A Formal Descriptive Semantics of UMLLijun Shan Dept of Computer ScienceNational University of Defense Technology,Changsha, China

Hong ZhuDepartment of Computing and ElectronicsOxford Brookes University, Oxford, UK

Outline

What is descriptive semantics? The concept and motivation

How to specify descriptive semantics? The formal framework Mappings from diagrams to first order logic (FOPL) Application to UML class, interaction and state

machine diagrams Application of descriptive semantics to

consistency checking of UML modelsConclusion

What is the meaning of UML diagrams? The system has two classes called

Member and Book, (There are two types of objects called member and book)

There is an association between them, which is called Borrow (Members can borrow books)

The upper limit of borrowing is 10 (Each member can only borrow up to 10 books at any time)

The upper limit for a book to be borrowed is 1 (Each book can only be borrowed by at most 1 member at any time)

Member

Book

Borrow

Library system

0..10

0..1

Instances of the class Member

Instances of the class Book

Semantics of modelling languages

‘A model is a set of statements about some system under study’ [Seidwitz, 2003]

The semantics of a modelling language is the satisfaction relationship between a model and a system.

s m

Software model, and models in all scientific disciplines

A system in the subject domain D, i.e. universe of systems that the language is modelling.

A well-formed model in the modelling language.

s is an instance of the model m, i.e. s satisfies the statements of the model m

Two types of semantics

Descriptive semantics Describe the system based on

a set of basic concepts, such as class, association multiplicity upper bound, etc.

Example: The system has two classes

called Member and Book

Functional semantics Define how the system funct

ions at run time. Example:

There are two types of objects called members and books.

Existing work on UML semantics often occurs as a combination of them. They are at ‘object level’.

Example: x.(Member(x) ||{y | Book(y) Borrows(x,y)}|| 10)

Class(Member), Class(Book), Asssociation(Borrows), …

Difficulties in the formalisation of UML Interpretation in different subject domains:

One model can be interpreted in several different subject domains Library system model:

• When interpreted on the subject domain of real world systems, the library of Oxford Brookes University is an instance

• When interpreted on the computer software, …

Extension with new concepts: New basic concepts and constructs can be introduced through

extension facilities, such as introduce new stereo types in profile definitions

Uses in different context: One model may have different meanings in different context

Example: Which of the following programs is the

correct semantics of the model on the right? There are exactly three different classes such that

…; There are at least three different classes such that

…; Which of the following programs can be

considered as satisfying the model?

Member

Staff Student

class Member{ … }class Staff extends Member{…}Class Student extends Member{…}

class MScStudent extends Student{…}

class Member { public enum MemberType {

Staff, Student }public MemberType

TypeOfMember;…

}

UML gives no definition on this issue!

Overview of our approach

This is consistent with the theory of institution proposed by Goguen and Burstall (1992) for formal specification languages.

The formal frameworkDefinition 1. (Semantics of a modelling language)

A formal semantic definition of a modelling language consists of the following elements.

A signature Sig, which defines a formal logic system; A set AxmD of axioms about the descriptive semantics, which is

in the formal logic system defined by Sig; A set AxmF of axioms about the functional semantics, which is

also in the formal logic systems defined by Sig; A mapping T from models to a set of formulas in the formal

logic system defined by Sig. The formulas are the statements for the descriptive semantics of the model;

A mapping H from models to a set of formulas in the formal logic system defined by Sig. The formulas represent the hypothesis about the context in which the descriptive semantics is interpreted.

Definition 2. (Semantics of a model)

Given a semantics definition of a modelling language as in Definition 1, the semantics of a model M under the hypothesis H, written SemH(M), is defined as follows.

SemH(M) = AxmD AxmF T(M) H(M)

where T(M) and H(M) are the sets of statements obtained by applying the semantic mappings T and H to model M, respectively. The descriptive semantics of a model M under the hypothesis H, written DesSemH(M), is defined as follows.

