# A Tensor-Based Formulation of
Hetero-functional Graph Theory

Amro M. Farid, Dakota Thompson, Wester Schoonenberg Amro M. Farid is an
Associate Professor of Engineering with the Thayer School of Engineering at
Dartmouth and a Visiting Associate Professor with the mechanical engineering
department at MIT, Cambridge, MA, USA<EMAIL_ADDRESS>Thompson is with
the Thayer School of Engineering at Dartmouth, Hanover, NH, USA.
<EMAIL_ADDRESS>C. Schoonenberg is with the Thayer
School of Engineering at Dartmouth, Hanover, NH, USA.
<EMAIL_ADDRESS>
###### Abstract

Recently, hetero-functional graph theory (HFGT) has developed as a means to
mathematically model the structure of large-scale complex flexible engineering
systems. It does so by fusing concepts from network science and model-based
systems engineering (MBSE). For the former, it utilizes multiple graph-based
data structures to support a matrix-based quantitative analysis. For the
latter, HFGT inherits the heterogeneity of conceptual and ontological
constructs found in model-based systems engineering including system form,
system function, and system concept. These diverse conceptual constructs
indicate multi-dimensional rather than two-dimensional relationships. This
paper provides the first tensor-based treatment of hetero-functional graph
theory. In particular, it addresses the “system concept” and the hetero-
functional adjacency matrix from the perspective of tensors and introduces the
hetero-functional incidence tensor as a new data structure. The tensor-based
formulation described in this work makes a stronger tie between HFGT and its
ontological foundations in MBSE. Finally, the tensor-based formulation
facilitates several analytical results that provide an understanding of the
relationships between HFGT and multi-layer networks.

##  I Introduction

One defining characteristic of twenty-first century engineering challenges is
the breadth of their scope. The National Academy of Engineering (NAE) has
identified 14 “game-changing goals”[1].

  1. 1.

Advance personalized learning

  2. 2.

Make solar energy economical

  3. 3.

Enhance virtual reality

  4. 4.

Reverse-engineer the brain

  5. 5.

Engineer better medicines

  6. 6.

Advance health informatics

  7. 7.

Restore and improve urban infrastructure

  8. 8.

Secure cyber-space

  9. 9.

Provide access to clean water

  10. 10.

Provide energy from fusion

  11. 11.

Prevent nuclear terror

  12. 12.

Manage the nitrogen cycle

  13. 13.

Develop carbon sequestration methods

  14. 14.

Engineer the tools of scientific discovery

At first glance, each of these aspirational engineering goals is so large and
complex in its own right that it might seem entirely intractable. However, and
quite fortunately, the developing consensus across a number of STEM (science,
technology, engineering, and mathematics) fields is that each of these goals
is characterized by an “engineering system” that is analyzed and re-
synthesized using a meta-problem-solving skill set[2].

######  Definition 1:

Engineering system[3]: A class of systems characterized by a high degree of
technical complexity, social intricacy, and elaborate processes, aimed at
fulfilling important functions in society.

TABLE I: A Classification of Engineering Systems by Function and Operand[3] Function/Operand |  Living Organisms |  Matter |  Energy |  Information |  Money
---|---|---|---|---|---
Transform |  Hospital |  Blast Furnace |  Engine, electric motor |  Analytic engine, calculator |  Bureau of Printing & Engraving
Transport |  Car, Airplane, Train |  Truck, train, car, airplane |  Electricity grid |  Cables, radio, telephone, and internet |  Banking Fedwire and Swift transfer systems
Store |  Farm, Apartment Complex |  Warehouse |  Battery, flywheel, capacitor |  Magnetic tape & disk, book |  U.S. Buillon Repository (Fort Knox)
Exchange |  Cattle auction, (illegal) human trafficking |  eBay trading system |  Energy market |  World Wide Web, Wikipedia |  London Stock Exchange
Control |  U.S. Constitution & laws |  National Highway Traffic Safety Administration |  Nuclear Regulatory Commission |  Internet engineering task force |  United States Federal Reserve

The challenge of _convergence_ towards _abstract_ and _consistent_
methodological foundations for engineering systems is formidable. Consider the
engineering systems taxonomy presented in Table I[3]. It classifies
engineering systems by five generic functions that fulfill human needs: 1.)
transform 2.) transport 3.) store, 4.) exchange, and 5.) control. On another
axis, it classifies them by their operands: 1.) living organisms (including
people), 2.) matter, 3.) energy, 4.) information, 5.) money. This
classification presents a broad array of application domains that must be
consistently treated. Furthermore, these engineering systems are at various
stages of development and will continue to be so for decades, if not
centuries. And so the study of engineering systems must equally support design
synthesis, analysis, and re-synthesis while supporting innovation; be it
incremental or disruptive.

