1.  FOAL 2010 Modeling Aspects by Category Theory Serge P. Kovalyov Novosibirsk, Russia

2. Aspect-oriented software development  Mission: explicit separation and composition of concerns  Motivation  Concerns are tangled  Concerns crosscut modular architecture bounds  Traceability is compromised (ability to determine what each fragment of the system is included into it for)  Proposed solution  Equip program models with traces of refinements that produce them from concerns (i.e. “label” programs by concerns)  Explicitly identify, compose (weave), and separate concerns  Application: enhance modular design technologies with aspect handling capabilities  FOAL 2010

3. Category-theoretic formalization  Explicit definition of intuitive notions  Objects (things)  Morphisms (connections)  Functors (translations)  Describing objects via relations with similar objects  Avoiding appeal to “interiors” of objects  Constructing object by systemic criteria  Universality (existence and uniqueness of connection with similar objects)  Naturality (independence of the result on the way it is reached)  Formal specification and verification of systemic properties  Complexity  Modularity  Traceability  FOAL 2010

4. Category of descriptions  Category c-DESC  Objects are formal models of programs (descriptions)  Morphisms are actions of integrating components into systems  Composition is multistep integration  Identity morphisms are “doing nothing”  Example: category of UML classes and inheritance relations  Scenario modeling  Category Pos  Objects are partially ordered sets (posets) of events ordered by causal dependence  Morphisms are poset homomorpisms (preserving events and ordering)  FOAL 2010

5. Diagrams  c-DESC-diagram  Functor  : X  c ‑ DESC  Graph of X labeled by c ‑ DESC-objects and c ‑ DESC-morphisms  Cocone  Natural transformation of a diagram (base) to a singleton (vertex)  Colimit of a diagram   Universal cocone with base   Minimal “container” that encapsulates objects of  respecting their interconnections  FOAL 2010

6. Configurations  Well-formed configuration  Is a c-DESC-diagram (of components and their interconnections)  Has a colimit (system built from interconnected components)  Satisfies structural constraints  Configurations of scenarios  Well-formed configurations are disjoint unions of cocones  Examples  FOAL 2010 well-formed: parallelism ill-formed: concurrency

7. Interfaces  Category SIG of interfaces and their integration actions  Functor sig : c-DESC  SIG  Default realization of any interface  Functor sig* : SIG  c-DESC  sig ◦ sig* = 1 SIG  Bijective map Mor(sig*(I), A) to Mor(I, sig(A)) by functor sig (i.e. sig* is left adjoint to sig with identity as the unit)  Example: signature of a program module  Scenario interface  Set of events  Forgetful functor |–| : Pos  Set  FOAL 2010 |–|

8. Refinements  Category r-DESC  Objects are models  Morphisms are refinements (individual component development steps)  Examples  Elaborating requirements  Implementing specification by means of a programming language  Scenario refinement  Replacing atomic events with subscenarios fully inheriting the order  Dual to a surjective homomorphism  FOAL 2010

9.  Tracing a refinement r : X  S  Labeling S by concerns that constitute X  Trace is a c-DESC-morphism t : S  X dual to a refinement  sig(t) has right inverse (to preserve traceability at subsequent integrating S into a larger system) Traceable refinements  FOAL 2010 r s : sig(t) ◦ s = 1 X S sig(X) sig(S) t = r op sig(t) sig  Every refinement of scenarios is traceable

10. Enhancing descriptions with aspects  Aspect-oriented description is a pair  A, l : sig(A)  L   A  Ob c ‑ DESC is a “modular” part  L  Ob SIG is an aspect structure  l labels sig(A) by aspects (sig(l) is a trace)  Morphism of AO-description  A, l  to  A', l '  is a pair  p, q   A, l : sig(A)  L  p  sig(p)   q  A', l ' : sig(B)  L'   Aspect-oriented scenarios  Object are pomsets (labeled posets)  Morphisms are homomorphisms that preserve labeling  FOAL 2010

11. Aspect-oriented design  AO-configurations are modular configurations that admit any labeling of components by aspects (i.e. have suitable colimit)  Interfaces of AO-descriptions  mod :  A, l  |  A (modular design interfaces)  asp :  A, l  |  l (aspect design interfaces)  int :  A, l  |  sig(A) (original interfaces)  AO-refinements are duals to such AO-morphisms that are produced from traces  Aspect-oriented scenario modeling  Configurations are disjoint unions of AO-cocones  Functor mod forgets labeling  Functor mod* labels each event by a unique label (event itself)  Refinement replaces events with subscenarios fully inheriting the order and detailing the labeling  FOAL 2010

12. Aspects  Aspects are “elementary” building blocks of AO-descriptions  An integration of an aspect into a system is an invertible embedding at the level of aspect structures  A is an aspect iff for every object A' and morphism  p, q  : A  A' q has left inverse (often a trace)  Aspects in scenario modeling  Aspect is a scenario with all events labeled by the same label  Aspect is precisely a pair  A, ! : |A|  1   FOAL 2010

13. Weaving  Specifying how to weave an advice W with a base program B  Connector: C  Pointcut descriptor: j : C  B  Entry points descriptor: e : C  W  Performing weaving  Pushout (colimit):  1 C, e  : C  C  W j   B  X  Weaving labeled scenarios  Weaving exists if a connector “tolerates” concurrency (i.e. it doesn’t impose specific order of executing different aspects of the advice bound to the same join point )  Weaving with an aspect preserves labeling of a base  FOAL 2010

14. Explication of aspect structure  Explication of aspect structure of an AO-description  A, l   Obtaining “actual” refinement from concerns  r ‑ DESC-morphism s : X  A where s op is a trace and sig(s op ) = l  An explication of  A, l  is universal if every AO-morphism  p, q  :  A, l    A', l '  has an explication (provided that  A', l '  has) p : A  A' s op   r op p' : X  X '  Explicating labeled scenarios  Every explication is universal  Every aspect is explicable  “Many” scenarios are inexplicable e.g. interleaving  FOAL 2010

15. Separation of concerns  Subaspect of an AO-description S  AO-morphism m : A  S where A is an explicable aspect  Explication m' of m is right inverse to a trace  Explication diagram of m is a pullback (i.e. mod(m) is a “preimage” of m' along an explication trace of S)  An aspect has no proper subaspects  If S is an aspect, then m is an isomorphism  Each explicable labeled scenario can be partitioned to subaspects  Each scenario can be labeled by linearly ordered subaspects  Maximal partition: assign a unique label to each event (i.e. apply mod*)  Minimal partition: factorize by linear equicomparability relation  FOAL 2010

16. Industrial application  Distributed measurement system (DMS) development  Main measurement cycle is linear order of separable aspects  Measurement automation infrastructure aspects are woven to it undermining separation of concerns  DMS scenario weaving schema measure  store  validate  compute  display      FOAL 2010 + metadata model + monitoring + security

17. Summary  Understanding systemic nature of AOSD concepts  Providing formal paradigm-neutral description of  Aspects  Weaving  Explication of aspect structure  Separation of concerns  Verifying structural properties of aspect-oriented operations  Applying in concurrency theory: labels are aspects  Applying in industry: large-scale DMS development  FOAL 2010

18. Thank you for your attention  FOAL 2010