Technical Report, SC_TR_00018, System ...

Technical Report, SC_TR_00018 System Ordering Software Joseph J. Simpson, Mary J. Simpson System Concepts LLC, January 30, 2018

I.

INTRODUCTION

System modeling software for structural integration modeling continues as the primary theme of this technical report. Structural models produced from a natural language relationship that create an ordering of the system objects is the focus of this document. Structural modeling processes and methods require the use of software-based tools to manage the information and data associated with the inference processes. There is a wide range of existing closed source software applications. The closed source nature of these tools impedes a detailed analysis and evaluation. This technical report uses the basic relationship and data associated with the GMU ISM DOMODEL and the GMU ISM DOPRIOR command examples in Appendix 2 of the Handbook of Interactive Management [Warfield and Cardenas, 2002]. In addition, this report updates and enhances a standard approach to the analysis and evaluation of structural modeling software functions that form the basis for the authors current analysis and development. It is important to note that this report is based on the information and relationship associated with the GMU ISM software DOMODEL and DOPRIOR functions. The DOMODEL function is presented as a general solution approach. The closed source nature of the GMU ISM software limits the evaluation of the DOMODEL function in this report to the specific relationship ‘is heavier than.’ Also included is an evaluation of the DOPRIOR function to support the overall investigation of the three GMU ISM software functions. Section II, Strict Ordering, presents a discussion of the existing strict ordering concepts. Section III, System Concepts Strict Ordering Software, provides an overview and discussion of the current System Concepts strict ordering software applications, including orders that contain clusters. Section IV, Basis for Resource Minimization, discusses the approaches taken to minimize structural modeling resource needs. Section V, Specific SIM Ordering Software Features, presents and discusses the specific features and functions required in the new clustering software applications. Section VI, Summary and Conclusions, presents a summary and short discussion of the document’s content. II.

STRICT ORDERING

The authors have addressed the modeling of system sequential structures over a period of years, mainly in combination with other system structural topics [Simpson, et.al., Mar, 2017; Dec, 2016; July, 2015; June, 2015-1; June, 2015-2]. A strict order is a specific kind of sequential system structure. [Sakharov, 2018] provides a definition of strict order: A relation < is a strict order on a set S if it is 1. Irreflexive: 𝒂𝒂 < 𝒂𝒂 does not hold for any 𝒂𝒂 ∈ 𝑺𝑺. 2. Asymmetric: if 𝒂𝒂 < 𝒃𝒃, then 𝒃𝒃 < 𝒂𝒂 does not hold. 3. Transitive: 𝒂𝒂 < 𝒃𝒃 and 𝒃𝒃 < 𝒄𝒄 implies 𝒂𝒂 < 𝒄𝒄

Note that transitivity and irreflexivity combined imply that if 𝒂𝒂 < 𝒃𝒃 holds, then 𝒃𝒃 < 𝒂𝒂 does not. A strict order is total if, for any 𝒂𝒂, 𝒃𝒃 ∈ 𝑺𝑺, either 𝒂𝒂 < 𝒃𝒃, 𝒃𝒃 < 𝒂𝒂, or 𝒂𝒂 = 𝒃𝒃 Every partial order ≤ induces a strict order 𝒂𝒂 < 𝒃𝒃 : 𝒂𝒂 ≤ 𝒃𝒃 ∧ 𝒂𝒂 ≠ 𝒃𝒃. Every strict order ≤ induces a partial order 𝒂𝒂 ≤ 𝒃𝒃 : 𝒂𝒂 < 𝒃𝒃 ∨ 𝒂𝒂 = 𝒃𝒃. Page 1 of 14

These mathematical relation rules establish the context for the evaluation of natural language relationships that generate an ordered system structure. A strict order induces a partial order and a partial order induces a strict order. The concept of a partial order introduces uncertainty associated with the reflexive and symmetric logical properties. There is a strong connection between a strict order and a partial order in a pure mathematical sense. The empirical activity of structural modeling gathers information from individuals and groups. The act of gathering empirical information eliminates the uncertainty associated with an object’s value. An object is either equal to another object or it is not. The empirical object information places the object into an ordered sequence. A new ordering concept addresses the uncertainty associated with the concepts of strict order and partial order. This new concept focuses on the structural integration realm where the abstract concepts of mathematics align with the empirical, substantive features of real world operations. In essence, this concept orders a collection of classes, containing one or more objects, in a manner similar to a strict ordering. Figure 1 presents an overview of logical relations. System

