=Paper=
{{Paper
|id=None
|storemode=property
|title=Structural Models in Axiomatic SysML
|pdfUrl=https://ceur-ws.org/Vol-745/paper_8.pdf
|volume=Vol-745
|dblpUrl=https://dblp.org/rec/conf/dlog/Graves11
}}
==Structural Models in Axiomatic SysML==
https://ceur-ws.org/Vol-745/paper_8.pdf
Modeling Structure in Description Logic
Henson Graves, Yvonne Bijan
Algos Associates,
2829 West Cantey Street,
Fort Worth, TX 76109 United States
Henson.graves@hotmail.com
Abstract. An overall goal of the INCOSE MBSE initiative is to provide SysML
with a formal semantics and to integrate reasoning services as part of system
engineering. UML class diagrams have been encoded as Knowledge Bases
(KB) within the Description Logic (DL), ALCQI. The encoding provides a
formal semantics for class diagrams which accords with the informal semantics.
The encoding applies to SysML which is a profile of UML. The SysML block
definition and internal block diagrams are not covered by the class diagram
encoding. These diagrams are essential for representing composite structure
such as manufactured products and molecular structures. The class diagram
encoding is extended to composite structure diagrams in the DL ALCQIbid. A
composite structure diagram describes structures in terms of part
decompositions and connections between objects. A SysML composite
structure diagram can be encoded in the language of OWL2, but is not an
OWL2 axiom set, as the diagrams contain property equations which violate the
regularity ordering constraints for complex property inclusions. Conditions are
given for an ALCQIbid KB which are sufficient to encode a SysML composite
structure diagram. Further conditions are given for a KB, called a template,
which ensure that within an interpretation all realizations of the composite
structure have the same graph structure.
Keywords: Description Logic, Ontology, OWL, SysML, UML, structural
modeling, molecular chemistry, human anatomy.
1 Introduction
Many engineering tasks involve reasoning on a description (a model in engineering
terminology) to determine consistency and to derive implicit knowledge. An overall
goal of the INCOSE MBSE initiative [4] is to provide the system engineering
modeling language SysML [10], a dialect of UML [11] with a formal semantics and
to integrate reasoning services as part of system engineering. SysML lacks a formal
logic-based semantics, but has a well-developed informal semantics. For reasoning to
give correct results, the formal semantics must be in accord with the informal
semantics of SysML. To provide a formal semantics, one may axiomatize SysML
directly [7] or encode SysML in a language which has a formal semantics such as
OWL2 [12]. The OWL2 semantics is based on the Description Logic (DL)[1],
SHOIQ [8].
UML class diagrams have been encoded as a Knowledge Base (KB) within the
DL, ALCQI [2], a sublogic of SHOIQ. In this encoding, UML classes are encoded as
concepts and UML associations are encoded as roles; to encode the additional
information contained in class diagrams, other KB assertions are needed. The result
is that an encoding of a UML class diagram is as a KB. The encoding provides a
formal semantics for class diagrams which conforms to their informal semantics; the
encoding is further validated by comparison of first order logic (FOL) axiomatizations
of the UML constructions with the FOL representation of description logic. A
consequence of the encoding for integration with reasoning is that a class diagram
(class model) corresponds to a knowledge base within a DL. The results of DL
consistency checking and derived classes and class inclusions can be reinterpreted
within UML.
The DL encoding for UML and its results carry over to SysML. In SysML, blocks
are a stereotype of class and SysML uses associations as does UML. SysML is well
suited for representing descriptions of composite structure [5]. The SysML diagrams
(models) used to represent composite structure are not fully covered by the UML
encoding. A composite structure consists of objects and part objects linked by
connectors. A structural description is a collection of classes and properties which
describe a structure. A structure which satisfies the description is called a realization
of the structure. A composite structure is represented in SysML with a Block
Definition Diagram (BDD) and an Internal Block Diagram (IBD). A variety of
specializations of associations are used to represent part properties and structural
connections. Both part properties and connections are binary properties. A BDD
describes a part decomposition structure. An IBD is a BDD with connection
properties and property equations. An IBD can be used for representing structures
which have, for example, multiple objects of the same class, which play different
roles in the description. For example, an automobile description may specify four
wheels with two front wheels which are driven by the engine and two rear wheels
which are not driven. A SysML IBD model of the human heart accords well with the
informal semantics and elucidates the distinction between the different kinds of
properties (parts and connections) used to describe a heart.
