=Paper=
{{Paper
|id=None
|storemode=property
|title=Defeasible Reasoning in ORM2
|pdfUrl=https://ceur-ws.org/Vol-969/paper1.pdf
|volume=Vol-969
}}
==Defeasible Reasoning in ORM2==
https://ceur-ws.org/Vol-969/paper1.pdf
Defeasible reasoning in ORM2
Giovanni Casini1 and Alessandro Mosca2
1
Centre for Artificial Intelligence Research, CSIR Meraka Institute and UKZN, South Africa
Email: GCasini@csir.co.za
2
Free University of Bozen-Bolzano, Faculty of Computer Science, Italy
Email: mosca@inf.unibz.it
Abstract. The Object Role Modeling language (ORM2) is one of the main con-
ceptual modeling languages. Recently, a translation has been proposed of a main
fragment of ORM2 (ORM2zero ) into the description logic ALCQI, allowing the
use of logical instruments in the analysis of ORM schemas. On the other hand, in
many ontological domains there is a need for the formalization of defeasible infor-
mation and of nonmonotonic forms of reasoning. Here we introduce two new con-
straints in ORM2 language, in order to formalize defeasible information into the
schemas, and we explain how to translate such defeasible information in ALCQI.
1 Introduction
ORM2 (‘Object Role Modelling 2’) is a graphical fact-oriented approach for modelling,
transforming, and querying business domain information, which allows for a verbal-
isation in a language readily understandable by non-technical users [1]. ORM2 is at
the core of the OGM standard SBVR language (‘Semantics of Business Vocabulary
and Business Rules’), and of conceptual modelling language for database design in Mi-
crosoft Visual Studio (VS). In particular, the Neumont ORM Architect (NORMA) tool
is an open source plug-in to VS providing the most complete support for the ORM2
notation.
On the other hand, in the more general field of formal ontologies in the last years a
lot of attention has been dedicated to the implementations of forms of defeasible reason-
ing, and various proposals, such as [2,3,4,5,6,7,8], have been made in order to integrate
nonmonotonic reasoning mechanisms into DLs.
In what follows we propose an extension of ORM2 with two new formal constraints,
with the main aim of integrating a form of defeasible reasoning in the ORM2 schemas;
we explain how to translate such enriched ORM2 schemas into ALCQI knowledge
bases and how to use them to check the schema consistency and draw conclusions.
In particular, the paper presents a procedure to implement a particular construction in
nonmonotonic reasoning, i.e. Lehmann and Magidor’s Rational Closure (RC)[9], that is
known for being characterized by good logical properties and for giving back intuitive
results.
2 Fact-oriented modelling in ORM2
‘Fact-oriented modelling’ began in the early Seventies as a conceptual modelling ap-
proach that views the world in terms of simple facts about individuals and the roles they
play [1]. Facts are assertions that are taken to be true in the domain of interest about
Fig. 1. A conceptual schema including an instantiation of most of the ORM2 constraints.
objects playing certain roles (e.g. ‘Alice is enrolled in the Computer Science program’).
In ORM2 one has entities (e.g. a person or a car) and values (e.g. a character string or
a number). Moreover, entities and values are described in terms of the types they belong
to, where a type (e.g. House, Car) is a set of instances. Each entity in the domain of
interest is, therefore, an instance of a particular type. The roles played by the entities
in a given domain are introduced by means of logical predicates, and each predicate
has a given set of roles according to its arity. Each role is connected to exactly one ob-
ject type, indicating that the role is played only by the (possible) instances of that type
((e.g. TYPE(isBy.Student,Student)) - notice that, unlike ER, ORM2 makes no use of
‘attributes’). ORM2 also admits the possibility of making an object type out of a rela-
tionship. Once a relation has been transformed into an object type, this last is called the
objectification of the relation.