SM

Concepts

Where ‘ ‘ Means Must Be Included In

Where ‘ ‘ Means May Be Included In

←4

Partial Order

Plus a ‘Selection’ Rule

Antisymmetric StrictTotalOrder

Strict Order

NonSymmetric

Asymmetric

Similarity

Symmetric

←3

Equivalence

NonReflexive

Irreflexive

Partial Equivalence

Reflexive

PreOrder

Transitive

NonTransitive

←2 ←1

Intransitive

Figure 1. Logical Relations. From a functional point of view, there needs to be at least two classes in the collection to be ordered. If there is only one class in the collection, ordering cannot occur. Each class must have one or more objects; an empty class has nothing to order. Given a number of objects to order, the new ordering concept effectively engages and creates the ordered system structure, without uncertainty. This new ordering

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concept eliminates the uncertainty associated with a strict order and partial order by placing the ordering operation and the equality operation at two different levels of aggregation: 1. Equivalence classes formed by organizing equivalent objects into classes, 2. An ordered sequence formed by ordering the equivalence classes. The different sets of logical operators are each applied to different constructs. Equivalence logical operators (reflexive, symmetric and transitive) apply to create the equivalence classes. The ordering logical group (irreflexive, asymmetric and transitive) apply to create the ordered sequence of the equivalence classes. Equivalence and ordering use distinct and separate concepts and operations. This clear distinction eliminates uncertainty and reduces complexity. These concepts form the foundation for the software code creation. III.

SYSTEM CONCEPTS STRICT ORDERING SOFTWARE

Developed as a part of the System Concepts web-based software suite, a structural modeling proof of concept application is available for download at the following link: https://github.com/jjs0sbw/smp_seagl_16. This web application contains two (2) ordering approaches: (1) strict order (one object per class) and (2) strict order (one or more objects per class). Designed to support algorithm development, the existing web applications provided the flexibility to explore various kinds of ordering algorithms. The new ordering web applications will only support a binary search based algorithm. Restricting the new web applications to a binary search based approach reduces the number of available operations. The new web applications will have a different user interface, with different operations and no input text boxes. The next version of the web applications will be deployed in two general application types similar to the existing software configurations. Figures 2 and 3 show the initial screens for the two existing applications.

Figure 2. Initial Screen of “Strict Order – One Object Per Class.”

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Figure 3. Initial Screen of “Strict Order – One Or More Objects Per Class.” Note that a primary difference between these two screens is the “Entries Equal?” question shown in Figure 3. These software applications were developed to support structural modeling algorithm development. Therefore, the user of this software must select the next set of objects for processing by the software as well as the type of processing needed. The next version of the software automatically selects the pair of objects to be addressed, using an algorithm whose primary features are presented in this document. A key aspect of structural modeling is the determination of the structure associated with an unknown system. The unknown system is the object of the structural modeling activity, and is usually a large-scale social or social-technical system of some type. Two simple ordering examples facilitate the demonstration of the structural modeling approach. Appendix 2, “GMU ISM SOFTWARE,” in Handbook of Interactive Management [Warfield, 2002] contains these examples. The first example is based on the DOMODEL command from section A2.2 of the handbook. This section of the handbook evaluates seven items. Here we expand the size of the group of items to fifteen (15.) Using fifteen items aligns the graphical matrix with other examples that address fifteen objects. The fifteen objects are: (1) feather, (2) Mack truck (semi tractor), (3) beer can, (4) Volkswagen bug, (5) small boy, (6) professional wrestler, (7) Universe, (8) molecule of water, (9) dining room chair, (10) the Milky Way, (11) Draft horse, (12) the solar system, (13) dinner plate, (14) pickup truck, and (15) train locomotive. These items use the natural language structuring relationship, ‘is lighter than’ rather than the ‘is heavier than’ structuring relationship used in the handbook. Either relationship will create the same structure. In this case, we selected the relationship that meets our operational goals. A. Strict Order – One Object Per Class The existing Strict Order – One Object Per Class, demonstrates a ‘step by step’ algorithm used to order the items by weight. Figure 2 shows the initial screen. Unordered items are assigned a numbered sequence at the beginning of the process as shown in Table 1. Step one (1) is to gather empirical information to determine if a feather is lighter than a Mack truck. Figure 4 shows the application screen after the empirical data entry. From empirical data, it is determined that a feather is lighter than a Mack truck. Since a feather is item one and a Mack truck is item two, the matrix cell [1,2] (column 1 and row 2) has a green background and the number one (1). The natural language for this matrix cell configuration is “A feather is lighter than a Mack truck.” Matrix cell [2,1] (column 2 and row 1) has a red background and the number zero. This matrix cell configuration means, “A Mack truck is not lighter than a feather.” The matrix now has two