Finding an appropriate Description Logic to represent the class of composite
structure diagrams which is sufficiently expressive and for which reasoning is
decidable and computationally tractable is challenging. A SysML IBD can be
encoded in the language of SROIQ, but in the direct encoding it is not an OWL2
axiom set, as the connection property equations of an IBD violate the regularity
ordering constraints on SROIQ axioms [8]. While it is possible to represent these
diagrams within SROIQ with a description graph extension [9], there are questions
regarding whether the description graphs correctly capture the intended semantics.
For a DG extension of OWL2, the FOL suggested semantics for the human heart
example in [9] is different from the DL semantics of a SysML IBD which appears to
capture the informal semantics faithfully.
The role equations needed to represent the constructions in a SysML IBD are very
restricted, even though they do not satisfy the regularity conditions of OWL2. The
roles which represent part properties are all atomic with specified domain and range
classes which are also atomic. The conditions satisfied by the part properties of a
SysML IBD ensure that the part role paths [3] are finite and are unique. For each part
role path, a new atom can be introduced to represent the path. Using these atomic path
roles, simple role hierarchy assertions are sufficient to represent the equalities found
in an IBD.
The UML class diagram encoding and its extension for conceptual modeling for
data integration [3] is used to encode SysML Block Definition and Internal Block
Diagrams using the description logic ALCQIbid . A part structure for a KB is defined
which encodes the essential features of a BDD. Conditions are given on a KB to
encode the property equation features of an IBD. Additional meta conditions are
given for an IBD KB, called a template, which ensure that within an interpretation, all
realizations of the KB have the same graph structure. The conditions for a template
are easily checked. A template is illustrated with a SysML model of the water
molecule. For a template, results computed from the structure of the KB are valid in
any realization. For example the weight of a structure can be computed from the IBD
as all realizations will have the same number of parts.
2 The Description Logic ALCQIbid
The specific description logic used to encode SysML Internal Block Diagrams is
ALCQIbid [3]. ALCQIbid is an expressive DL that extends the basic DL language AL
(attributive language) with negation of arbitrary concepts (indicated by the letter C),
qualified number restrictions (indicated by the letter Q), inverse of roles (indicated by
the letter I), boolean combinations of roles (indicated by the letter b), and
identification assertions (indicated by the subscript id). Concepts and roles in
ALCQIbid are formed according to the following syntactic rules [3]
C, C'→A| ¬C | C ∩ C' | C U C' | R.C | R.C | ≥n Q.C | ≤n Q.C (1)
R,R'→ P | P- | R ∩ R' | R U R' | R \ R' (2)
where A denotes an atomic concept, P an atomic role, P− the inverse of an atomic
role, C an arbitrary concept, and R, R’ arbitrary roles. Furthermore, ¬C, C ∩ C’, C U
C', R.C, and R.C denote negation of concepts, concept intersection, concept union,
value restriction, and qualified existential quantification on roles, respectively. We
then use Q to denote basic roles, which are those roles that may occur in expressions
of the form ≥ n Q.C and ≤ n Q.C. A basic role can be an atomic role or its inverse, or
a role obtained combining basic roles through set theoretic operators, i.e., intersection
(“∩”), union (“U”), and difference(“\”). W.l.o.g., we assume difference applied only
to atomic roles and their inverses.
Abbreviations are introduced for terms and assertions. Thing denotes the top
concept which can be defined as C U¬C for a concept C, and Nothing the bottom
concept. The concept kQ.C is an abbreviation of ≤kQ.C ∩ ≥kQ.C. The empty denotes
the role p\p for a role p. The notation (funct p) is used for the assertion that p is
functional, i.e., as an abbreviation for Thing 1 p.Thing. The notation p:(A,B) is an
abbreviation for the assertions
A p.B and Bp-.A (3)
which capture the property that A is the domain and B is the range of p. For a
functional role with p:(A,B), we have A 1p.B. For a role p:(A,B) that is functional,
we use the notation p:(A,B)[1]. Two roles p and q are disjoint
p ┴ q IFF p∩q=empty (4)
A path is given by the production rule
1, 2S| D? | 1. 2 (5)
where S denotes an atomic role or the inverse of an atomic role, D denotes a concept,
and π1.π2 denotes the composition of paths π1 and π2. The expression D? is called a
test role, it denotes the identity relation on instances of the concept D. If is a path,
the length of , denoted length(), is 0 if has the form C?, is 1 if has the form S,
and is length(1) + length(2) if has the form 1.2.