DesSemH(M) = AxmD T(M) H (M)

Satisfaction relation

Definition 4. (Subject domain)

A subject domain Dom of signature Sig with an interpretation Eva is a triple <D, Sig, Eva>, where D is a collection of systems on which the formulas of the logic system defined by Sig can be evaluated according to a specific evaluation rule Eva. The value of a formula f evaluated according to the rule Eva in the context of system sD, written as Eva(f, s), is called the interpretation of the formula f in s. We write s|=Evaf, if a formula f is evaluated to true in a system sD, i.e. s|=Evaf iff Eva(f, s)= true. Definition 5. (Satisfaction of a model)

Let Sig be a given signature and Dom a subject domain of Sig. A system s in D satisfies a model M according to a semantic definition SemH(M) if s|= SemH(M), i.e. for all formulas f in SemH(M), s|=f.

Signature MappingThe signature mapping S from a metamodel M to a signature

= S (M) is defined by a set of signature rules so that statements representing the descriptive semantics of models are -sentences of first order predicates.

Signature RulesS1. For each metaclass named MC in the metamodel, we define a unary atomic predicate MC(x). S2. For each metaassociation from a metaclass X to a metaclass Y in a metamodel, if MA is the association end on the Y side of the metaassociation, a binary predicates MA(x, y) is defined. S3. For each metaattribute named MAttr of type MT in a metaclass MC in a metamodel, a unary function MAttr(x) is defined with domain MC and range MT. S4. For each enumeration value EV given in an enumeration metaclass ME in a metamodel, a constant EV is defined.

A Simplified Metamodel of Class Diagram

MultiplicityElement

ValueSpecification

+upperValue+lowerValue

TypedElement

Type+type

Classifier

+isAbstract: BooleanGeneralisation

+general

DirectedRelationshipFeature

+isStatic: Boolean

StructuralFeature BehaviouralFeature

Parameter

+direction: ParameterDirectionKind

+ownedParameter

Operation ClassProperty

+aggregation: AggregationKind

+ownedAttribute0..1*

Association

+memberEnd2..*

+ownedOperation

0..1*

Relationship

ParameterDirectionKind<<enumeration>>

+in+out+inout+return

AggregationKind<<enumeration>>

+none+shared+composite

Boolean<<enumeration>>

+t+f

DataType

Signal

+specific

Interface

+ownedOperation

+ownedAttribute

NamedElement

VisibilityKind<<enumeration>>

+public+private+protected+package

+associateTo

Example:

Signature elements: unary predicates :

Class(x), Classifier(x), Generalisation(x)

binary predicates: general(x, y) specific(x, y)

Classifier GeneralisationClass

+general

+specific

PersonWoman

Metamodel

Translation mappingTranslation mapping is defined by a set of translation rules that

generate formulas in the signature from a UML model M.

Translation Rules T1: Classification of elements. For each identifier id of concrete type MC, a formula in the form of MC(id) is generated. T2: Properties of elements. For each element a in the model and every applicable function MAttr that represents a metaattribute MAttr, a formula in the form of MAttr(a)=v is generated, where v is a’s value on the property. T3: Relationships between elements.

For each pair (e1, e2) of elements related by relationship R, a formula in the form of R(e1, e2) is generated to specify the relationship by applying binary predicate R(x1, x2).

Example:

Signature elements: unary predicates : Class(x), Classifier(x), Generalisation(x) binary predicates general(x, y) and specific(x, y)

Classifier GeneralisationClass

+general

+specific

PersonWoman

Metamodel

Model

Statements of the model: Class(Woman), Class(Person), Generalisation(wp) specific(wp, Woman), general(wp, Person)

Classifier GeneralisationClass

+general

+specific

PersonWoman

Interpretation in different contexts The context in which a model is interpreted can be specified as

hypothesis. A hypothesis can be defined as a rule that maps from a model to

formulas in the signature.

Hypothesis Rules H1: Distinguishability of elements.