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sections: (1) an ordered section, containing items one and two; and (2) an unordered section containing items three (3) through fifteen (15). Table 1. Ordered and unordered items of Figure 4.

Ordered Items feather Mack truck Unordered Items beer can Volkswagen bug small boy professional wrestler Universe molecule of water dining room chair the Milky Way Draft horse the solar system dinner plate pickup truck train locomotive

Assigned number 1 2 Assigned number 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 4. Result after First Empirical Data Entry Item three (beer can) is the next object in the unordered section of the matrix. Item three needs to be placed in the proper area of the ordered section of the matrix. Empirical information is gathered to determine the correct ordered position of item three. The goal is to minimize the empirical information needed to properly order the system. The use of binary search techniques minimizes the necessary information. In the given case, there are only two ordered items. Either item will serve as an acceptable comparison target. However, the goal of this exercise is to create an algorithm to automatically select the elements for evaluation. The algorithm must have two basic parts: 1. A rule to select the unordered item, and 2. A rule to select the ordered item. Part one of the algorithm is a simple rule that directs the system to select the unordered item at the top of the stack - in this case item three. Part two of the algorithm is a basic binary search that selects elements from the ordered section for evaluation against the item from the unordered section. The rule needs to know the length of the ordered matrix section - in this case it is two (two items are in the ordered matrix section). Then the length of the ordered section is divided by two, resulting in an outcome of one. Add a one (1) to the outcome of this division to get the initial ordered item number two. As a result, the next pair of objects to evaluate are items two and three: “Is a beer can lighter than a Mack truck?” Empirical data indicates that this is true. Figure 5 shows the items in the ordered section placed in the proper sequence.

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Table 2. Ordered and unordered items of Figure 5.

Ordered Items feather beer can Mack truck Unordered Items Volkswagen bug small boy professional wrestler Universe molecule of water dining room chair the Milky Way Draft horse the solar system dinner plate pickup truck train locomotive

Assigned number 1 3 2 4 5 6 7 8 9 10 11 12 13 14 15

Figure 5. Result of initial application of the binary search algorithm.

Additional empirical information is collected. Figure 6 shows the matrix configuration after completing the ordering of the first nine items. The new web applications will have a different user interface, with different operations and no input text boxes. The current case used thirteen (13) questions to properly order nine (9) items. Table 3. Ordered and unordered items of Figure 6. Ordered Items molecule of water feather small boy beer can dining room chair professional wrestler Volkswagen bug Mack truck Universe Unordered Items the Milky Way Draft horse the solar system dinner plate pickup truck train locomotive

Assigned number 8 1 5 3 9 6 4 2 7 10 11 12 13 14 15

Figure 6. Matrix Reflects Nine Objects Prioritized.

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A strict order with only one object per category has a key operational advantage. When large groups of people are engaged in the evaluation process, it is a challenge to find and articulate specific distinctions among objects. The use of one object per category forces these distinctions to be made. B. Strict Order – One Or More Objects Per Class The existing “Strict Order – One Or More Objects Per Class” web application is used next to execute the DOPRIOR structuring process. In this case of ‘The Handbook,’ the natural language structuring relationship is: “Is Element A of equal or higher priority in organizing our discussion agenda than Element B?” The referenced publication included ten (10) items for prioritization – shown in Table 4. An additional five (5) items are added to the list to make the example matrix the same size as previous examples Table 4. Prioritization Items Lack of adequate and reliable community communication Motivation and consistency of employees Maintenance of self-worth of the members Revival of lost tribal culture and traditions Pride of tribal participation Fast growth of the tribe Work load increase without additional personnel Lack of dedication of our younger generation Survival for the tribe Lack of recreation facilities Added Items Priority A Priority B Priority C Priority D Priority E

Assigned number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 7 reflects the initial steps of the algorithm.