An ALCQIbid knowledge base (KB) is a pair ‹T, A›, where T is a TBox and A is
an ABox. A TBox is a finite set of assertions of the form C ≤ C' with C and C'
arbitrary concepts, or of the form R ≤ R' with arbitrary roles R and R', or an
identification constraint. An identification constraint is an assertion of the form (id C
1, . . . , n) where C is a concept, n ≥ 1, and π1, . . . , n are paths (called the
components of the identifier) such that length(i) ≥ 1 for all i {1, . . . , n} and
length(i) = 1 for at least one i {1, . . . , n}. Intuitively, an identification constraint
asserts that for any two different instances o, o' of C, there is at least one i such that o
and o' differ in the set of their i-fillers. An ABox is a finite set of membership
assertions of the form A(a), P(a, b), and a ≠ b, with A and P respectively an atomic
concept and an atomic role occurring in T, and a, b constants. The condition for role
inclusions is weaker than the standard condition for a KB in ALCQIbid.
The semantics of ALCQIbid concepts and roles is given in terms of interpretations,
where an interpretation is defined as a correspondence of the KB concepts and roles
[3] with classes and properties in a domain for which all of the KB assertions are
satisfied. The semantics of a path is defined in terms of the reverse of the
composition of the roles occurring in the path. This device allows us to express well-
formed path equations as role assertions within an ALCQIbid KB.
The semantics of an ALCQIbid KB K = is the set of models of K, i.e., the
set of interpretations satisfying all assertions in T and A. As noted in [3], checking
whether an assertion holds in every model of a KB, is decidable in deterministic
exponential time.
While ALCQIbid works for encoding SysML block diagrams it seems likely that
some new variant could be devised which could be tailored more precisely for
representing composite structure models.
3 Encoding SysML Block diagrams in a DL
We review the principles of encoding class diagrams established in [2] as applied to
SysML block diagrams. We use a simple illustration of a water molecule model to
show how role equations naturally occur in a composite structure diagram. For the
water example we show that all realizations have the same structure. The next section
will generalize the concept of a KB which abstracts the properties of a composite
structure and show that a kind of KB called a template enjoyes the properties that all
of its realizations are isomorphic.
The SysML language uses blocks which are classes and associations with several
predefined kinds of specializations. The molecular unit of SysML is called a model.
A SysML model is a collection of declarations which introduce constants of a
signature and specify typing relations. A SysML model may contain subclasses and
limited kinds of role assertions and may be composed and presented using multiple
visual diagrams. However, all of the diagrams that constitute a model use the same
block and property symbols.
A class diagram in UML is a restricted kind of SysML model. The encoding of
UML class diagrams [2] carries over to SysML which is a UML profile developed for
systems engineering. The Description Logic ALCQI is used to provide the encoding.
This encoding accords with the informal semantics of UML. Classes (SysML Blocks)
and associations are translated into DL concepts and roles [2]. The translation of a
class diagram is as a role assertion. However, SysML models are not covered by the
encoding in [2]. In particular, a SysML Block Definition Diagram and an Internal
Block Diagram are not covered. An undirected association p is identified with a role
p. The diagram of boxes labeled A and B connected by a line becomes the assertion
An aggregation property p from A to B with cardinality restriction 1 is
represented in DL this becomes A ≤ (p). A property p with p:(A,B) is mandatory if
A ≤ k p. B. for an atomic functional role p with domain A and range B we use the
abbreviation p:(A,B)[1].
