Modeling Aspects
by Category Theory
Serge P. Kovalyov
Novosibirsk, Russia

FOAL 2010
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
Category-theoretic formalization

Explicit definition of intuitive notions




Describing objects via relations with similar objects



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




Avoiding appeal to “interiors” of objects
Constructing object by systemic criteria


Objects (things)
Morphisms (connections)
Functors (translations)
Complexity
Modularity
Traceability
FOAL 2010
Category of descriptions

Category c-DESC






Example: category of UML classes and inheritance relations
Scenario modeling




Objects are formal models of programs (descriptions)
Morphisms are actions of integrating components into systems
Composition is multistep integration
Identity morphisms are “doing nothing”
Category Pos
Objects are partially ordered sets (posets)
of events ordered by causal dependence
Morphisms are poset homomorpisms
(preserving events and ordering)
FOAL 2010
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
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
well-formed: parallelism

FOAL 2010
ill-formed: concurrency
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* = 1SIG
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
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
Traceable refinements
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)
r
X
S
t = rop


sig
s : sig(t) ◦ s = 1
sig(X)
sig(S)
sig(t)
Every refinement of scenarios is traceable
FOAL 2010
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
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
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
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):
1C, 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
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 sop is a trace and sig(sop) = 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'
sop   rop
p' : X  X '

Explicating labeled scenarios




Every explication is universal
Every aspect is explicable
“Many” scenarios are inexplicable e.g. interleaving
FOAL 2010
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
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





+ metadata model
+ monitoring
+ security

FOAL 2010
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
Thank you for your attention

FOAL 2010