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Table 5. Ordered and unordered items of Figure 7. Ordered Items Lack of adequate … Maintenance of … Unordered Items Pride of tribal … Fast growth of … Work load increase … Lack of dedication … Survival for the … Lack of recreation … Priority A Priority B Priority C Priority D Priority E

Assigned number 1 3 Assigned number 5 6 7 8 9 10 11 12 13 14 15

Figure 7. Matrix Reflects Initial steps of Algorithm. As shown in Figure 7, items one (1) and two (2) are equal. Items three (3) and four (4) are equal as well, and item one (1) has a higher priority than item three (3). The empirical data gathering continues until the first ten (10) items are properly prioritized, as shown in Figure 8. This prioritization process required twelve (12) questions to be answered. Table 6. Ordered and unordered items of Figure 8. Ordered Items Lack of adequate … Maintenance of … Pride of tribal … Fast growth of … Work load increase … Lack of dedication … Lack of recreation … Unordered Items Priority A Priority B Priority C Priority D Priority E

Assigned number 1 3 5 6 7 8 10 Assigned number 11 12 13 14 15

Figure 8. Matrix Reflects First Ten Objects Prioritized

An approach that uses a strict order with one or more objects per category combines the resource utilization efficiencies of both binary search and single category clustering. The natural language structuring

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relationship logical attributes must align with the selected modeling approach. These logical attributes must align between the structural integration modeling activity and the ISM (real world) activities. This alignment is similar to the alignment of the structural integration modeling methods (SIM ordering) and the mathematics area. IV.

BASIS FOR RESOURCE MINIMIZATION

Resource minimization occurs in two basic ways. The first basic minimization technique is process complexity reduction. A clearly articulated analysis and evaluation process results in reduced efforts to communicate required activities among the project team members. The new type of ordering documented in this and multiple technical reports related to Structural Integration Modeling is called SIM Order. A SIM Order consists of a group of two or more categories, with each category containing one or more objects. A SIM Order eliminates the confusion associated with the concepts of an ordered set and a partially ordered set. As detailed earlier, a SIM Order does not order the elements of a set, it orders two or more categories present in a collection. Each category must contain one or more objects, these objects are not elements in a set. The logical properties of irreflexive, asymmetric and transitive are assigned to a SIM Order. The logical properties of reflexive, symmetric and transitive are assigned to the collection in each category. These are clear, distinct concepts that are not part of, or induced upon each other. This clarity reduces uncertainty, and therefore complexity and resource consumption. The second basic way that resource minimization occurs is associated with efficient structuring and ordering methods. In Technical Report 16, Clustering Software, the most efficient method of clustering was associated with the case where all the elements were the same [Simpson and Simpson, Dec, 2017]. This, and other System Concepts research publications show that the binary search ordering algorithm is very efficient. In addition, using a clustering technique to create categories for ordering increases the potential to minimize the required resources. Structural integration modeling techniques that use a SIM Order, and a clustering technique to create the categories to be ordered, reduce resource consumption. V.

SPECIFIC SIM ORDERING SOFTWARE FEATURES

The next version of the structural modeling ordering software will consist of two applications. The first application supports a SIM Order with only one object per category. The second application supports a SIM Order with one or more objects per category. Each new web application will use a ‘binary search’ based algorithm to select two objects for evaluation. The application that has only one object per category presents the user with two radio buttons: one marked YES and one marked NO. The user will select the applicable radio button to input the correct answer into the software application. The application that allows one or more objects per category will have three (3) radio buttons: one marked YES, one marked NO, and one marked SAME. The SAME radio button indicates that the objects are in the same category as each other. This new software application does not need to provide 'enter data', 'swap entries', or 'infer information' functions. Removal of these functions will decrease the software size and complexity. The modified application software architecture will contain redesigned and rewritten software modules. VI.