This encoding of a SysML model as a KB is illustrated with a SysML water
molecule model. The water molecule is represented as a SysML model using two
kinds of diagrams, a Block Definition Diagram (BDD) to represent the decomposition
structure and an Internal Block Diagram (IBD) to represent relationships among the
parts within the structure. The language elements in both diagrams are part of the
same SysML model. In Figure 1, the top half shows the decomposition structure for
water. The BDD shows that water has three part properties whose range classes are
oxygen and hydrogen. The shared Association (open diamond headed arrow) is a part
property in SysML. There are two kinds of part properties in SysML. The shared
Association property is used because the atoms can be a part of any molecule. The
two arrows pointing at hydrogen mean that there are two parts of type hydrogen
within water. The diamond arrow pointing to oxygen shows that there is one part of
type oxygen within a water molecule. The numbers on the arrows in this diagram
represent the cardinality restriction on the number of parts that a water molecule can
have. In this case, the numbers are all 1, which says that an individual water molecule
has exactly one oxygen and two hydrogen atoms as parts.
Fig. 1. SysML model for water molecule
The bottom diagram, called an Internal Block Diagram, shows the bonding
relationships between the part properties. The arrows from the BDD are represented
as rectangles with dashed lines. In this diagram, the rectangles are not blocks; they
represent part properties of Water. The rectangle is labeled with the name of the part
property and the range type of the property, as well as the cardinality restriction of the
properties. The diagram shows the oxygen part has a covalent bond with each of the
hydrogen parts. The diagram title box signifies that the IBD is within the scope of
water.
The informal semantics of the water model is that any realization of a water
molecule has exactly three atoms: two hydrogen atoms and one oxygen atom. Further,
we expect that the covalent bonds from oxygen atom are connected to distinct
hydrogen atoms. This fact will follow from the declaring the covalentBond property
to be functional and declaring that the hasHydrogenAtom1 and hasHydrogenAtom2
are disjoint. Informally, a realization of a water molecule in this theory is a tree with a
root w1 corresponding to the water molecule and three other nodes, an oxygen atom
o1, and two hydrogen atoms h1, h2. The tree has edges { , ,
, , }. The first three edges correspond to the part properties
and the last two correspond to the bond properties.
For a KB used to encode a SysML composite diagram a realization of a KB is a
collection of ABox assertions of the form objects Cj(ai) where the Cj are atomic and
for any atomic concept C in the KB there is at least one ai with C(ai). Also, for any
atomic role p in the KB there is a pair with p(ai,aj). There may be multiple ai
with C(ai). A realization is an internal model of the KB. In general, a realization may
not be finite. With the restrictions that will be used on a KB to encode an IBD, one
can construct realizations of the KB by adding individuals. A template KB has finite
realizations. Also, a model may contain multiple realizations.
In the first order logic representation of a DL we replace an existential assertion
p:(A,B)[1] with a Skolem function and use the symbol p for the Skolem function as
well as the role. We use the notation a.p for the value of the Skolem function p. This
notation allows us to write p(a, p(a)) as p(a,a.p). More generally for p1 and p2
functional atomic roles then from the composition semantics for roles one has
a.(p1.p2) = (a.p1).p2.
The KB encoding the SysML water molecule model has as atomic roles, Water,
Oxygen, and Hydrogen and as atomic roles, hasOxygen, hasHydrogen1,
hasHydrogen2, covalentBond1, covalentBond2. Using the abbreviations the KB
contains the assertions:
hasOxygen:(Water, Oxygen)[1] (6)
hasHydrogen1:(Water, Hydrogen)[1] (7)
hasHydrogen2:(Water, Hydrogen)[1] (8)
covalentBond1:(Oxygen, Hydrogen)[1] (9)
covalentBond2:(Oxygen, Hydrogen)[1] (10)
the equational role equations
hasOxygen.covalentBond1 = hasHydrogen1 (11)
hasOxygen.covalentBond2 = hasHydrogen2 (12)
and the disjointness assertions
Oxygen ┴ Hydrogen (13)
hasHydrogen1 ┴ hasHydrogen2 (14)
covalentBond1 ┴ covalentBond2 (15)
It is easy to show that any realization of the water KB has the same structure. For any
ABox w with w.Water, we iterate the constructions to obtain the set
{w, w.hasOxygen, w.hasHydrogen1, w.hasHydrogen,
w. hasOxygen.covalentBond1, w.hasOxygen.covalentBond2 }.
