Technical Report, SC_TR_00023, System Structure ...

Technical Report, SC_TR_00023 System Structure and Behavior Joseph J. Simpson, Mary J. Simpson System Concepts LLC, July 2, 2018

I.

INTRODUCTION

Created by John N. Warfield, structural modeling supports structured group learning. The focus of the group learning activity spans a wide range of topics; however, large-scale organizational problem structures appear to be the most popular topic. Structural modeling has multiple types of process implementations. In this technical report, the authors identify four general types of structural modeling approaches and discuss the similarities and differences between and among them. These general types are: 1. 2. 3. 4.

Classification, Ordering, Behavior, and Everything else.

Each of these types has unique software implementation approaches, algorithms, and processes. Additional adjustment and configuration of these unique processes target either system discovery mode or system design mode activities. To discover an unknown or poorly defined system structure, structural modeling activity starts with no known system structure and moves forward to create a well-defined system structure. Structural modeling uses a group of knowledgeable individuals to effectively explore and discover the system structure of interest. All four general types (classification, ordering, behavior, and ‘everything else’) need an information management system to manage the data and artifacts produced by the group. General type 1 (classification) and type 2 (ordering) require a specialized computer-based logical analysis component to address these specific types of human judgments. General type 3 (behavior) and type 4 (‘everything else’) may be engaged without this type of distinct, computer-based logic unit. With all four types, after creating the initial system structure, logic-based knowledge management software components are a viable means for further analysis and development. This technical report also outlines a conceptual process and software architecture associated with these four general types of structural modeling approaches. Section II, Historical and Current Operational Context, discusses and contrasts the historical modeling context and the current context. Section III, System Structure, discusses structural modeling processes and techniques. Section IV, System Behavior, discusses system dynamics and other frameworks. Section V, Knowledge Management, outlines a conceptual process and software architecture for system information management in this structural modeling domain. Section VI, Summary and Conclusions, presents a summary and short discussion of the document’s content. II.

HISTORICAL AND CURRENT OPERATIONAL CONTEXT

In the late 1950's and early 1960's, Warfield created and designed structural modeling as a group learning process [Warfield, 1959]. The early use of computer-based tools and processes augmented human reasoning and information processing capabilities. The initial structural modeling approach addressed two primary types and sources of information: (1) empirical data and (2) logical inference. The original software algorithms and implementation processes reflected the state of computer hardware and network architectures available in that time period. The original implementations assigned almost all data management to manual, hard-copy-based information repositories.

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Today, sixty plus years after the creation of structural modeling, the ubiquitous availability of on-demand, networked computing capability presents a substantially different environment for the implementation and deployment of structural modeling processes. The current structural modeling operational environment contains vast amounts of digital data distributed across a wide range of document types and storage mechanisms. This high density of ambient data availability, creates a situation wherein computer-based learning algorithms and data analysis techniques can contribute substantially to the preparation before a structural modeling event. Current computer-based technology has the capability to augment the human learning process during a structural modeling event. Updated structural modeling methods and techniques can, and should, fully utilize current knowledge management techniques. III.

SYSTEM STRUCTURE

Creating a system structure is a very efficient way to communicate a vast amount of information. The challenges associated with generating a previously unknown system structure are many and varied. Structural modeling processes and Interpretive Structural Modeling (ISM) methods and techniques assist in overcoming these challenges. A primary feature of structural modeling is the natural language structuring relationship used to associate and organize a collection of unstructured objects into a welldefined system structure. Some structuring relationships cluster objects into identifiable classifications. Other structuring relationships place objects into a prescribed order. Clustering and ordering are the two kinds of system structure supported by the legacy GMU ISM software. Different kinds of discovery and documentation also uncover system structure. Existing structural modeling logical software components directly support natural language structuring relationships that create clusters and/or strict orders. The use of a proper sequence of the existing clustering and strict ordering software components provides indirect support to natural language structuring relationships that create a partial order. It is not clear that other types of natural language structuring relationships can take advantage of the logical operational efficiencies associated with the existing structural modeling software. In fact, the use of the structural modeling logical software component when considering other types of natural language structuring relationships is not clear and is subject to ongoing research. At this time, it appears that the classical logical software component is unnecessary and may well be the source of many factual and conceptual issues if used in areas outside clustering and strict ordering. A structural modeling process begins by collecting empirical data from a selected group of experts. If the natural language relationship is the proper type, then formal logic may infer missing data. Given the proper context, reduction of empirical collection costs occurs. Given the reduction in empirical collection costs, larger-scale system structuring activities are enabled. The authors have addressed the structural modeling process and different types of natural language structuring relationships in previous technical reports. Technical report SC_TR_0011, Local Attribute of the Structuring Relationship, identifies and discusses the logical and context specific characteristics of a natural language structuring relationship in terms of the Augmented Model-Exchange Isomorphism (AMEI) [Simpson and Simpson, July, 2017]. Technical report SC_TR_0012, Defining Structuring Relationships, continues the analysis and discussion of natural language relationships properties and attributes [Simpson and Simpson, August, 2017]. These two reports begin to illuminate the connections between logical property groups and natural language relationships that support the reduction of structural modeling data collection costs. A primary feature of these cost-reducing natural language relationships is notable; they belong in only one of the AMEI logical group categories. A primary feature of those natural language relationships that do not provide ‘logical cost reduction benefits,’ is that those relationships may belong in any one, or all twenty-seven, of the AMEI logical group categories.