According to the ORM2 design procedure, after the specification of the relevant
object types (i.e. entity and value types) and predicates, the static constraints must be
considered. The rest of this section is devoted to an informal introduction of the con-
straint graphical representation, together with their intended semantics. Fig. 1 shows an
example of an ORM2 conceptual schema modelling the ‘academic domain’ (where the
soft rectangles are entity types, the dashed soft rectangles are value types, and the se-
quences of one or more role-boxes are predicates). The example is not complete w.r.t.
the set of all the ORM2 constraints but it aims at giving the feeling of the expressive
power of the language. The following are among the constraints included in the schema
(the syntax we devised for linearizing them is in square brackets):
1. Subtyping (depicted as thick solid and dashed arrows) representing ‘is-a’ re-
lationships among types. A partition, made of a combination of an exclu-
sive constraint (a circled ‘X’ saying that ‘Research&TeachingStaff, Ad-
min, Student are mutually disjoint’), and a total constraint (a circled dot for
‘Research&TeachingStaff, Admin, Student completely cover their common super-
type’). [O-SETTot ({Research&TeachingStaff, Admin, Student},UNI-Personnel)]
2. An internal frequency occurrence saying that if an instance of Research&TeachingStaff
plays the role of being lecturer in the relation isGivenBy, that instance can play the role
at most 4 times [FREQ(isGivenBy.Research&TeachingStaff,(1,4))]. A frequency occur-
rence may span over more than one role, and suitable frequency ranges can be specified. At
2
most one cardinalities (depicted as continuos bars) are special cases of frequency occurrence
called internal uniqueness constraints [e.g. FREQ(hasWeight.Course,(1,1))].
3. An external frequency occurrence applied to the roles played by Student and Course,
meaning that ‘Students are allowed to enrol in the same course at most twice’.
[FREQ(isIn.Course,isBy.Student,(1,2))]
4. An external uniqueness constraint between the role played by Course in isIn and
the role played by Date in wasOn, saying that ‘For each combination of Course
and Date, at most one Enrollment isIn that Course and wasOn that Date’.
[FREQ(isIn.Course,wasOn.Date,(1,1))]
5. A mandatory participation constraints (graphically represented by a dot),
among several other, saying that ‘Each Course isGivenBy at least one in-
stance of the Research&TeachingStaff type’ (combinations of manda-
tory and uniqueness translate into exaclty one cardinality constraints).
[MAND(isGivenBy.Research&TeachingStaff,Research&TeachingStaff)]
6. A disjunctive mandatory participation, called inclusive-or constraint (depicted as
a circled dot), linking the two roles played by the instances of AreaMan-
ager meaning that ‘Each area manager either works in or heads (or both)’.
[MAND({worksIn.AreaManager,heads.AreaManager},AreaManager)]
7. An object cardinality constraint forcing the number of the Admin instances to be less or
equal to 100 (role cardinality constraints, applied to role instances, are also part of ORM2).
[O-CARD(Admin)=(0,100)]
8. An object type value constraint indicating which values are allowed in Credit (role value
constraints can be also expressed to indicate which values are allowed to play a given role).
[V-VAL(Credit)={4,6,8,12}]
9. An exclusion constraint (depicted as circled ‘X’) between the two roles played by the
instances of Student, expressing the fact that no student can play both these roles. Ex-
clusion constraint can also span over arbitrary sequences of roles. The combination of
exclusion and inclusive-or constraints gives rise to exclusive-or constraints meaning that
each instance in the attached entity type plays exactly one of the attached roles. Exclu-
sion constraints, together with subset and equality, are called set-comparison constraints.
[R-SETExc (worksFor.Student,collaborates.Student)]
10. A ring constraint expressing that the relation reportsTo is asymmetric.
[RINGAsym (reportTo.Admin,reportTo.AreaManager)]
A comprehensive list of all the ORM2 constraints, together with their graphical
representation, can be found in [1].
3 The ALCQI encoding of ORM2zero
With the main aim of relying on available tools to reason in an effective way on ORM2
schemas, an encoding in the description logic ALCQI for which tableaux-based rea-
soning algorithms with a tractable computational complexity have been developed [10].
ALCQI corresponds to the basic DL ALC equipped with qualified cardinality restric-
tions and inverse roles, and it is a fragment of the OWL2 web ontology language (a
complete introduction of the syntax and semantics of ALCQI can be found in [11]).
We also introduce in the ALCQI language the expression ‘C ⊃ D’ as an abbreviation
for the expression ‘¬C t D’.
