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A Calculus of Design Patterns

A Calculus of Design Patterns. Hong Zhu Dept. of Computing and Electronics Oxford Brookes University Oxford OX33 1HX, Uk Email: hzhu@brookes.ac.uk. Acknowledgement. Ian Bayley and Hong Zhu, A formal Language of Pattern Composition , Proc. of PATTERNS 2010, Nov. 2010. (in press)

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A Calculus of Design Patterns

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  1. A Calculus of Design Patterns Hong Zhu Dept. of Computing and Electronics Oxford Brookes University Oxford OX33 1HX, Uk Email: hzhu@brookes.ac.uk

  2. 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) Seminar: A Calculus of Design Patterns

  3. 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 Seminar: A Calculus of Design Patterns

  4. 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) Seminar: A Calculus of Design Patterns

  5. Formal Specification of DPs extension of BNF for graphical modelling languages • 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. Seminar: A Calculus of Design Patterns

  6. Example: The Object Adapter DP • In GoF book: In general, there can be a set of such methods! Indicate where messages can be sent to rather than a component of the pattern! Seminar: A Calculus of Design Patterns

  7. Formal Spec of the Object Adapter DP Seminar: A Calculus of Design Patterns

  8. Example 2: The Composite DP • GoF Diagram: In general, there can be many leaves! Seminar: A Calculus of Design Patterns

  9. Spec of the Composite DP Seminar: A Calculus of Design Patterns

  10. 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. Seminar: A Calculus of Design Patterns

  11. 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 Seminar: A Calculus of Design Patterns

  12. Example of Pattern Composition Seminar: A Calculus of Design Patterns

  13. Overview of Our Approach • We define a set of sixoperators 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) Seminar: A Calculus of Design Patterns

  14. 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, Seminar: A Calculus of Design Patterns

  15. 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 Seminar: A Calculus of Design Patterns

  16. 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. Seminar: A Calculus of Design Patterns

  17. 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. Seminar: A Calculus of Design Patterns

  18. 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. Seminar: A Calculus of Design Patterns

  19. 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 Px\x', is the pattern that has the following property. where the power set of T We also write Px Seminar: A Calculus of Design Patterns

  20. Example • The single-leaf variant of the Composite pattern Composite1 can also be defined as follows. Composite1=Composite Leaves\ Leaf Seminar: A Calculus of Design Patterns

  21. 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 1 X'={x'1, …, x'n} X'Vars(P)= . • Notation: PX \ X' to denote Px1\ x'1 … xn\ x'n. We also write PX Seminar: A Calculus of Design Patterns

  22. 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 Seminar: A Calculus of Design Patterns

  23. Example • We can define the Composite pattern as a generalisation of the single-leaf variant Composite1: Composite=Composite1Leaf \ Leaves Seminar: A Calculus of Design Patterns

  24. Lift Operator the existentially quantified class components • 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 PX, is the pattern defined as follows. the remainder of the predicate Seminar: A Calculus of Design Patterns

  25. Example Seminar: A Calculus of Design Patterns

  26. 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] Seminar: A Calculus of Design Patterns

  27. 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) Seminar: A Calculus of Design Patterns

  28. Results of The Case Study Seminar: A Calculus of Design Patterns

  29. Seminar: A Calculus of Design Patterns

  30. 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. Seminar: A Calculus of Design Patterns

  31. Algebraic Laws of the Operators Let 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 Seminar: A Calculus of Design Patterns

  32. 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. • Lemma • For all patterns P, Q and R, we have that Seminar: A Calculus of Design Patterns

  33. 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. Seminar: A Calculus of Design Patterns

  34. 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 Seminar: A Calculus of Design Patterns

  35. 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. Seminar: A Calculus of Design Patterns

  36. Laws of Flattening and Generalisation Seminar: A Calculus of Design Patterns

  37. 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 Seminar: A Calculus of Design Patterns

  38. 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) Seminar: A Calculus of Design Patterns

  39. Seminar: A Calculus of Design Patterns

  40. where Seminar: A Calculus of Design Patterns

  41. 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 Seminar: A Calculus of Design Patterns

  42. 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 Seminar: A Calculus of Design Patterns

  43. Example of Proving Equivalence (2) • Two ways of composition of the Composite and the Adapter patterns • Express the compositions as expressions that consist of the operators • Apply the algebraic laws to prove that they are equivalent Seminar: A Calculus of Design Patterns

  44. 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). Seminar: A Calculus of Design Patterns

  45. Seminar: A Calculus of Design Patterns

  46. 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. Seminar: A Calculus of Design Patterns

  47. 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 laws • The completeness of the algebraic laws Seminar: A Calculus of Design Patterns

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