Self-Adaptation in Collective Adaptive Systems

Abstract

An adaptive system is currently on spot: collective adaptive system (CAS), which is inspired by the socio-technical systems. CASs are characterized by a high degree of adaptation, giving them resilience in the face of perturbations. In CASs, highest degree of adaptation is self-adaptation. The overarching goal of CAS is to realize systems that are tightly entangled with humans and social structures. Meeting this grand challenge of CASs requires a fundamental approach to the notion of self-adaptation. To this end, taking advantage of the categorical approach we establish, in this paper, a firm formal basis for modeling self-adaptation in CASs.

Appendices

Appendix

We recall some concepts from the category theory [1–5] used in this paper.

What is a category?

■ A category C can be viewed as a graph (O b j(C), A r c(C),s,t), where

• O b j(C) is the set of nodes we call objects,

• A r c(C) is the set of edges we call morphisms and

• s,t:A r c(C)→O b j(C) are two maps called source (or domain) and target (or codomain), respectively.

We write $f:\mathcal {X} \longrightarrow \mathcal {Y}$ when f is in A r c(C) and $s(f)=\mathcal {X}$ and $t(f)=\mathcal {Y}$.

An object in the category is an algebraic structure such as a set. We are probably familiar with some notations for finite sets: {Student A, Student B, Student C } is a name for the set whose three elements are Student A, Student B, Student C. Note that the order in which the elements are listed is irrelevant.

A morphism f in the category consists of three things: a set $\mathcal {X}$, called the source of the morphism; a set $\mathcal {Y}$, called the target of the morphism and a rule assigning to each element x in the source an element y in the target. This y is denoted by f(x), read “f of x”. Note that the morphism is also called the map, function, transformation, operator or arrow. For example, let $\mathcal {X}=\{$ Student A, Student B, Student C }, $\mathcal {Y}=\{$ Math, Physics, Chemistry, History } and let f assign each student his or her favorite subject. The following internal diagram is an illustration.

(22)

This states that the favorite subject of the Student C is History, written by f(Student C) =History, while Student A and Student B prefer Chemistry. There are some important properties of any morphism

• From each element in the source {Student A, Student B, Student C}, there is exactly one arrow leaving.

• To an element in the target {Math, Physics, Chemistry, History}, there may be zero, one or more arrows arriving.

It is possible that the source and target of the morphism could be the same set. The following internal diagram is an example.

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and, in the case, the morphism is called an endomorphism whose representation is available as in

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■ Associated with each object $\mathcal {X}$ in O b j(C), there is a morphism $1_{\mathcal {X}} = \mathcal {X} \longrightarrow \mathcal {X}$, called the identity morphism on $\mathcal {X}$, and to each pair of morphisms $f:\mathcal {X} \longrightarrow \mathcal {Y}$ and $g:\mathcal {Y} \longrightarrow \mathcal {Z}$, there is an associated morphism $f;g: \mathcal {X} \longrightarrow \mathcal {Z}$, called the composition of f with g. The representations in Eq. 25 include the external diagrams of identity morphism and composition of morphisms.

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Here are the corresponding internal diagrams of the identity morphism.

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Or

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And here, the composition of morphisms is described in the internal diagram

(28)

Or, in the external diagram By the diagram (28), we can obtain answers for the question “What should each student support to his or her favorite classmate for subject?”. In fact, the answers are such as “Student A likes Student B, Student B likes Chemistry, so Student A should support Chemistry”, “Student B likes Student C, Student C likes History, so Student B should support History” and “Student C likes Student B, Student B likes Chemistry, so Student C should support Chemistry”.

The composition of two morphisms e and f means that e and f are combined to obtain a third morphism This is represented in the following internal diagram.

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where, for example, e;f(Student B)=History is read as “the favorite subject of the favorite classmate of Student B is History”.■ The following equation must hold for all objects $\mathcal {X}$, $\mathcal {Y}$ in O b j(C) and morphism $f:\mathcal {X} \longrightarrow \mathcal {Y}$ in A r c(C):

$$ {\kern2pc}\text{\textit{Identity}}:\quad\quad\quad\quad 1_{\mathcal{X}};f = f = f;1_{\mathcal{Y}} $$

(30)

The following equation must hold for all objects $\mathcal {X}$, $\mathcal {Y}$ and $\mathcal {Z}$ in O b j(C) and morphisms $f:\mathcal {X} \longrightarrow \mathcal {Y}$, $g:\mathcal {Y} \longrightarrow \mathcal {Z}$ and $h:\mathcal {Z} \longrightarrow \mathcal {T}$ in A r c(C):

$$ {\kern2pc}\text{\textit{Associativity}}:\quad\quad (f;g);h = f;(g;h) $$

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Isomorphism

A morphism $f:\mathcal {X} \longrightarrow \mathcal {Y}$ in the category C is an isomorphism if there exists a morphism $g: \mathcal {Y} \longrightarrow \mathcal {X}$ in that category such that $f;g = 1_{\mathcal {X}}$ and $g;f = 1_{\mathcal {Y}}$.

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That is, if the following diagram commutes.

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Element of a set

For any set A, x∈A iff (or x:1→A) where 1 denotes a singleton set. Focus on one element of {Math, Physics, Chemistry, History }, say {subject}, and call this set “1”. Let us see what the morphisms from 1 to {Math, Physics, Chemistry, History } are. There are exactly four of them.

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By this way, we can write (or $2: 1 \longrightarrow \mathbb {N}$ ) for $2 \in \mathbb {N}$, (or $i: 1\longrightarrow \mathbb {N}$) for $i \in \mathbb {N}$ and so on.

Functor

Functor is a special type of mapping between categories. Functor from a category to itself is called an endofunctor. Note that the functors are also viewed as morphisms in a category, whose objects are smaller categories.

T-algebra

Let C be a category, $\mathcal {A}$ an object in O b j(C), T:C→C an endofunctor and f a morphism $\textsf {T}(\mathcal {A}) \stackrel {f}{\longrightarrow } \mathcal {A}$; then T-algebra is a pair $\langle \mathcal {A},f\rangle$. O b j(C) is called a carrier of the algebra and T a signature of the algebra.