A hypothesis that the elements of type MC in the model are all different from each other can be generated as formulas in the form of ei ej, for ij {1,2,…,k}. H2: Completeness of elements.

A hypothesis on the completeness of elements of type MC can be generated as a formula in the following form.

x. MC(x) (x = e1) (x = e2) … (x = ek) H3: Completeness of relations.

A hypothesis on the completeness of relation R in the model can be generated as a formula in the following form.

x1,x2.R(x1,x2)((x1=e1,1)(x2=e1,2))((x1= e2,1) (x2= e2,2))

… ((x1= en,1) (x2= en,2))

Axiom mappingIt is defined by a set of rules that maps metamodel to -sentence

s as axioms of the models. These rules represents the semantics of OO concepts applied to meta-modelling.

Axiom rules:A1: Completeness of classification.

Let MC1, MC2, …, MCn be the set of concrete metaclasses in a metamodel. We have an axiom x. MC1(x) MC2(x) … MCn(x)A2: Disjointness of classification.

Let MC1, MC2, …, MCn be the set of concrete metaclasses in a metamodel. For each pair of different concrete metaclasses MCi and MCj, ij, we have an axiom x. MCi(x) ¬ MCj(x). A3: Logical implication of inheritance.

For a generalisation relation from metaclass MA to MB in a metamodel, we have an axiom x. MA(x) MB(x).

Strict metamodelling principle Axiom rules also contain ‘hypothesis’ on the uses of class diagra

ms as metamodels, such as the strict meta-modelling principle, which is to ensure that a metamodel is a well-defined abstract syntax of modelling language.

Strict Metamodelling:

In an n-level modelling architecture M0, M1, …, Mn, every element of an Mm-level model must be an instance-of exactly one element of an Mm+1-level model, for all 0 m < n-1, and any relationship other than the instance-of relationship between two elements X and Y implies that level(X) = level(Y).

Axiom Rules (Continue)

A4: Completeness of specialisations.

Let MA be a metaclass in a metamodel and MB1, MB2, …, MBk be the set of metaclasses specialising MA. We have an axiom x. MA(x) MB1(x) MB2(x) … MBk(x).

A5: Types of parameters of predicates.

For each binary predicate MA(x, y) derived from an association from metaclass MC1 to MC2 in a metamodel, we have an axiom x, y. MA(x, y) MC1(x) MC2(y).

A6: Domain and range of functions.

For each function MAttr(x) derived from a metaattribute MAttr of type MT in a metaclass MC, we have an axiom x,y. MC(x) (MAttr(x) = y) MT(y).

A7: Multiplicity of binary predicate. For each binary predicate MA(x, y) derived from an association from metaclass MC1 to MC2 in a metamodel, let Mul be the multicity value specified on the association end MA, we have axioms in the following form.If Mul = 0..1: x, y, z. MC1(x) MA(x, y) MA(x, z) (y = z)If Mul = 1 or unspecified: x. MC1(x) y. MA(x, y) and

x, y, z. MC1(x) MA(x, y) MA(x, z) (y = z)If Mul = 1..*: x. MC1(x) y. MA(x, y)If Mul = 2..*: x. MC1(x) y, z. MA(x, y) MA(x, z) (y z)If Mul = 0..2: x, y, z, u. MC1(x) MA(x, y) MA(x, z) MA(x, u)

(y = z) (y = u) (u = z)

Axiom Rules (continue)A8: Multiplicity of function.

For each function MAttr(x) derived from a metaattribute MAttr of type MT in a metaclass MC, let Mul be the multicity value of the metaattribute MAttr, we have axioms:If Mul = 0..1: x, y, z. MC(x) (MAttr(x) = y) (MAttr(x) = z) -> (y = z)If Mul = 1: x. MC(x) -> y. (MAttr(x) = y) and x, y, z. MC(x) (MAttr(x) = y) (MAttr(x) = z) -> (y = z)If Mul = 1..*: x. MC(x) -> y. (MAttr(x) = y)A9: Distinguishability of the literal constants.

