a calculus of design patterns hong zhu dept. of computing and electronics oxford brookes university...

A Calculus of Design Patterns

Hong ZhuDept. of Computing and ElectronicsOxford Brookes UniversityOxford OX33 1HX, UkEmail: [email protected]

Seminar: A Calculus of Design Patterns 2Aug. 2010

Acknowledgement Ian Bayley and Hong Zhu, A formal

Language of Pattern Composition, Proc. of PATTERNS 2010, Nov. 2010. (in press)

Hong Zhu and Ian Bayley, Laws of Pattern Composition, Proc. of ICFEM 2010, Nov. 2010. (in press)

Ian Bayley and Hong Zhu, Formal specification of the variants and behavioural features of design patterns, Journal of Systems and Software Vol.83, No.2, pp209–221 (Feb 2010)


Overview Introduction Operators on patterns Case study 1: The expressiveness

GoF suggested compositions Laws of pattern composition Case study 2: the proof of equivalence

Representation of composition with overlaps Soundness of the operators Future work


Introduction Software Design Patterns (DP)

codified reusable solutions to recurring design problems,

increasingly important role in the development of software systems

Research on DP identified and catalogued (e.g. Gang of Four book) formally specified (Mik 1998, Taibi 2003, Gasparis et

al. 2008, Bayley&Zhu 2009) included in software tools (e.g. PINOT, Mapelsden et

al. 2002)


Formal Specification of DPs Formal approaches

Eden's approach• Formal logic predicates on the structural features of the source code

of object-oriented programs (Gasparis et al. 2008) Taibi's approach

• Formal logic predicates on the structural features of the source code • Temporal logic statements for behavioural features

Our predicates (Bayley & Zhu, 2007, 2008, 2009)• Formal logic predicates for both structural and behavioural features

of designs in UML class and sequence diagrams• The primitive predicates are induced from GEBNF definition of UML

Semi-formal approaches Uses of graphic meta-modelling languages For example: RBML, DPML, PDL, etc.

extension of BNF for graphical modelling languages


Example: The Object Adapter DP In GoF book:

Indicate where messages can be sent to rather than a component of the pattern!

In general, there can be a set of such methods!


Formal Spec of the Object Adapter DP


Example 2: The Composite DP GoF Diagram:

In general, there can be many leaves!


Spec of the Composite DP


General Definition In general, a design pattern P can be defined abstractly

as an ordered pair <V, Pr>, where Pr is a predicate on the domain of some representation of

software systems. It specifies the structural and behavioural features of the pattern

V is a set of declarations of variables free in Pr. It specifies the components of the pattern.

Notations: Let V = { v1:T1, …, vn:Tn } The semantics of the specification is a ground predicate in the

form:

v1:T1 … vn:Tn . (Pr) Spec(P) to denote the predicate above, Vars(P) for the set of variables declared in V, and Pred(P) for the predicate Pr. m P : a design model m conforms to pattern P

vi are variables that range over the type Ti of software elements.


Pattern Composition Why compose patterns

Patterns are usually to be found composed with each other in real applications

In practice: Annotation:Role representation: In graphic diagram

Taibi’s work 2006 pattern composition is a combination of formal statements of

structural and behavioural features of patterns Illustrated by an example on how two patterns can be composed

Bayley &Zhu, 2008 One DP composition operator with a notion of overlaps, which

can be• One-to-One overlaps• One-to-Many overlaps• Many-to-many overlaps


Example of Pattern Composition


Overview of Our Approach We define a set of six operators on DPs so that

compositions of DPs can be formally represented as expressions Restriction: P Pr P Superposition: P P P Extension: P V Pr P Flatten: P V P Generalisation: P V P Lift: P V P

We propose a set of algebraic laws to reason about the equivalence of DP compositions (i.e. expressions)


Restriction Operator P[c] Let P be a given pattern and c be a

predicate defined on the components of P. A restriction of P with constraint c, written as P[c], is the pattern obtained from P by imposing the predicate c as an additional condition on the pattern. Formally,


Example

A variant of the Composite pattern in which there is only one leaf can be formally defined as follows.

