Communication Equivalence Classes in
Networks
Gabriel Ciobanu Mihai Gontineac
Abstract
The quotient structure defined by a specific equivalence re-lation allows to simplify the study of complex communications between processes in computer networks. The quotient also pre-serves the initial algebraic structure.
This paper describes an algebraic structure modelling the commu-nication in computer networks; it continues the study initiated in [1]. Given two communicating processes, we consider the translations be-tween their communication ports, and introduce an equivalence relation between them in order to express that semantics of the communication between processes does not depend on the permutations of the com-munication ports. Considering a propertyP preserved by this abstract semantics of the communication, we first define P-translations, and then provide conditions for permutation-independent communication to preserve the property P. The quotient structure defined by this equivalence relation allows to simplify the study of a complex com-munication between processes to the study of the equivalence classes representatives. The quotients preserve the initial algebraic structure. A communication structure is given by the following elements:
(i) P – the set of processes;p, q, s, . . . range overP;
(ii) for every process p ∈ P we associate the set of communication portsH(p), and a permutation groupSHp(p);
c
(iii) for every process p ∈ P, and for every subset of handles K ⊆ H(p), we associate a permutation subgroup overK,SKp =S(K)∩ SHp(p), where S(K) is the permutation group over K; obviously,
S(K) can be viewed as a subgroup ofS(H(p)).
The reader can find more information on the algebraic notions and re-sults in [3] and [4]. Since H(p) is the set of all communication ports of a server process p, the permutations of SHp(p) describe the commu-nication symmetry of p, which expresses the possibility to change the communication ports without affecting the communication. We can remark that:
1. IfH ⊆K ⊆H(p), then SHp ≤SKp.
2. If H⊆K ⊆H(p), ρ∈SKp and ρ/K\H =idK\H, thenρ ∈S(H).
Clearly,ρ is also inSHp(p); thereforeρ∈S(H)∩SHp(p) =SHp. 3. The fact thatH ⊂K does not imply thatSHp < SKp; it is possible
to have SHp =SKp.
Let P and Q be two communication structures; a correspondence between two processes p∈ P and q ∈ Qis a triple (SHp, ϕ, SHq′), where H ⊆H(p), H′⊆H(q) andϕ:SHp →SHq′ is a group morphism. Ifϕis
one-to-one, we call it afaithcorrespondence. If ϕis onto, we say that the correspondence isfull.
We denote byC(p, q) the set of all correspondences betweenp∈ P and
q∈ Q.
C(P,Q) =S{C(p, q)|p∈ P, q∈ Q.
Let P, Q and S be three communication structures. The compo-sition of correspondences is a partial binary operation ◦ : C(P,Q)× C(Q,S)−→ C(P,S) defined by
(SHp , ϕ, SHq′)◦(S
q
H′, ϕ′, SHs′′) = (SHp, ϕ′◦ϕ, SHs′′),
(i) the objects pH are pairs (p, H), where p∈ P and H⊆H(p); (ii) amorphism ϕ:pH →p1H1 is a correspondence (SHp, ϕ, S
p1
H1).
The notation ϕ ∈ Comm(P)(pH, p1H1) means that ϕ : pH → p1H1
is a morphism of the category. The category Corresp(P,Q) of the correspondences from the structureP to the structureQ is defined by the following elements:
(i) the objects are triples (pH, ϕ, qH′), where pH ∈ |Comm(P)|,
qH′ ∈ |Comm(Q)|, and (SHp, ϕ, SqH′) is a correspondence from p
toq;
(ii) a morphism from (pH, ϕ, qH′) to (p
1H1, ϕ1, q1H1′) is a pair
(α, β) of morphisms α : pH → p1H1, β : qH′ → q1H1′ such
thatβ ϕ =ϕ1 α.
Let P and Q be two communication structures, and Comm(P) and
Comm(Q) their associated categories. We say that P and Q are iso-morphic, and we denote this by P ≃ Q, if their associated categories are isomorphic, namely there is a functorF :Comm(P)→Comm(Q) that is an isomorphism of categories.
