IEEE Systems, Man and Cybernetics Magazine - July 2018 - 17

element e is written as e.a, and this has the (data)
type D a .
◆ R is a finite collection of relations in T. For each
R ! R, R is a k ary relation in T1 # g # Tk, where
Ti ! T, i = 1, f, k. The proposition that a collection of
elements e 1, e 2, f, e k of types T1, T2, f, Tk satisfies relation R is written as R (e 1, e 2, f, e k) .
◆ Ax is a finite collection of axioms about elements and
their relations. Each axiom Ax ! Ax is a well-formed
logic formula that is constructed from expressions of
the form of e.a and R (e 1, e 2, f, e k), as described above,
but also from operations on and relations among the
attribute data types ^eg., + , - , 1, 2 h, equality (i.e., =
and !), set constructions in the form of " x 1, f, x n , or
" x | p (x) ,, set operators (i.e., + , , , -) and set relations (i.e., !, 1, 3, 2, 4), logic connectors and ^/h, or
^ 0 h, not ^ J h, implication ^& h, equivalence ^+ h, and
quantifiers for-all ^6 h and exists ^7 h .
For the sake of readability, in this article we use floor
tile layout patterns to explain and illustrate our notions
and notation rather than software design patterns or
cyberpatterns, both of which are less comprehensible.
Example 1: The Domain of a Floor Tile Layout.
Consider the layout of floor tiles as the subject domain.
The elements are all of one type of tile with the following attributes:
◆ Shape : " Square, Rectan gle ,, indicating whether the
tile is in the shape of a square or a rectangle

the point of view of a pattern-oriented research methodology, much more work is needed to understand how patterns interrelate and interact not only within a domain but
also, and more importantly, across domains.
Relationships between patterns in a domain include one
pattern being the subpattern of another or one pattern
being, through composition, a part of another, more complex pattern. Relationships across domains, between patterns of different types, are of great importance to the
understanding of cyberspace and the effective protection
of Internet resources and infrastructure. Figure 1 is a first
attempt at depicting some of these relationships.
There are two key research questions for a pattern-oriented research methodology that aims at developing theories of a subject domain as human knowledge rather than
just machine-readable data. One is how to describe, document, and specify patterns in a comprehensible, testable,
verifiable, and reusable form. However, since we also
require machine processing of the knowledge captured by
patterns, a second question relates to how cyberpatterns
can be stored, retrieved, and applied automatically with
the support of computer software. To lay a solid foundation for answering these questions, in the next two sections we generalize the formal theory of patterns beyond
just software design and demonstrate that it can be
applied to a domain completely unconnected with software design.
Pattern as a Knowledge
Representation Formalism
A pattern in a specific subject domain consists of a collection of elements of various
types. Each type typically represents a
kind of artifact or atomic event in a subject domain. Each element can be characterized by a number of attributes or
features, and the elements are connected
by relations.
Subject Domain
A subject domain can, accordingly, be
characterized as a collection of all of the
possible element types and the relationships that may hold between them. The
elements and relationships may have some
properties universally satisfied by all of
the elements and relations. These are
called the axioms of the domain.
Definition 1 (domain): A subject
domain (or simply domain) D is a triple
T, R, Ax , where
◆ T is a finite collection of element
types. For each type T ! T, there is a
finite set A T of attributes associated
with the elements of T. For each
attribute a ! A T, the value of a for

User Requirements Pattern/
Environment Pattern

Aggregates into

Workload
Pattern

Determines
Aggregates into

User Input Data Pattern/
Environment Data

Aggregates into

Network Traffic
Pattern

Is a Subtype of

Attack Pattern

Social Network
Pattern

Targets at

Vulnerability
Pattern

Detects

Investigates

Mitigates

Prevents

Forensic Pattern

Is Context of

Security Design
Pattern
Is a Subtype of
Software Design
Pattern

Figure 1. the potential relationships between different types of

cyberpatterns.

Ju ly 2018

IEEE SyStEmS, man, & CybErnEtICS magazInE

Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - July 2018