SUMMARY AND CONCLUSIONS

SIM ordering introduces a new type of system structural ordering as a part of systems integration modeling. This new ordering approach reduces resource consumption in two basic ways: (1) reducing complexity and

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(2) using ‘binary search’ based algorithms. Clarity associated with the definition of two distinct aspects of structural ordering reduces complexity and increases communication potential. Using ‘binary search’ based algorithms makes a direct connection between system science structural modeling techniques and basic computer science techniques. These connections highlight common areas of practice and common algorithms. Common areas allow a large group of individuals, with a technical background, to make further contributions to the methods and techniques used in structural integration modeling. Implementation of the ‘binary search’ based structural modeling approach requires new structural modeling software applications. The existing software applications support general algorithm development, not binary search algorithms specifically. The new software applications are the focus of the next few technical reports from System Concepts. Similar to the effort to further define and clarify the role of mathematics in structural integration modeling, the effort to define natural language attributes and properties is focused on the natural language description of the real world activity of interest. Figure 9 presents logical mathematical aspects of structural integration modeling that must align with the natural language properties.

stem Concepts LLC

Figure 9. Logical Attributes of Natural Language These natural language concepts need further clarification and testing. The new System Concepts web applications are designed to support these activities.

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References K.C. Bausch, With reason and vision, structured dialogic design, Ongoing Emergence Press, Cincinnati, Ohio, 2015. F.M. Brown, Boolean reasoning, the logic of boolean equations, second edition, Dover Publications, Inc, Mineola, NY, 2003. K.E. Boulding, General systems theory - the skeleton of science, Management Science, Linthicum, Maryland, April, 1956, 2: 197-208. A.D. Hall III, Metasystems methodology, a new synthesis and unification, Pergamon Press, New York, NY, 1989. Sakharov, Alex. "Strict Order." From MathWorld--A Wolfram Web Resource Jan 10, 2018, created by Eric W. Weisstein. http://mathworld.wolfram.com/StrictOrder.html J.J. Simpson, C. Dagli, A. Miller, System evaluation and description using abstract relation types (ART), Proc 1st Annual 2007 IEEE Systems Conference, Honolulu, Hawaii, April, 2007. J.J. Simpson and M.J. Simpson, System integration frameworks, Proc Fifteenth Annu IS of INCOSE, "Systems Engineering: Bridging Industry, Government, and Academia", Rochester, New York, July, 2005. J.J. Simpson and M.J. Simpson, Formal systems concepts, Proc Fourth Annu Conf Syst Eng Res, Los Angeles, April, 2006. J.J. Simpson and M.J. Simpson, A pragmatic complexity framework, Proc INCOSE Spring 2009 Conference, "Systems Engineering - Affordable Success", Suffolk, Virginia, April, 2009. J.J. Simpson and M.J. Simpson, Complexity reduction: a pragmatic approach, Systems Engineering Journal, Chicago, Illinois,Volume 14, Issue 2, June, 2011. J.J. Simpson and M.J. Simpson, Structural modeling, Technical Report SC_TR_0001, September, 2014. J.J. Simpson and M.J. Simpson, Augmented model-exchange isomorphism Version 1.1, Data, February, 2015. DOI: 10.13140/2.1.3948.6241 J.J. Simpson and M.J. Simpson, Objects, relations and clusters for system analysis, Proc 25th Annual INCOSE International Symposium (IS2015), Seattle, WA June, 2015 DOI: 10.13140/RG.2.1.4654.7047 https://www.researchgate.net/publication/279205089_Objects_Relations_and_Clusters_for_System_Analysis J.J. Simpson and M.J. Simpson, Structural modeling framework, Proc 25th Annual INCOSE International Symposium (IS2015), Seattle, WA June, 2015 DOI: 10.13140/RG.2.1.2033.2647 https://www.researchgate.net/publication/279204981_Structural_Modeling_Framework J.J. Simpson and M.J. Simpson, Foundational aspects of system complexity reduction, Proc 25th Annual INCOSE International Symposium (IS2015), Seattle, WA July, 2015. J.J. Simpson and M.J. Simpson, Structural integration modeling – strengthening the foundations of systems science and systems engineering, Technical Report SC_TR_0004, December, 2016. DOI: 10.13140/RG.2.2.27173.29924 https://www.researchgate.net/publication/311984805_Structural_Integration_Modeling_Strengthening_the_Foundat ions_of_Systems_Science_and_Systems_Engineering_--_Technical_Report_SC_TR_0004