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Natural language structuring relationships that only belong in the reflexive-symmetric-transitive (RST) AMEI logical group category, support the clustering (or classification) general type. Natural language structuring relationships that only belong in the irreflexive-asymmetric-transitive (IAT) AMEI logical group category, support the ordering general type. At this time, it appears that all other natural language structuring relationships do not directly support logical inference techniques that reduce the cost of empirical data collection. IV.

SYSTEM BEHAVIOR

Natural language structuring relationships that indicate system behavior (interaction between two system objects) do not appear to provide the logical configuration needed to infer additional structural data about a given unknown system of interest. As listed in Technical Report SC_TR_0012, the following system behavior-based natural language relationships do not provide the required logical inference power. • • • • • • • • •

Causes Affect Aggravate Enhance Support Confirm Weakens Strengthens Influences

System structures developed using these and other similar types of system structuring relationships cannot effectively use the logical inference component of the structural modeling approach to reduce activity cost. System dynamics causal loop diagrams are an example of system structural types that belong in the system behavior-based category. Other components of the structural modeling approach can organize, manage, communicate, and present empirical data associated with a structural modeling activity that uses a system behavior-based natural language structuring relationship. Behavior-based structural modeling provides a direct connection to other types of system science and system thinking methods and processes. Using this direct connection as an architectural guide, the authors believe it is possible to create a modular structural modeling software suite that is configurable. This adaptive software system provides the components necessary to properly address system structure-based relationships as well as system behavior-based relationships. Adaptable software that supports group learning and knowledge development opens the door for system dynamics applications distributable across a wide range of individuals and problem types. The current structural modeling software does not directly support system behavior-based natural language structuring relationships. This is an area rich with opportunity for the system science, system dynamics and system engineering communities. V.

KNOWLEDGE MANAGEMENT

A key benefit of the structural modeling approach is the detailed, optimized processes used to engage large groups of subject matter experts. This knowledge development and management benefit is available to all processes – those that use the original ISM logical inference and those that do not. A Handbook of Interactive Management is an early example of knowledge management associated with structural modeling [Warfield and Cardenas, 2002]. Warfield and Cardenas provided example process templates and

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detailed processes in an effort to create a curated set of repeatable, effective operational methods and techniques. These knowledge management and development techniques provide an excellent guide for the development of a software based, multiuser, distributed system knowledge development and management system. Presently vast quantities of data and information are available in digital, network-based computing systems. These networked resources provide the foundation upon which computer-based learning techniques may be based. Integration of classical system engineering and science techniques such as structural modeling and constraint theory with other knowledge management techniques enhance the effectiveness and application of system engineering and science. The authors are currently exploring the following techniques and areas for potential integration: • • •

Computer-based learning systems, with emphasis on the Theme One program developed by Jon Awbrey. Theme One addresses two problem types: (1) learning arbitrary formal languages and (2) modeling arbitrary formulas in propositional calculus. Union rule configuration fuzzy logic approach, developed by William E. Combs, that focuses on eliminating rule explosion in computer-based reasoning systems. Constraint theory as an approach to mathematical model management, developed by George J. Friedman, which addresses the internal consistency and computational allowability of any given mathematical model.

