APPENDIX 9
SEVEN WAYS TO PORTRAY COMPLEXITY
(Front Material Only. The complete document
requires 39 pages. Given here are the cover page, the Contents page, and an Abstract)
John N. Warfield
George Mason University
MS 1B2
Fairfax, Virginia 220304444
"sum, ergo cogito"
This document was prepared specifically to supplement and support a presentation at the
George Washington University Notational Engineering Laboratory,
February 28, 1996
© John N. Warfield, 1996
CONTENTS
Abstract 
3 
Seven Ways to Portray Complexity 
4  INTRODUCTION TO COMPLEXITY 

Abstract 
5 
ASchools of Thought About Complexity 
6 
Abstract 
7 
The BehaviorOutcomes Matrix 
8 
Summary Data from Applications on Complexity Measures 
9 
STRUCTURAL FIELDS 

Symbolizing a FourLevel Inclusion Hierarchy (Quad) 
1012 
Options Field for fulfillment of anticipated needs of children
and their families 
1315 
Attributes Field for Analytical Powertrain 
1621 
Problems Field for Analytical Powertrain 
2227 
STRUCTURAL PROFILES 

Options Profile for Planning an Interactive Management Workshop 
28 
ELEMENTRELATION DIAGRAM (based on Friedman Constraint Theory) 

(Photocopy of Page 55 from Friedman Dissertation (UCLA, 1967) 
29 
PARTITION STRUCTURE 

Lattice of Partitions of a threeelement set 
30 
TOTAL INCLUSION STRUCTURE 

Lattice of Subsets of a 3element set 
31 
Lattice of Communication Alternatives 
32 
DELTA CHART 

DELTA Chart of Options Profile Methodology 
33 
ARROWBULLET DIAGRAMS 

Problematique for Industrial Development in the
State of Nuevo Leon, Mexico 
34 
Superimposed Plausibility Structures for Strategic Planning
Purposes in Mexico 
35 
Problematique for Joint Planning and Execution Process (JOPES) 
36 
Problematique for PolicyOriented PhD Research (scores and shading) developed by PhD students at George Mason University 
37 


REFERENCES 
38 
ABSTRACT
SEVEN WAYS TO PORTRAY COMPLEXITY
John N. Warfield
George Mason University
Prose alone is inadequate to portray complexity. Mathematics is often unavailable because mathematical language is restricted to a small percent of the population.
For this reason, language components comprised of integrated prosegraphics representations enjoy unique potential for representing complexity.
Because of the desirability of taking advantage of computers to facilitate the development and production of such integrated representations, it is best if the prosegraphics representations are readily representable in computer algorithms, even if their utility for general communication is limited. Mappings from mathematical formats to graphical formats can often be readily done, although manual modification of graphics for readibility may be necessary.
The following specific graphical representations have proved useful in representing complexity:
 ArrowBullet Diagrams (which are mappable from square binary matrices, and
which correspond to digraphs)
 ElementRelation Diagrams (which are mappable from incidence matrices, and
which correspond to bipartite relations)
 Fields (which are mappable from multiple, square binary matrices, and which
correspond to multiple digraphs)
 Profiles (which correspond to multiple binary vectors, and also correspond to
Boolean spaces)
 Total Inclusion Structures (which correspond to distributive lattices and to
power sets of a given base set)
 Partition Structures (which correspond to the nondistributive lattices of all
partitions of a base set)
 DELTA Charts (which are restricted to use with temporal relationships, and
which sacrifice direct mathematical connections to versatility in
applications)
Each of these will be described briefly (detailed descriptions are given in the References), and at least one example of the use of each in an application will be given. All structures from applications were developed participatively by persons intimately engaged with the relevant issues.
May, 1996