Now, the discrepancy between ORM2 and ALCQI poses two main obstacles that
need to be faced in order to provide the encoding. The first one, caused by the absence of
n-ary relations in ALCQI, is overcome by means of reification: for each relation R of
3
arity n ≥ 2, a new atomic concept AR and n functional roles τ (R.a1 ), . . . , τ (R.an ) are
introduced. The tree-model property of ALCQI guarantees the correctness encoding
w.r.t. the reasoning services over ORM2. Unfortunately, the second obstacle fixes,
once for all, the limits of the encoding: ALCQI does not admit neither arbitrary
set-comparison assertions on relations, nor external uniqueness or uniqueness involving
more than one role, or arbitrary frequency occurrence constraints. In other terms, it can
be proven that ALCQI is strictly contained in ORM2. The analysis of this inclusion
thus led to identification of the fragment called ORM2zero which is maximal with
respect to the expressiveness of ALCQI, and still expressive enough to capture the
most frequent usage patterns of the conceptual modelling community. Let ORM2zero =
{TYPE, FREQ− , MAND, R-SET− , O-SETIsa , O-SETTot , O-SETEx , OBJ} be the
fragment of ORM2 where: (i) FREQ− can only be applied to single roles, and (ii)
R-SET− applies either to entire relations of the same arity or to two single roles. The
encoding of the semantics of ORM2zero shown in table 1 is based on the S ALCQI
signature made of: (i) A set E1 , E2 , . . . , En of concepts for entity types; (ii) a set
V1 , V2 , . . . , Vm of concepts for value types; (iii) a set AR1 , AR2 , . . . , ARk of concepts
for objectified n-ary relations; (iv) a set D1 , D2 , . . . , Dl of concepts for domain
symbols; (v) 1, 2, . . . , nmax + 1 roles. Additional background axioms are needed here
in order to: (i) force the interpretation of the ALCQI knowledge base to be correct
w.r.t. the corresponding ORM2 schema, and (ii) guarantee that that any model of the
resulting ALCQI can be ‘un-reified’ into a model of original ORM2zero schema.
The correctness of the introduced encoding is guaranteed by the following theorem
(whose complete proof is available at [12]):
Theorem 1. Let Σ zero be an ORM2zero conceptual schema and Σ ALCQI the ALCQI
KB constructed as described above. Then an object type O is consistent in Σ zero if and
only if the corresponding concept O is satisfiable w.r.t. Σ ALCQI .
Let us conclude this section with some observation about the complexity of reason-
ing on ORM2 conceptual schemas, and taking into account that all the reasoning tasks
for a conceptual schema can be reduced to object type consistency. Undecidability of the
ORM2 object type consistency problem can be proven by showing that arbitrary com-
binations of subset constraints between n-ary relations and uniqueness constraints over
single roles are allowed [13]. As for ORM2zero , one can conclude that object type consis-
tency is E XP T IME-complete: the upper bound is established by reducing the ORM2zero
problem to concept satisfiability w.r.t. ALCQI KBs (which is known to be E XP T IME-
hard) [14], the lower bound by reducing concept satisfiability w.r.t. ALC KBs (which is
known to be E XP T IME-complete) to object consistency w.r.t. ORM2zero schemas [15].
Therefore, we obtain the following result:
Theorem 2. Reasoning over ORM2zero schemas is E XP T IME-complete.
4 Rational Closure in ALCQI
Now we briefly present the procedure to define the analogous of RC for the DL lan-
guage ALCQI. A more extensive presentation of such a procedure can be found in [4]:
it is defined for ALC, but it can be applied to ALCQI without any modifications. RC
is one of the main construction in the field of nonmonotonic logics, since it has a solid
4
Table 1. ALCQI encoding.