Online, automated analysis, based on one or more of the above techniques, creates an opportunity for effective, efficient knowledge development and communications. Decision makers can consume information products from the knowledgeable management systems as they prepare for structural modeling events. The authors present an integration of knowledge management and structural modeling through the use of a specific example from Constraint Theory, Multidimensional Mathematical Model Management [Friedman, 2005; Friedman and Phan, 2017]. This example is the same in both the first edition (2005) and the second edition (2017.) In Chapter 1, Motivations, the example discusses the encoding of expert knowledge into a collection of mathematical relations. Each mathematical relation is associated with a specific subject matter expert in the area of system development. The key challenge is the efficient, effective integration of this loose collection of mathematical relations into a focused, integrated system design metric. Figure 1 depicts the example details as reflected in Friedman's “Example of Low Dimension.” Constraint Theory is an interesting endeavor with a range of evaluation tools. The example, as provided in Chapter 1, does have a few issues. The first issue is the uncertainty associated with equation variables not listed in the original variable set (Dmax and x). The second issue is the use of the term “degrees of freedom” for a mathematical model composed of both linear and nonlinear equations. The third issue is the representation of a variable, S, as both a linear equation as well as a nonlinear equation. These issues, combined with a fairly abstract set of terms, operations, and methods, make the precise meaning of the material associated with the example in Chapter 1 elusive. The authors use structural modeling as an alternative analytical and evaluation technique in this case. The system considered for analysis is composed of the model variables. The natural language structuring relationship used to organize the system model is ‘dependent on.’ This natural language relationship has the following three logical attributes: (1) irreflexive, (2) asymmetric, and (3) transitive. This structural modeling method creates a dependency tree structure of the model variables. The use of a method that addresses only the model variables, creates a structured graphic approach that is less involved than the original bipartite graphs from constraint theory that include both variables and relations. The structural model focuses on one kind of object and one natural language structuring relationship and provides a system graphic view that is easy to comprehend and interpret. This application of structural modeling to the

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structure of mathematical relations is different than more classical applications wherein structural modeling techniques apply directly to text-based knowledge statements from subject matter experts.

Figure 1. Friedman’s Example of Low Dimension Applied in a step-wise manner starting with the total systems optimization criterion, T, the structural modeling technique shows the outcome of the first six evaluation steps in Figure 2, Analysis Steps. Step 1 shows that T depends on P, E and C. Step 2 shows that E depends on M, A, D, Dmax and that C depends on D and S. The dependency analysis continues and creates the dependency tree shown in Figure 2. Figure 2 shows which structural threads terminate without a cycle as well as the structural threads that never terminate because of cyclical dependencies. The structure of the dependency tree may yield a wealth of information. However, the information required in this case is immediately apparent directly from Figure 2. The allowable computations are associated with the variables circled in green. The constrained computations are associated with the variables contained in recursive cycles (highlighted in orange.) While the Constraint Theory text indicates that there are only three allowable computations [Friedman and Phan, 2017, pp. 6-7]: 1. T = f9(E,P), 2. T = f10(M,P), and 3. T= f11(A,P), there may be a different interpretation from the graphic in Figure 2. While P, M, A, Dmax and x are terminal variables, E is not a terminal variable and is composed of M, A, D, and Dmax. To say that T = f9(E,P) is

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allowable means that P and some combination of the variables that make up E are allowable computations. Since Dmax and x were not included in the list of variables, it is difficult to understand the precise meaning of T = f9(E,P) is allowable. One alternative evaluation of the allowed computations, based on the analysis steps shown in Figure 2 is: T = f(P, M), T = f(P, A), T = f(P, Dmax), T = f(P, M, A), T = f(P, M, Dmax), T = f(P, A, Dmax.), and T = f(P, M, A, Dmax). The variable D (as a component of variable E) is constrained due to the recursive cycles in the analysis graph.

Figure 2. Analysis Steps Many, if not all, of the unaddressed mathematical issues encountered in the example of ‘Low Dimension’ are not important in the structural modeling natural language analysis. What to do about the ‘missing’ variables? How to apply the concept of ‘degrees of freedom’ to a collection of equations that contain both linear and nonlinear equations? How to handle a variable represented as a linear equation as well as a nonlinear equation? These questions and issues disappear using a natural language relationship to structure the variables from the collection of equations. In this manner, structural modeling bodes well as an alternative validation method associated with the application of Constraint Theory. A computational program like Theme One could contribute to the evaluation of the detailed computational structure of constraint theory models. The

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combination of appropriate structural modeling and knowledge management techniques has the capability to greatly reduce complexity and increase our ability to design effective systems. Development of an open source, modular, web-based, multiuser, configurable, knowledge management system could support the complete range of structural modeling activities. At this time, only the logical inference components for clustering and strict ordering have been prototyped and demonstrated. These independent software components will be refactored into software modules for integration into a larger, adaptable knowledge development and management system. There are a number of areas associated with logical inference and system behavior relationships that need further research and development. This research would benefit from the creation of standard, formal processes for structural modeling. These documented processes would be similar to those in A Handbook of Interactive Management, except that encoding these processes in electronic forms will ensure that processes remain open to user community feedback, analysis, and continuing development. These processes would also include direct links to open source software. Most current structural modeling activities focus on the system discovery mode. The improved, modular structural modeling software suite will support system design processes and methods. The improved capability provides direct support for evolutionary computation, neural networks, and other artificial intelligence techniques. Modular integration of these capabilities allows adaptive process configuration based on information and decisions made previously in the structuring process. VI.