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J.J. Simpson, M.J. Simpson, and T.B. Kercheval, Threshold metric for mapping natural language relationships among objects, Proc Conference on Systems Engineering Research, "Disciplinary Convergence: Implications for Systems Engineering Research", Redondo Beach, California, March, 2017. J.J. Simpson and M.J. Simpson, Structural modeling project – overview, Technical Report, SC_TR_0010, June, 2017. DOI: 10.13140/RG.2.2.29542.63041 https://www.researchgate.net/publication/317954151_Technical_Report_SC_TR_0010_Structural_Modeling_Projec t_-_Overview J.J. Simpson and M.J. Simpson, Local Attribute of the Structuring Relationship, Technical Report, SC_TR_0011, July, 2017. DOI: 10.13140/RG.2.2.10979.86565 https://www.researchgate.net/publication/318583668_Technical_Report_SC_TR_0011_Local_Attribute_of_the_Str ucturing_Relationship J.J. Simpson and M.J. Simpson, The Problematique Application Structural Type, Technical Report, SC_TR_0013, October 3, 2017. DOI: 10.13140/RG.2.2.16159.66723 https://www.researchgate.net/publication/320194953_Technical_Report_SC_TR_0013_The_Problematique_Applic ation_Structural_Type J.J. Simpson and M.J. Simpson, The Interconnection Matrix, Technical Report, SC_TR_00014, October 22, 2017. DOI: 10.13140/RG.2.2.21348.07047 https://www.researchgate.net/publication/320557156_The_Interconnection_Matrix J.J. Simpson and M.J. Simpson, Structural modeling software, Technical Report, SC_TR_00015, November 26, 2017. DOI: 10.13140/RG.2.2.12089.47203 https://www.researchgate.net/publication/321304816_Structural_Modeling_Software J.J. Simpson and M.J. Simpson, Clustering software, Technical Report, SC_TR_00016, December 30, 2017. DOI: 10.13140/RG.2.2.11046.63041 https://www.researchgate.net/publication/322152428_Technical_Report_SC_TR_00016_Clustering_Software C. Spanoude, ISM algorithm, Working paper for structural modeling working group, Cyprus, June, 2015. J.N. Warfield, Binary matrices in system modeling, IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-3, No. 5, September, 1973. J.N. Warfield, Structuring complex systems, monograph no. 4, Battelle Memorial Institute, Columbus, Ohio, 1974. J.N. Warfield, Societal systems: planning policy and complexity, John Wiley & Sons, Inc., New York, NY 1976. J.N. Warfield, A science of generic design, managing complexity through system design, second edition, Iowa State University, Ames, 1994. J.N. Warfield and A.R. Cardenas, A handbook of interactive management, second edition, AJAR Publishing Company, Palm Harbor, FL, 2002. Weinberg, G.M., An introduction to general systems thinking, Dorset House Publishing, New York, NY, 2001, p 18.