Composite1 = Composite [ ||Leaves|| = 1]

called Composite1 in the sequel


Superposition Operator P * Q Let

P and Q be two patterns. the component variables of P and Q are disjoint, i.e.,

Vars(P) Vars(Q) = . Definition:

The superposition of P and Q, written P * Q, is a pattern that consists of both pattern P and pattern Q as is formally defined as follows.


Example The superposition of Composite and Adapter

patterns: Composite * Adapter

It requires each instance to contain one part that satisfies the Composite pattern and another that satisfies the Adapter pattern.

These parts may or may not overlap, The following expression does enforce an overlap,

requiring that a class in Leaves of the Composite be the target of an Adapter.


Extension Operator P#(V c) Let

P be a pattern, V be a set of variable declarations that are disjoint

with P's component variables, i.e.,

Vars(P) V = c be a predicate with variables in Vars(P) V.

Definition:The extension of pattern P with components V and

linkage condition c, written as P#(V c), is defined as follows.


Flatten Operator P x\ x' Let

P be a pattern, Vars(P)= {x: P(T), x1:T1, …, xk:Tk},

Pred(P)=p(x,x1,…,xk), and x' Vars(P).

Definition:The flattening of P on variable x, written Px\ x',

is the pattern that has the following property.

where

the power set of T

We also write Px


Example The single-leaf variant of the Composite

pattern Composite1 can also be defined as follows.

Composite1 = Composite Leaves\ Leaf


Notations From the definition, we have the following

property. For x1 x2 and x1’ x2’,

We can overload the operator to a set of component variables Let X be a subset of P's component variables all of

power set type, i.e., X={x1:P(T1), …, xn:P(Tn)} Vars(P), n 1X'={x'1, …, x'n} X' Vars(P) = .

Notation: P X \ X' to denote P x1\ x'1 … xn\ x'n.

We also write PX


Generalisation Operator P x\x' Let

P be a pattern, x Vars(P)={x:T, x1:T1, …, xk:Tk}.

Definition:The generalisation of P on variable x,

written P x\x', is defined as follows.

Notation: Similar to the operator, we also write P x, and

P X


Example We can define the Composite pattern as a

generalisation of the single-leaf variant Composite1:

Composite = Composite1 Leaf \ Leaves


Lift Operator Let

P be a pattern CVars(P)={x1:T1, …, xn: Tn}, n>0

OPred(P)=p(x1, …, xn).

X={x1, …, xk}, 1 k < n, be a subset of the variables in the pattern.

Definition:The lifting of P with X as the key, written PX, is the

pattern defined as follows.

the existentially quantified class components

the remainder of the predicate


Example


Example

Notation:• P[x' := x]: systematically renaming x to x’;• P[v:= x=y]: the syntactic sugar for P[x=y][v:= x][v:=y];• P[v:= x y]: abbreviates P[x y][v:= x]


Case Study Goal:

to demonstrate the expressiveness of the operators Subjects of the case study:

In the GoF book, the documentation for each pattern concludes with a brief section entitled Related Patterns.

• It compares and contrasts patterns, • it makes suggestions for how other patterns may be used with the

one under discussion. Example: page 106 of the GoF book

“A Composite is what the builder often builds''. This can be formally specified as follows.

(Builder * Composite) [Product = Component]. Method:

to formalise them all as expressions with • The operators from this paper • Formal logic predicates specifying the patterns in our previous work

(Bayley&Zhu 2009)


Results of The Case Study


Summary of Case Study Results Five new arrows have been added to the GoF diagram

(numbered in bold font) discussed in GoF main text but omitted from its diagram We were able to formalise them

Four arrows from the original diagram were not formalised (labelled with asterisks) Composite and Interpreter: is a specialisation relation, which can

be formally proved; see (Bayley & Zhu 2007). Decorator and Strategy: is a comparison of the two, not a

composition suggestion, Strategy and Template Method: as above. Iterator and Visitor: it is mentioned in GoF only on the diagram,

but not expanded upon in the main text.