SUMMARY AND CONCLUSIONS

Structure and behavior are primary characteristics of any system. Detailing the structure of an unknown system is an obvious first step in the identification of the unknown system. Structural modeling software that creates clusters or strict orders have been available for multiple decades. When applying this type of software behavior-based system attributes, the process outcome is not reproducible or efficient. System structure based on system behavior needs a new, reproducible, verifiable analytical process. In the example provided, the authors show that structural modeling techniques help validate and elaborate knowledge already encoded in mathematical equations. Structural modeling methods, techniques and supporting software tools should be integrated in a manner that provides open source access to all methods and techniques. Open source tools provide the foundation for effective, verifiable, and adaptive structural modeling techniques.

References 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. K. Dye, Majority rule and erroneous priorities effect (EPE), 1999 a. http://www.futureworlds.eu/wiki/Erroneous_Priorities_Effect https://en.wikipedia.org/wiki/Majority_rule#Erroneous_priorities K.M. Dye, Dye's law of requisite evolution of observations, How People Harness their Collective Wisdom and Power, Christakis, A.N. and Bausch, K. (Eds.), Information Age Publishing, Greenwich, CT, 1999 b, pp.166–169.

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D.B. Fogel, Evolutionary computation: toward a new philosophy of machine intelligence, 3rd edition, IEEE by John Wiley & Sons, Inc, Hoboken, NJ, 2006. G.J. Friedman, Constraint theory, multidimensional mathematical model management, second edition, Springer International Publishing, AG 2005. G.J. Friedman and P. Phan, Constraint theory, multidimensional mathematical model management, second edition, Springer International Publishing, AG, 2017. A.D. Hall III, Metasystems methodology, a new synthesis and unification, Pergamon Press, New York, NY, 1989. P.J. Ossenbruggen, Systems analysis for civil engineers, John Wiley & Sons, New York, NY, 1984. 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, Structural modeling, Technical Report SC_TR_0001, September, 2014. J.J. Simpson and M.J. Simpson, System analysis and identification: objects, relations and clusters, Technical Report SC_TR_0002, October, 2014. DOI: 10.13140/2.1.2821.1201 J.J. Simpson and M.J. Simpson, Objects, relations and clusters for system analysis, Proc 25th Annual INCOSE International Symposium, Seattle, July, 2015. 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, Structural modeling project – overview, Presentation to INCOSE Seattle Metropolitan Chapter of SC_TR_00010, June 21, 2017. J.J. Simpson and M.J. Simpson, Local Attribute of the Structuring Relationship, Technical Report, SC_TR_00011, 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, Defining Structuring Relationships, Technical Report, SC_TR_0012, August, 2017. DOI: 10.13140/RG.2.2.16587.28964 https://www.researchgate.net/publication/319326831_Technical_Report_SC_TR_0012_Defining_Structuring_Relat ionships

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J.J. Simpson and M.J. Simpson, System concepts and theories, Technical Report, SC_TR_00020, March 31, 2017. DOI: 10.13140/RG.2.2.25841.17765 https://www.researchgate.net/publication/324138184_System_Concepts_and_Theories J. Singh, Consumer’s behaviour: cardinal utility analysis, May, 2018 http://www.economicsdiscussion.net/cardinal-utility-analysis/consumers-behaviour-cardinal-utility-analysisexplained-with-diagram/1111 M. Thompson, Pragmatic philosophy of C.S. Peirce, University of Chicago, Chicago, IL, 1953. R.C. Tryon and D.E. Bailey, Cluster analysis, McGraw-Hill Book Company, New York, 1970. J.N. Warfield, Introduction to electronic analog computers, Prentice-Hall, Inc, Englewood Cliffs, N.J., 1959. 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. J.N. Warfield and A.N. Christakis, Dimensionality, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Volume II, pp. 1118-21, 1986.

Appendix A: Definitions Abduction generates theories, conjectures, hypotheses, and explanations not yet verified by induction or deduction [Simpson and Simpson, 2018, March]. Augmented human intelligence is the proper alignment of computer-based resources focused on providing the information humans need, in a format that humans can process, to perform the value analysis tasks at which humans excel. 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]. Deduction is a process of formal, mathematical reasoning that produces a conclusion based on a set of assumptions given as true [Simpson and Simpson, 2018, March]. 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.

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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. Induction generates conclusions based on observations of experimental data [Simpson and Simpson, 2018, March]. Inference reduces uncertainty and develops clear objective views of the world [Simpson and Simpson, 2018, March]. 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]. 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].

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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. A ‘construction rule’ definition; that is, a system is a relationship mapped over a set of objects [Simpson and Simpson, Oct 2014]. The ‘functional rule’ definition of a system is “a constraint on variety,” wherein constraint identifies and defines the system of interest [Simpson and Simpson, 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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