Background domain axioms: Ei v ¬(D1 t · · · t Dl ) for i ∈ {1, . . . , n}
Vi v Dj for i ∈ {1, . . . , m}, and some j with 1 ≤ j ≤ l
Di v ulj=i+1 ¬Dj for i ∈ {1, . . . , l}
> v A>1 t · · · t A>nmax
> v (≤ 1i.>) for i ∈ {1, . . . , nmax }
∀i.⊥ v ∀i + 1.⊥ for i ∈ {1, . . . , nmax }
A>n ≡ ∃1.A>1 u · · · u ∃n.A>1 u ∀n + 1.⊥ for n ∈ {2, . . . , nmax }
AR v A>n for each atomic relation R of arity n
A v A>1 for each atomic concept A
TYPE(R.a, O) ∃τ (R.a)− .AR v O
FREQ− (R.a, hmin, maxi) ∃τ (R.a)− .AR v ≥ min τ (R.a)− .AR u ≤ max τ (R.a)− .AR
MAND({R1 .a1 , . . . , R1 .an , O v ∃τ (R1 .a1 )− .AR1 t · · · t ∃τ (R1 .an )− .AR1 t · · · t
. . . , Rk .a1 , . . . , Rk .am }, O) ∃τ (Rk .a1 )− .ARk t · · · t ∃τ (Rk .am )− .ARk
(A)
R-SET− Sub (A, B) AR v AS (A) A = {R.a , . . . , R.a }, B = {S.b , . . . , S.b }
1 n 1 n
(A)
R-SET− Exc (A, B) AR v A>n u ¬AS
(B)
R-SET− Sub (A, B) ∃τ (R.ai )− .AR v ∃τ (S.bj )− .AS (B) A = {R.a }, B = {S.b }
i j
(B)
R-SET− Exc (A, B) ∃τ (R.ai )− .AR v A>n u ¬∃τ (S.bj ).AS
O-SETIsa ({O1 , . . . , On }, O) O1 t · · · t On v O
O-SETTot ({O1 , . . . , On }, O) O v O1 t · · · t On
O-SETEx ({O1 , . . . , On }, O) O1 t · · · t On v O and Oi v un j=i+1 ¬Oj for each i = 1, . . . , n
OBJ(R, O) O ≡ AR
logical characterization, it generally maintains the same complexity level of the under-
lying monotonic logic, and it does not give back counter-intuitive conclusions; its main
drawback is in its inferential weakness, since there could be desirable conclusions that
we won’t be able to draw (see example 2 below).
As seen above, each ORM2zero schema can be translated into an ALCQI TBox. A
TBox T for ALCQI consists of a finite set of general inclusion axioms (GCIs) of form
C v D, with C and D concepts. Now we introduce also a new form of information,
the defeaible inclusion axioms C < ∼ D, that are read as ‘Typically, an individual falling
under the concept C falls also under the concept D’. We indicate with B the finite set of
such inclusion axioms.
Example 1. Consider a modification of the classical ‘penguin example’, with the concepts
P, B, F, I, F i, W respectively read as ‘penguin’, ‘bird’, ‘flying’, ‘insect’, ‘fish’, and ‘have wings’,
and a role P rey, where a role instantiation (a, b):P rey read as ‘a preys for b’. We can define a
defeasible KB K = hT , Bi with T = {P v B, I v ¬F i} and B = {P ∼ < ¬F , B ∼ < F,
P∼ < ∀P rey.F i, B ∼< ∀P rey.I, B ∼ < W }.
In order to define the rational closure of a knowledge base hT , Bi, we must first of
all transform the knowledge base hT , Bi into a new knowledge base hΦ, ∆i, s.t. while
T and B are sets of inclusion axioms, Φ and ∆ are simply sets of concepts. Then, we
shall use the sets hΦ, ∆i to define a nonmonotonic consequence relation that models the
rational closure. Here we just present the procedure, referring to [4] for a more in-depth
explanation of the various steps.
Transformation of hT , Bi into hΦ, ∆i. Starting with hT , Bi, we apply the following
steps.
5
Step 1. Define the set representing the strict form of the set B, i.e. the set Bv = {C v D |
C ∼ < D ∈ B}, and define a set AB as the set of the antecedents of the conditionals in B,
i.e. AB = {C | C ∼ < D ∈ B}.
Step 2. We determine an exceptionality ranking of the sequents in B using the set of the an-
tecedents AB and the set Bv .
Step 2.1. A concept is considered exceptional in a knowledge base hT , Bi only if it is classi-
cally negated (i.e. we are forced to consider it empty), that is, C is exceptional in hT , Bi
only if
T ∪ Bv |= > v ¬C
where |= is the classical consequence relation associated to ALCQI. If a concept is
considered exceptional in hT , Bi, also all the defeasible inclusion axioms in B that have
such a concept as antecedent are considered exceptional. So, given a knowledge base
hT , Bi we can check which of the concepts in AB are exceptional (we indicate the set
containing them as E(AB )), and consequently which of the axioms in B are exceptional
(the set E(B) = {C ∼ < D | C ∈ E(AB )}).