If P and Q are two communication structures, a communica-tion map between P and Q is a mapping F from |Comm(P)| to
|Corresp(P,Q)|. We say that a communication mapF iscompatible with the correspondencesfromP ifF is a (covariant) functor from
Comm(P) toCorresp(P,Q). This means that
1. for each pH there is a unique F(pH) = (pH, ϕ, qH′)∈ ∈ |Corresp(P,Q)|,
2. for each morphismα:pH →p1H1 ofComm(P) there is a unique
morphismF(α) in Corresp(P,Q), and
3. F(α α1) =F(α)◦ F(α1) for allα, α1∈Comm(P).
LetP be a structure, andComm(P) its communication category. (i) For two given processes p, q ∈ P, a translation δ : H → H′
between their communication ports is a bijection from pH to
qH′, where pH and qH′ are objects in Comm(P). We denote such a translation bypH
δ
−→qH′ (or simply byδ whenever there
(ii) two translationspH δ1
−→qH′ and pH −→δ2 qH′ are equivalent (and
we denote this byδ1∼δ2) if there exist two permutationsρ∈SHp,
ρ′ ∈Sq
H′ such that δ2◦ρ = ρ′◦δ1.
For each communication structure P we can define the category
T rans(P) of its translations, with the same objectspHas inComm(P) (namely pairs (p, H) with p∈ P and H ⊆H(p)), and the translations between communication ports as morphisms between objects.
Let us consider a property P of the translations, and ℜ = {δ ∈ T(P)|δ satisfiesP}. We say thatℜis aP-translationover the com-munication structure P if for each translation δ in ℜ, its equivalence class [δ] ∈ ℜ, and if pH
δ
−→ pH ∈ ℜ, then δ ∈ SHp. The following
result provides a condition for ≈ℜ to become an equivalence. In this way |Comm(P)|/≈ℜ defines classes of objects having similar
commu-nication.
Theorem 1 Ifℜis aP-equivalence over the communication structure
P, then ≈ℜ is an equivalence relation over |Comm(P)|. Theorem 2 If pH ≈ℜ qH′, then the groups SHp and S
q
H′ are
isomor-phic.
Let P be a communication structure, ℜ a P-equivalence over the communication structureP, and≈ℜthe corresponding equivalence re-lation. More results of the quotient set|Comm(P)|/≈ℜ are presented
in [2]; for instance, we prove that a communication structure defined on the equivalence classes does not depend on the classes representatives. Thus any two communication structures determined by two different representative choices are isomorphic. It makes sense to talk about the quotient communication structure with respect to aP-equivalence. Let
ℜbe aP-equivalence and≈ℜ the associated equivalence relation. The following communication structure on|Comm(P)|/≈ℜ is well-defined:
• processes are given by the equivalence classes {[pH] | p ∈ P, H ⊆H(p)};
• the set of handles H([pH]) isH;
This communication structure is called thequotient communication structure ofP with respect to theP-equivalence ℜ, and we denote it by P/ℜ.
We can affirm that such a structure reflects the complexity of the communication along the computer networks. The need of a concep-tual and formal framework which make possible the study of network communication problems and properties is necessary (in our opinion). This paper is a step to such a suitable framework. It defines and studies the communication structures, formal tools which are comple-mentary to process algebras, and can describe, analyze, and provide a semantic framework to complex distributed systems involving process communication. The algebraic structure suggests constructive solu-tions for communication problems by considering the group acsolu-tions on the communication ports, the existence of limits in our communica-tion categories, and so on. A wealth of research topics remains to be considered.
References
[1] G. Ciobanu, F.E. Olariu. Abstract Structures for Communication between Processes. In D.Bj¨orner, M.Broy, A.Zamulin (Eds.): 3rd PSI (A.Ershov) Conference, LNCS 1726, Springer Verlag, pp.220-226, 1999.
[2] G. Ciobanu, V.M. Gontineac. Algebraic Constructions for Com-munication Structures. Fuzzy Systems and Artificial Intelligence, vol.10(2), Romanian Academy, pp.29-40, 2004.
[3] A. Kurosh. Cours d’alg`ebre sup´erieure, MIR, Moscou, 1980. [4] J.J. Rotman,Theory of Groups. An Introduction, Allyn and Bacon,
1979.
Gabriel Ciobanu, Mihai Gontineac, Received November 10, 2004