However,
w. hasHydrogenAtom1 ≠ w. hasHydrogenAtom2 (16)
by axiom (12). By axiom (9)
w. hasOxygen.covalentBond2= w. hasOxygen.covalentBond2 (17)
The structure only contains the root and part instances. We can verify that the role
instance relations hold. This KB implies that any realization of Water has the
expected component parts with the expected connections between them. The next task
is to identify the properties of part roles which enable these arguments to be
generalized.
4 Block Definition Diagrams and Internal Block Diagrams
The KBs which are used to encode SysML BDDs and IBDs are described below. An
abstract Block Definition Diagram (ABDD) is a KB with a subset of the KB signature
pi : i{1,…n} called part roles. Part roles satisfy the constraints that each pi is
declared with a domain and range type with a numeric multiplicity, the domain and
range concepts of the part roles are in the KB signature and are atomic. There may be
multiple part properties with the same domain and range types. For the following
discussion, we restrict part properties to be functional. We use the abbreviation
p:Part(A,B) for a part role p with p(A,B). The concepts that occur as the domain or
range of a part role are called Part concepts. A part concept which is not the range
concept of any part property is a root. We assume that the part class has a root and
that it is unique. A part path is a well-formed composition of part properties p1. p2
.…pn were the range(pi) = domain (pi+1) for all i. A part path is called acyclic if
domain(pi) ≠ range(pi) for any pi in the path.
(P0) Root(A), for some atomic class A and the class is unique.
(P1) If p is a part path then p is acyclic
(P2) If p:Part(A,B) and p2:Part(C,B) then p1 ┴ p2
(P3) PartClass(A) IFF Root(A) or p:Part(B,A) and PartClass(B)
(P0) identifies a concept as the root. (P1) ensures that part paths do not contain
multiple occurrences of a part properly and so have finite length. (P2) states that any
two part roles with the same range are disjoint. The (P3) implies that all part concepts
are connected the root by a part path. The meta-properties (P0) through (P3) are easily
in a KB.
For an ABDD, the directed graph whose nodes are the root together with the
expressions p:A, for a part property p with A = Range(p) and whose edges are the part
roles is a tree. As there may be multiple parts with the same range, class labeling the
class with the part role using the expressions of the form p:A makes the nodes
distinct. Each part concept is reachable by a part path p1…pn where pn = p from the
root by (P4). For any two property paths p1…pn and q1…qk which terminate at the
node p:A, then the domain of pn and qk are equal. By (P2), the part roles pn and qk
are disjoint and so the two paths cannot be equal. Thus, the part property path is
unique. The ABBD is used to encode a SysML BDD. The conditions used to define
an ABDD are enforced in a BDD.
An Abstract IBD (AIBD) is an ABDD together with a finite set of function role
(connections) whose domain and range are part concepts, and a set of path equations
of the form
p1…pn = q1…qk.c (18)
where the pi and qj are part paths and c is in {c1,..,ck}. The connection roles encode
the arrows between the boxes of an IBD. Conversely an AIBD can be displayed as a
graph. A new atomic role p.c is introduced for any part path followed by a connection
role. Note that p1…pn:(A,B) where domain(p1) = A and range(pn) = B. We extend
this notation to p1….pn.c for a connection role c.
A path of atomic roles p1,…pn is well-formed if range(pi) = domain(pi+1) for all i
{1, . . . , n}. For each well-formed path, , of atomic roles, we introduce a new role
atom . For any path of length 1, the new role is identified with the atomic role that
formed the path. As there is a finite number of part paths, we define a new atomic role
for each part path p1….pn. Recall that when the atoms in a path are functional, we
write a.p for the unique individual b with p(a,b). This notation simplifies the
application of a path p applied to a. We also use the notation p:Path(A,B) for a path
with domain A and range B.
While the DL ALCQIbid does not permit composition directly in the role inclusion
assertions, we simulate composition with the atomic path roles. For any connection
equation p1…pn = q1…qk.c in the IBD, we replace it with role inclusion assertion
q1…qk.c ≤ p1…pn. However, for any a,b , q.c(a,b) implies p(a,b). However, if a.A
then as p is functional, a.p = a.c.b. Thus, q.c = p. So the inequalities in an AIBD are
actually equalities. A template is a AIBD where
(P5) p1:Part(A,B) and p2:Part(A,C) then p ┴ q or p = q
The template axiom says that no parts can be reused in a part decomposition. This
statement can be made precise in the first order logic theory of the KB.