So, given a knowledge base hT , Bi we can construct iteratively a sequence E0 , E1 , . . . of
subsets of B in the following way:
– E0 = B
– Ei+1 = E(Ei )
Since B is a finite set, the construction will terminate with an empty set (En = ∅ for
some n) or a fixed point of E.
Step 2.2 Using such a sequence, we can define a ranking function r that associates to every
axiom in B a number, representing
its level of exceptionality:
i if C ∼< D ∈ Ei and C ∼ 1 , A>2
f1, f2, f3
Background axioms: > v A>1 t A>2
> v (≤ 1f1.>), > v (≤ 1f2.>)
∀f1.⊥ v ∀f2.⊥, ∀f2.⊥ v ∀f3.⊥
A>2 ≡ ∃f1.A>1 u ∃f2.A>1 u ∀f3.⊥
WorksFor v A>2 , Manages v A>2
Employee v A>1 , Manager v A>1 ,
AreaManager v A>1 , TopManager v A>1
Typing Constraints: WorksFor v ∃f1− .Employee, WorksFor v ∃f2− .Project
Manages v ∃f1− .TopManager, Manages v ∃f2− .Project
Frequency Constraints: ∃f1− .Manages v = 1 f1− .Manages
Mandatory Constraints: Employee v ∃f1− .WorksFor
TopManager v ∃f1− .Manages
Project v ∃f2− .WorksFor
Project v ∃f2− .Manages
Exclusion Constraints: ∃f1− .WorksFor v A>2 u ¬∃f1.Manages
Subtyping Constraints: Manager v Employee u (AreaManager t TopeManager)
AreaManager v ¬TopeManager
a change we obtain a knowledge base as the one above, but with the defeasible inclusion
axiom Employee ∼< ∃f1− .WorksFor instead of the axiom Employee v ∃f1− .WorksFor.
Introducing such constraints, we introduce the forms of defeasible subsumptions
appropriate for modeling nonmonotonic reasoning. In particular:
– A subclass relation, as the ones in example 3, is translated into an inclusion axiom
C v D, and correspondingly we translate the defeasible connection C ; D into a
defeasible inclusion axiom C < ∼ D.
– Analogously, consider the strict form of the example 4. The mandatory participation
of the class B to the role AN is translated into the axiom B v ∃f1− .AN . If we use
the defeasible mandatory participation constraint, we simply translate the structure
using the defeasible inclusion <
∼, obtaining the axiom B < ∼ ∃f1− .AN .
9
Hence, from a ORM graph with defeasible constraints we obtain an ALCQI knowl-
edge base K = hT , Bi, where T is a standard ALCQI Tbox containing concept inclu-
sion axioms C v D, while the set B contains defeasible axioms of the form C < ∼ D.
Once we have our knowledge base K, we apply to it the procedure presented in the
previous section, in order to obtain the rational closure of the knowledge base.
Consistency. In ORM2, and in conceptual modeling languages in general, the notion
of consistency is slightly different from the classical form of logical consistency. That
is, generally from a logical point of view a knowledge base K is considered inconsistent
only if we can classically derive a contradiction from it; in DLs that corresponds to
saying that K |= > v ⊥, i.e. every concept in the knowledge base results empty. Instead,
dealing with conceptual modeling schemas we generally desire that our model satisfies a
stronger form of consistency constraint, that is, we want that none of the classes present
in the schema are forced to be empty.
Definition 2 (Strong consistency). A TBox T is strongly consistent if none of the
atomic concepts present in its axioms are forced to be empty, that is, if T 6|= > v ¬A
for every atomic concept A appearing in the inclusion axioms in T .
As seen above, the introduction of defeasible constraints into ORM2zero allows to
build schemas that in the standard notation would be considered inconsistent, but that,
once introducing the defeasible constraints, allow for an instantiation such that all the
classes result non-empty. Hence it is necessary to redefine the notion of consistency
check in order to deal with such situations.