For each pair of different literal values a and b of an enumeration type, we have an axiom a b. A10: Type of the literal constants.

For each enumeration value a defined in an enumeration metaclass ME, we have an axiom in the form of ME(a) stating that the type of a is ME. A11: Completeness of the enumeration.

An enumeration type only contains the listed literal constants as its values, hence for each enumeration metaclass ME with literal values a1, a2, …, ak, we have an axiom in the form ofx. ME(x) -> (x = a1) (x = a2) … (x = ak). Axiom Rule 12: Well-formedness rules.

For each WFR formally specified in OCL, we have a corresponding axiom in the first order language.

Semantics of Interaction and State Machine Same signature, axiom and formula mappings are applied to the

meta-models of UML state machine and interaction diagrams. Additional Axiom Rules for inter-metamodel connectionsAxiom Rules

A10: Cross metamodel association and inheritance.

For each cross metamodel inheritance from metaclass MA to external metaclass MB, we have an axiom in the form of x. MA(x) -> MB(x).

A2’: Completeness of specialisations across metamodels.

Let MA be a metaclass depicted in two metamodels MM1 and MM2. Let metaclasses MB1, MB2, …, MBk be the set of metaclasses that specialise MA in metamodel MM1, and MC1, MC2, …, MCp be the set of metaclasses that specialise MA in metamodel MM2. We have the following axiom when a model is defined by MM1 and MM2.

x. MA(x) -> MB1(x) … MBk(x) MC1(x) … MCp(x)

A Simplified Metamodel of Interaction Diagram

Interaction

Lifeline Message

+lifeline +message

ConnectableElement

+represents

MessageEvent

SendOperationEvent SendSignalEventOperation (from Kernel) Signal (from Kernel)

+operation +signal

+event

+sender

+receiver+after

Behaviour BehaviouralFeature (from Kernel)Classifier(from Kernel)

+specification

+context

TypedElement(from kernel)

A Simplified Metamodel of State Machine

StateMachine

ProtocolStateMachine Vertex

StatePseudoState

+kind: PseudostateKind

Transition

+vertex

+transition

+source

+target

Trigger Constraint

+trigger +guard

ProtocolConformance

DirectedRelationship(from Kernel)

+generalMachine+specificMachine

Behaviour (from Interaction)

+exit +doActivity+entry

+effect

PseudostateKind<<enueration>>

+initial+final+deepHistory+shallowHistory+join+fork+junction+choice

BehaviourStateMachine

StateBehaviour

The Tool LAMBDESLAMBDES stands for Logic Analyser of Model/Metamodel Based on Descriptive Semantics

User InterfaceLAMBDES

Logic system for metamodel

Modelling tool StarUML

Axiom Generator

Metamodel

Metamodel in XMI

Theorem prover SPASS

Inference result

Statements

Model

Model in XMI

Signature Generator

Formula Generator

Signature Axioms

Proof Goal

Hypothesis

Modelling Context

Hypothesis Generator

Domain Generator

Auxiliary constants &

formulas

Conjecture Generator

Conjecture

Logic system for model

Design Pattern Specification

Design Pattern Spec

Repository

Applications of Descriptive Semantics

Consistency checking of UML models Validation of consistency constraints for UML models Consistency checking of UML meta-models Conformance checking of designs to design patterns Consistency checking of specification of design patterns

Definition 3. (Properties of a model)

Let SemH(M) be the semantics of a model M. M has a property P (represented as a formula in the logic system defined by Sig) under the semantics definition SemH(M) and the hypothesis H, if and only if AxmDAxmFT(M) H(M) P in the formal logic system. Similarly, we say that M has a property P in descriptive semantics, if and only if AxmDT(M) H(M) P in the formal logic system.

Consistency checking of UML models

Definition 6. (Logical consistency)

Let SemH(M) = AxmD AxmF T(M) H(M) be the semantics of a model M. Model M is said to be logically inconsistent in the semantic definition SemH(M) if SemH(M)|-false; otherwise, we say that the model is logically consistent.