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Appendix A: Definitions Boolean reasoning is based on Boolean equations, and not on the predicate calculus. Boolean reasoning is based on the Blake canonical form, and syllogistic reasoning. The Boolean reasoning used in this paper is similar to, but different than, switching theory or Boolean minimization approaches [Brown, 2003]. Warfield’s use of a zero (0) to represent either the formal logical notion of false (mathematical concept) or the empirical real-world state of unknown (empirical knowledge), is a key operation that integrates the operations of the system ‘metalanguage’ (natural language relationships) and system ‘object language’ (mathematical relations). Complexity is defined as the measure of the difficulty, effort and/or resources required for one system to effectively observe, communicate, and/or interoperate with another system [Simpson and Simpson, 2009, p. 2]. Degenerative structural thread is defined as a malformed or degenerative structural thread that is composed of a single object. Given a single object there is no other object to support a relationship connection. Degree of focus: A relationship’s ‘degree of focus’ is a numerical value that indicates how many AMEI categories are associated with any given natural language structural relationship. The minimum dispersion value is two (2) and the maximum dispersion value is twenty-seven (27) [Simpson and Simpson, August, 2017]. Dispersed natural language structuring relationship attribute: A dispersed natural language structuring relationship is dispersed if it maps to more than one logical property group within the AMEI. The dispersed attribute’s numerical values range from two (2) groups to twenty-seven (27) groups. This numerical value provides a dispersion metric that describes the ‘degree of focus.’ [Simpson and Simpson, August, 2017]. Global attribute of a system structuring relationship structures a system using a mediating artifact between and among the system objects [Simpson and Simpson, June, 2017, p. 10] Identity matrix is a square matrix that has a one (1) in each cell on the matrix diagonal. Local attribute of a system structuring relationship operates directly between and among the system objects without a mediating artifact [Simpson and Simpson, June, 2017, p. 10]. Mathematical model relations are the formal mathematical constructs used to represent the natural language structuring relationships in a well-defined formal manner. Mathematical relations focus mainly on the relations of sets, and set members. Warfield built on Hilbert’s ‘language pair’ concept of metalanguage and object language to view mathematical relations as the ‘object language,’ while viewing natural language relationships as the ‘metalanguage’ [Warfield, 1994:47]. Natural language relationship is a term used in human conversation and contextual discourse to indicate some type of order, structure or other manner in which two or more objects are associated between and among themselves. A natural language relationship conveys substantive real-world knowledge, and is an interpretive relationship. Warfield identified six categories of interpretive relationships: 1) definitive, 2) comparative, 3) influence, 4) temporal, 5) spatial, and 6) mathematical [Warfield, 1994:60-61].

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Order: If a binary relation, R, is reflexive and transitive, and R and the complement of R, are antisymmetric, then R is called an order [Simpson and Simpson, 2014]. Partial Order: If the complement of R is not antisymmetric and all other conditions are met, then R is called a partial order [Simpson and Simpson, 2014]. Reachability matrix: A reachability matrix is a square, transitive, reflexive, binary matrix (M), which serves as a model matrix for a matrix model whose model relation is 'is antecedent to’ [Warfield, 1976, p. 231]. Reflexivity: Reflexivity involves one individual object. The logical properties constituent to the Reflexivity grouping are the reflexive, irreflexive, and nonreflexive property. If a relation is reflexive, then an object bears this relation to itself (xRx). An irreflexive relation states that no object bears this relation to itself (x'Rx). The nonreflexive logical property is a composite property, which states that in a set of objects, some objects are reflexive and some objects are irreflexive [Simpson, Simpson, and Kercheval, Mar 2017]. Specific natural language structuring relationship attribute: A specific natural language structuring relationship is specific if it maps to only one logical property group within the AMEI [Simpson and Simpson, August, 2017]. Structural thread: Artifacts created on structural graphs when two or more objects are connected using a relationship link; a term used to identify a pattern of relationship connections in a structural graph. [Simpson and Simpson, June, 2017, p. 11] Symmetry: Symmetry involves two individual objects. The symmetric, asymmetric and nonsymmetric logical properties belong to the Symmetry grouping. A symmetric relation requires that if object x bears a relation to object y, then object y also bears a relation to object x ((if xRy, then yRx) and (x != y)). An asymmetric relation states that if object x bears a relation to object y, then object y does not bear a relation to object x ((if xRy, then y'Rx) and (x != y)). The nonsymmetric logical property is a composite property and can only exist when a set of objects have both symmetric and asymmetric relations mapped among them [Simpson, Simpson, and Kercheval, Mar 2017]. System may be defined in a number of ways. This paper uses a ‘construction rule’ definition; that is, a system is a set of two or more objects with a structural relationship (or relationships) mapped over the object set [Simpson and Simpson, April, 2006]. Transitivity: Transitivity involves three or more individual objects. Transitive, intransitive, and nontransitive relations all belong to the transitivity grouping. Transitive relations state that if object x bears a relation to object y and object y bears a relation to object z, then object x also bears a relation to object z ((if (xRy and yRz), then xRz) and (x != y != z)). Intransitive relations state that if object x bears a relation to object y and object y bears a relation to object z, then object x does not bear a relation to object z ((if (xRy and yRz), then x'Rz) and (x != y != z)). The nontransitive logical property is a composite property and may only exist where a set of objects have both transitive and intransitive relations mapped among them [Simpson, Simpson, and Kercheval, Mar 2017].

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