Such a consistency check is not problematic, since we can rely on the ranking pro-
cedure presented above. Consider a TBox T obtained by an ORM2zero schema, and
indicate with C the set of all the atomic concepts used in T . It is sufficient to check the
exceptionality ranking of all the concepts in C with respect to T : if a concept C has
an exceptionality ranking r(C) = n, with 0 < n < ∞, then it represents an atypical
situation, an exception, but that is compatible with the information conveyed by the de-
feasible inclusion axioms. For example, in the above examples the penguins and the top
managers would be empty classes in the classical formalization, but using the defeasi-
ble approach they result exceptional classes in our schemas, and we can consider them
as non-empty classes while still considering the schema as consistent. The only case
in which a class has to be considered necessarily empty, is when it has ∞ as ranking
value: that means that, despite eliminating all the defeasible connections we can, such
a concept still results empty. Then, the notion of strong consistency for ORM2zero with
defeasible constraints is the following:
Definition 3 (Strong consistency with defeasible constraints). A knowledge base K =
hT , Bi is strongly consistent if none of the atomic concepts present in its axioms are
forced to be empty, that is, if r(A) 6= ∞ for every atomic concept A appearing in the
inclusion axioms in K.
Given a defeasible ORM2zero schema Σ, eliminate from it all the defeasible con-
straints (call Σstrict the resulting schema). From the procedures defined above, it is
immediate to see that if Σstrict is a strongly inconsistent ORM2zero schema, in the ‘clas-
sical’ sense, then Σ is a strongly inconsistent defeasible schema: simply, if the negation
of a concept is forced by the strict part of a schema, it will be necessarily forced at each
ranking level, resulting in a ranking value of ∞.
10
On the other hand, there can be also strongly inconsistent defeasible schemas in
which inconsistency depends not only on the strict part of the schema, but also on the
defeasible part. For example, the schema in figure 4 is inconsistent, since the class A
results to have a ranking value of ∞ (the schema declares that the class A is directly
connected to two incompatible concepts). Now, we can check the results of the defined
procedure in the examples presented.
Example 5. Consider example 3. From the translation of the defeasible form of the schema we
conclude that the axiom Penguin v NonFlyingObject has rank 1, while Bird v WingyObject
and Bird v FlyingObject have rank 0, that means that we end up with two default concepts:
d
– δ0 := {Penguin ⊃ NonFlyingObject, Bird ⊃ WingyObject, Bird ⊃ FlyingObject};
– δ1 := Penguin ⊃ NonFlyingObject
We can derive the same kind of conclusions as in example 2, and again we can see the limits of
the rational closure, since we cannot derive the desirable conclusion that Penguin|∼WingyObject.
Example 6. Consider the knowledge base obtained in the example 4. We have only a defeasible
inclusion axiom Employee ∼ < ∃f1− .WorksFor, and, since Employee does not turn out to be an
exceptional concept, we end up with a single default concept in B:
– δ0 := {Employee ⊃ ∃f1− .WorksFor};
Since TopManager is not consistent with all the strict information contained in the schema
plus δ0 , we cannot associate δ0 to TopManager and, despite we have the information that for
non-exceptional cases an employee works for a project, we are not forced to conclude that for the
exceptional class of the top managers.
6 Conclusions and further work
In this paper we have presented a way to implement a form of defeasible reasoning into
the ORM2 formalism. Exploiting the possibility of encoding ORM2zero , that represents a
big portion of the ORM2 language, into the description logic ALCQI on one hand, and
a procedure appropriate for modeling one of the main forms of nonmonotonic reasoning,
i.e. rational closure, into DLs on the other hand, we have defined two new constraints,
a defeasible subclass relation and a defeasible mandatory participation, that are appro-
priate for modeling defeasible information into ORM2, and that, once translated into
ALCQI, allow for the use of the procedures characterizing rational closure to reason
about the information contained into an ORM2zero schema.
The present proposal deals only with reasoning on the information contained in the
TBox obtained from an ORM2 schema, but, once we have done the rational closure of
Fig. 4. Inconsistent schema.
11
the TBox, we can think also of introducing an ABox, that is, the information about a
particular domain of individuals. A first proposal in such direction is in [4]. Actually we
still lack a complete semantic characterization of rational closure in DLs, but hopefully
we shall obtain soon such a result (a first step in such a direction is in [3]). Another future
step will be the implementation of nonmonotonic forms of reasoning that extend rational
closure, overcoming its inferential limits (see example 2), such as the lexicographic
closure [16] or the defeasible inheritance based approach [5].
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