Development and Application of Abstract Relation Types for Use in Systems and System-of-Systems Design and Evaluation

Abstract relation types (ART) are developed to represent, describe and establish a computational framework for a system. An abstract relation type is closely related to and builds upon two fundamental ideas. The first idea is the binary relation and structural modeling techniques developed by John N. Warfield. The second idea is the concept of abstract data types. These two ideas are combined to create an abstract relation type that provides a structured representation and computational method for systems and system components. The complete system description approach is based on six abstract relation types: context, concept, functions, requirements, architecture, and test (CCFRAT). When combined with digraphs and other graphical representations of the matrix form, ART provides a powerful tool for the communication of complex system interactions to large system design teams. Introduction. System evaluation is a task that is becoming more difficult due to the increasing number of existing systems, interfaces and operations. At a basic level, a system can be defined in one of two general ways: a functional definition, or a “construction rule” definition. The functional definition of a system is “a constraint on variation.” The construction rule definition of a system is “a relationship mapped over a set of objects.” Abstract relation types assist in evaluation of systems that are defined or described in either of these two forms (Klir, 1969). A primary component of the ART construct, binary relations, have a long history of application in the systems science and engineering fields (Klir, 1991). Binary relations can be represented as matrices, directed graphs or lattices (Gratzer, 2003). The strong ties between the matrix form and graphical form of binary relations provide an excellent tool for system evaluation computation and communication. Structured group techniques that are based on binary relations and graph theory have been successfully used in the evaluation of large systems (Warfield, 2006). ART are an important tool used to group contextual system relations with a set of visual and computational techniques to reason and communicate about systems. A well defined set of ART constructs will provide a common foundation for a systems engineering language and approach. Abstract Relation Types. Two basic ideas are combined to create the ART. The first basic idea of structuring complex systems was developed by John N. Warfield (Warfield, 2003). This basic concept included the use of binary matrices, binary relationships and contextual relationships. A matrix N is represented by four sets; N = {Is, It, Is x It, EIsxIt}: where Is is an ordered vertical index set It is an ordered horizontal index set Is x It is the set of all ordered pairs of Is and It EIsxIt is the entry set (or content) of the matrix. A binary relation, as defined by Warfield, is represented by a binary matrix A that defines a two-block partition {R; ~R}on Is x It, such that all ordered pairs in Is x It for which the entry is 1 are in the first block, and all other ordered pairs are in the second block. The first block is the binary relation R on Is x It and the second block is the complementary relation ~R on Is x It. A binary relation R on Is x It defines a binary matrix A indexed by Is and It. Some contextual relations developed by Warfield are: “is included in”, “is antecedent to”, “is subordinate to” and “is adjacent to.” These relationships are mapped in a binary matrix and are used in the analysis of the structure of complex interactions and systems. (Warfield, 1974) The second basic idea is the abstract data type. An abstract data type is used to organize the structure and specify the operations of a specific data type. The abstract nature of the data type ensures that the data type is independent of the underlying information types. (Rosen, 2000) Abstract relation types are not independent of the underlying relations. However, they use a predefined set of operations for each specific ART. An ART is composed of two primary components: a binary relation matrix and operations on the binary relation matrix. The operations are grouped into three basic types: those operations associated with the semantics of the relation, the binary matrix construction and evaluation, and the organization and display of the matrix information. ART extend the detail used in describing and defining the contextual relationship as developed by Warfield (1974). Binary Relations. Warfield used relations that connect elements of the same set A, usually called relations on A. However binary relations can also apply to two sets, set A and set B. A binary relation between set A and set B links specific elements of set A with specific elements of set B. Binary relations have been applied in many areas in the practice of systems engineering. These areas include the practice of Interactive Management, N Squared Diagrams and Design Structure Matrices, which mostly use a set A on set A relation mapping. Systems analysis, software evaluation and knowledge development areas use a binary, relation-based approach called Formal Concept Analysis. (Ganter, Wille) A formal context is constructed from two sets, A and B, and a relation I between A and B. The elements of set A are called objects and the elements of set B are called attributes of the context. The relation I is called the incidence relation of the context. The formal context structure is used to create a formal concept definition. These binary, relation-based structures have been used in software and knowledge engineering applications. Active research is continuing in the application of this technology in many engineering domains. The formal context binary relation shown in Figure 1 is used in the Context ART (XART). In this arrangement the context for any system is bounded by the systems that are “visible” to the system of interest. As Figure 1 shows, the context for system 1A is the collection of systems, 1B, 1C and 1D.