The case study has demonstrated that the operators defined in this paper are expressive to define compositions of design patterns.


Algebraic Laws of the OperatorsLet P and Q be design patterns. Definition: (Specialisation relation)

Let P and Q be design patterns. Pattern P is a specialisation of Q, written P Q, if for all models m, whenever m conforms to P, then, m also conforms to Q.

Definition: (Equivalence relation)Pattern P is equivalent to Q, written P = Q, if P Q and Q P.

Lemma:

m |= P


The TRUE and FALSE Patterns Definition:

Pattern TRUE is the pattern such that for all models m, m |= TRUE.

Pattern FALSE is the pattern such that for no model m, m |= FALSE.

LemmaFor all patterns P, Q and R, we have that


Laws of Restriction Let vars(p) denote the set of free variables in a predicate p. For all predicates c, c1, c2 such that vars(c), vars(c1) and

vars(c2) Vars(P), the following equalities hold.


Laws of Superposition For all patterns P and Q, we have the following equations.

From this and reflexivity of , it follows that superposition is idempotent.

TRUE and FALSE are the unit and zero of superposition.

Superposition is also commutative and associative


Laws of Extension Let

U be any set of component variables that is disjoint to Vars(P), and c1, c2 be any given predicates such that vars(ci) Vars(P) U,

i=1,2.

Let U and V be any sets of component variables that are disjoint to

Vars(P) and to each other, c1 and c2 be any given predicates vars(c1) Vars(P) U and vars(c2) Vars(P) V.


Laws of Flattening and Generalisation


Laws Connecting Several Operators For all predicates c such that vars(c) Vars(P),

For all X Vars(P),

For all X Vars(P) Vars(Q),

where


For all sets of variables X such that

X vars(P) =, and all predicates c such that

vars(c) (Vars(P) X),

where

For all x Vars(P) such that x : P(T)


where


Example of Proving Equivalence (1) We have seen the following as examples of the

operators: Composite1 = Composite [ ||Leaves|| = 1]

Composite1 = Composite Leaves\ Leaf

Composite = Composite1 Leaf \ Leaves

We can prove that: Composite [ ||Leaves|| = 1]= Composite Leaves\ Leaf Composite= (Composite Leaves\ Leaf) Leaf \ Leaves Composite1=(Composite1 Leaf \ Leaves) Leaves\ Leaf


Proof of Composite [ ||Leaves|| = 1] = Composite Leaves\ Leaf

For all sets X, we have that

||X||=1 x(X={x})

Therefore,

Composite[ ||Leaves||=1]

= Composite [ Leaf (Leaves={Leaf}) ]

= Composite # (Leaf Leaves={Leaf})

= Composite Leaves \ Leaf


Example of Proving Equivalence (2) Two ways of composition of the Composite

and the Adapter patternsExpress the compositions as expressions that

consist of the operatorsApply the algebraic laws to prove that they

are equivalent


First Way of Composition One of the Leaves in the Composite pattern is the Target in

the Adapter pattern. This leaf is renamed as the AdaptedLeaf.

Lift the adapted leaf to enable several of these Leaves to be adapted. lift the OneAdaptedLeaf pattern with AdaptedLeaf as the key flatten those components in the composite part of the pattern

(i.e. the components in the Composite pattern remain unchanged).


Second Way of Composition Lift the Adapter with target as the key Superpose it to the Composite patterns

We can prove that the two ways are equivalent by applying the algebraic laws.


Conclusion Work in progress and future work

The uses of theorem provers for automated reasoning about the compositions of design patterns

The soundness of the algebraic lawsThe completeness of the algebraic laws

a calculus of design patterns hong zhu dept. of computing and electronics oxford brookes university...

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