KB in DL-Litec Texp adopted for the application to sports entertainment. Moreover, A\Box3 and A\Box4 are less trivial than A\Box2 (again, < 1; 0; 0; 0 > < 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 A\Box3 is weakly less trivial than A\Box4, since A\Box4 A\Box5 and < 1; 0; 0; 0 > < 0; 1; 2; 0 >. This allows to conclude that A\Box3 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
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
We have moved a first step in the direction of an alternative semantics for
preferential 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-LitecTexp. We have chosen the logic DL-LitecTmin as the base logic of our
approach also thanks to its computational properties: in [
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 [
The above mentioned alternative approach suggests the opportunity of studying
extensions of Description Logics of typicality with restrictions on the cardinality of
concepts. This task is of its own interest. As far as we know, no other non-monotonic
extension of DLs (DLS+default rules [
In this work we have tried to tackle a problem coming from sports entertainment, however the logic DL-LitecTexp of typicality with degree of expectedness can find several alternative applications. As an example, it could be applied in healthcare and medical 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-LitecTexp could be used in order to formulate a “mystery” diagnosis, alternative to the standard one.
As a further direction, we aim at extending our approach to other Description
Logics. On the one hand, we want to take into account other lightweight DLs, for instance
the logics of the E L family, allowing for conjunction (u) and (qualified) existential
restriction (9R:C). Despite their relatively low expressivity, they are relevant for several
applications, in particular in the bio-medical domain; for instance, small extensions
of E L 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 [