To prove properties about the models of an AIBD KB we use the full first order
representation of the DL. In the theory generated by the KB existential assertions of
the form p:(A,B)[1] are replaced by a first order Skolem function. Properties that hold
in this theory will hold in any model of the KB. Note that the part path roles become
functions. Thus, the notation a.p1..pn is meaningful and we have the associativity law
a.(p.q) = (a.p).q.
Definition. For a template KB with root A, a:A, and t:B for a part class B, let
Partof(a,t) IFF t = a or a.p1….pn (19)
for a part path.
Lemma. For a template with root A, and a:A: If t:B for a part class B and t is a part of
a, then the part decomposition is unique.
If t is a part of a, then t has a decomposition of the form t = a.p1….pk for some
p1,…,pk. If t = a.p and t = a.q, for two part properties, then by the template property
p disjoint q or they are equal, and so a.p = a.q. The argument is repeated for the
successive individuals (a.p1).p2…pn. With the first order logic of the KB extended
with an abstraction construction which allows terms of the form
{ t : P(t)} (20)
constructed from a predicate where the predicate is restricted to equalities with
Boolean connectives, then we can define the notion of a realization of a template
within the extended theory of the axiom set. The axioms for the abstraction
construction include
P(a) IFF a:{ b : P(b)} (21)
with usual rules for variables. Using the extended logic, a realization of a template is
an abstraction type G = {t : Partof(a,t) } for a:Root. From the axioms for the part
property declarations, a realization of a root instance has a unique part decomposition.
A tree structure can be defined with the individuals in G as the nodes and for
a part property. Connection edges can be added similarly. A graph isomorphism can
be inductively defined between any two realizations.
Theorem. For a template, the instances of the type G = {t : Partof(a,t)} for a:Root are
the nodes of a tree with root a. The edges where range(t) = domain(p). The
correspondence defined by mapping a to the root and a node of the form t.p to
p:Range(p) corresponds the nodes of the parts structure with the nodes of the BDD
together with the mapping of an edge to the edge p in the BDD defines an
isomorphism of the parts structure with the BDD. Any parts tree has the same number
of parts.
The axioms given do not prohibit a structure from sharing individuals with another
structure. This property can be added with the axiom:
(P6) p:Part(A,B) implies p.p*=id(A)
An interpretation of an axiomatic SysML theory is a mapping of the individuals,
pairs, classes, and properties, and other types which preserves the sort structure, the
logical axioms, and the declarations. In particular, classes are mapped to subclasses of
the mapping of Thing and individuals are instances of Thing. In any valid
interpretation of the theory of a model, the unique decomposition will hold.
5 Conclusion
The encoding of a UML class diagram as an ALCQI KB gives an encoding of Class
diagrams into OWL2. This encoding is extended to an encoding of a SysML Internal
Block Diagram as a KB within the OWL2 language, but not as an OWL2 KB. Each
atomic property in the Block Diagram is an atomic role and the well-formed property
paths are finite. The encoding correctly captures the part decomposition which
ensures that the models are tree like. SysML model development tools enforce the
axioms that define a part structure. Conversely, the correspondence between the block
diagrams and the DL language constructions provides a graphical syntax for DL.
SysML dos not have individuals, i.e., ABoxes. The Encoding makes clear how
individuals can be added to SysML. With the encoding all derivations of
inconsistency and concept inclusions can be exported back into SysML.
It is easily to check whether the additional axioms for a template are present.
These axioms correspond to manufacturing assumptions that ensure implementations
of a design have the same parts and connection structure. For example, water is a
subclass of molecules which have an oxygen part. However, Oxygen is not a subclass
of the things bonded to hydrogen, only the oxygen molecules which are parts of water
have this property. The use of a template KB enables the development of SysML
models for which all realizations are isomorphic. This is very useful as computations
of the model hold for all of its realizations. It seems likely that graph defined for an
abstract IBD, is a Description Graph in the sense of [9].
The axioms given for the water model provide only structural information and are
incomplete in terms of constraints on the bonds needed to determine a 3D
visualization of a water molecule and do not address the dynamic behavior of water
such as how it changes when it freezes. Much more complete axiomatic models of
water can be given which address these properties. These SysML models require
further extensions to DL to be addressed.
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