Definition 7. (Consistent interpretation in a subject domain)

Let Dom=<D, Sig, Eva> be a subject domain as defined in Definition 4. The interpretation of formulas in signature Sig is consistent with first order logic if and only if for all formulas q and p1, p2, …, pk that p1, p2, …, pk |- q, and for all systems s in D that Eva(pi, s) =true for i=1,2,…, k, we always have Eva(q, s) =true.

Validity of consistency checkingTheorem 1. (Unsatisfiability of inconsistent model)

A model M that is logically inconsistent in the semantic definition SemH(M) is not satisfiable on any subject domain whose interpretation of formulas is consistent with first order logic.

Inconsistent model => it cannot be implemented (not satisfiable) Consistency model => not necessarily implementable Other issues effect satisfiability:

Property of the subject domain: e.g. whether the programming language is powerful enough to implement

Non-logic property of the model: e.g. whether the system is feasible

Validation of consistency constraints Consistency constraints:

Logic statements about models, e.g. ‘a life line must represent an instance of a class’ x, y, z. Lifeline(x) represent(x,y) type(y, z) Class(z)

Definition 8. (Consistency w.r.t. consistency constraints)

Given a set of consistency constraints C={c1, c2, …, cn}, the consistency of a model M with respect to the constraints C under the semantics definition SemH(M) is the consistency of the set U = SemH(M) C of formulas. In particular, we say that a model fails on a specific constraint ck, if SemH(M) is consistent, but SemH(M) {ck} is not.

Results of experiments:• A number of sample UML models were checked to be

consistent.• Mutants of the models were checked and inconsistency in

mutants were detected.

Definition of validity and effectiveness

Definition 9. (Validity of consistency constraints)

Let AxmD and AxmF be the sets of axioms for descriptive semantics and functional semantics, respectively. A set C={c1, c2, …, cn} of consistency constraints is descriptively valid if AxmDC is logically consistent. The set C of consistency constraints is functionally valid AxmDAxmFC is logically consistent. Definition 10. (Effectiveness of consistency constraints)

Let A be a set of semantics axioms. A set C={c1, c2, …, cn} of consistency constraints is logically ineffective with respect to the set A of axioms if A C.

Results of Experiments:A set of 5 consistency constraints were validated by using LAMBDES tool. They were proved valid and effective.

Consistency checking of meta-modelsDefinition 11 (Inconsistency of meta-model).

A meta-model M is inconsistent if AxmD(M) is logically inconsistent.

Experiments: Subjects:

Simplified UML, UML 2.0 metamodel, AspectJ profile Findings:

Simplified metamodel: consistent UML 2.0:

16 inconsistencies due to voilation of strict meta-modelling 1 inconsistently as abstract and concrete metaclasses in different metamodel class diagrams.

AspectJ: Incomplete definition of 7 meta-classes2 inconsistency due to violation of strict meta-modelling

Conclusion Separation of functional semantics from descriptive semantics ca

n simplify the formal semantic definition of UML and it is scalable Applicable for all types of diagrams defined in meta-model uniformly Applicable to multiple views defined by multiple meta-models, and addre

ssed the extendability issues Addressed the issue due to flexibility in the uses of modelling language in

different development context Addressed the issue for different interpretation of modelling languages in

different subject domains Reasoning about models in first order logic can be feasible and u

seful in model drive software development Can be automated by tools such as LAMBDES + SPASS + StarUML Can reason about properties that are not possible for semantics at object le

vel, such as The validation of consistency constraints, The consistency of meta-model, etc. The conformance of designs to patterns,

Future work Further development of the axioms for functional

semantics; Theoretical analysis of the logic properties of the

semantics definition Soundness of the rules: yes Consistency of the rules: yes Completeness of the rules:

In what sense? How to prove?

Case studies on reasoning about other properties of designs, such as Platform specific models, platform independence, etc. Transformation of models,