###  I-A Background Literature

Two fields in particular have attempted to traverse this convergence
challenge: systems engineering and network science. Systems engineering, and
more recently model-based systems engineering (MBSE), has developed as a
practical and interdisciplinary engineering discipline that enables the
successful realization of complex systems from concept, through design, to
full implementation[4]. It equips the engineer with methods and tools to
handle systems of ever-greater complexity arising from greater interactions
within these systems or from the expanding heterogeneity they demonstrate in
their structure and function. Despite its many accomplishments, model-based
systems engineering still relies on graphical modeling languages that provide
limited quantitative insight (on their own)[5, 6, 7].

In contrast, network science has developed to quantitatively analyze networks
that appear in a wide variety of engineering systems. And yet, despite its
methodological developments in multi-layer networks, network science has often
been unable to address the explicit heterogeneity often encountered in
engineering systems[7, 8]. In a recent comprehensive review Kivela et. al [8]
write: “The study of multi-layer networks $\ldots$ has become extremely
popular. Most real and engineered systems include multiple subsystems and
layers of connectivity and developing a deep understanding of multi-layer
systems necessitates generalizing ‘traditional’ graph theory. Ignoring such
information can yield misleading results, so new tools need to be developed.
One can have a lot of fun studying ‘bigger and better’ versions of the
diagnostics, models and dynamical processes that we know and presumably love –
and it is very important to do so but the new ‘degrees of freedom’ in multi-
layer systems also yield new phenomena that cannot occur in single-layer
systems. Moreover, the increasing availability of empirical data for
fundamentally multi-layer systems amidst the current data deluge also makes it
possible to develop and validate increasingly general frameworks for the study
of networks.

$\ldots$ Numerous similar ideas have been developed in parallel, and the
literature on multi-layer networks has rapidly become extremely messy. Despite
a wealth of antecedent ideas in subjects like sociology and engineering, many
aspects of the theory of multi-layer networks remain immature, and the rapid
onslaught of papers on various types of multilayer networks necessitates an
attempt to unify the various disparate threads and to discern their
similarities and differences in as precise a manner as possible.

$\ldots$ [The multi-layer network community] has produced an equally immense
explosion of disparate terminology, and the lack of consensus (or even
generally accepted) set of terminology and mathematical framework for studying
is extremely problematic.”

In many ways, the parallel developments of the model-based systems engineering
and network science communities intellectually converge in _hetero-functional
graph theory_ (HFGT)[7]. For the former, it utilizes multiple graph-based data
structures to support a matrix-based quantitative analysis. For the latter,
HFGT inherits the heterogeneity of conceptual and ontological constructs found
in model-based systems engineering including system form, system function, and
system concept. More specifically, the explicit treatment of function and
operand facilitates a structural understanding of the diversity of engineering
systems found in Table I. Although not named as such originally, the first
works on HFGT appeared as early as 2006-2008[9, 10, 11, 12]. Since then, HFGT
has become multiply established and demonstrated cross-domain
applicability[13, 7]; culminating in the recent consolidating text[7].

The primary benefit of HFGT, relative to multi-layer networks, is the broad
extent of its ontological elements and associated mathematical models[7]. In
their recent review, Kivela et. al showed that _all_ of the reviewed works
have exhibited at least one of the following modeling constraints[8]:

  1. 1.

Alignment of nodes between layers is required [8, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62]

  2. 2.

Disjointment between layers is required [8, 63, 64, 56, 65, 66, 67, 68, 51,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80]

  3. 3.

Equal number of nodes for all layers is required [8, 25, 38, 44, 19, 34, 35,
36, 18, 31, 32, 33, 65, 26, 43, 40, 22, 14, 15, 67, 47, 81, 27, 51, 20, 45,
39, 41, 82, 29, 71, 48, 72, 59, 61, 42, 30, 17, 24, 50, 23, 21, 46, 28, 37,
49, 16]

  4. 4.

Exclusively vertical coupling between all layers is required [8, 25, 38, 44,
19, 34, 35, 36, 18, 31, 32, 33, 26, 43, 66, 40, 83, 22, 84, 14, 15, 47, 27,
51, 20, 45, 39, 41, 82, 29, 48, 59, 61, 42, 30, 17, 24, 50, 23, 21, 46, 28,
37, 49, 85, 16, 86]

  5. 5.

Equal couplings between all layers are required[8, 25, 38, 19, 34, 35, 36, 18,
31, 32, 33, 26, 66, 40, 83, 22, 84, 47, 27, 51, 20, 45, 39, 41, 82, 29, 48,
59, 61, 30, 17, 24, 50, 23, 21, 46, 28, 37, 49, 85, 16, 86]

  6. 6.

Node counterparts are coupled between all layers[25, 38, 19, 34, 35, 36, 18,
31, 32, 33, 26, 66, 40, 83, 84, 47, 27, 51, 20, 45, 39, 41, 82, 29, 48, 59,
61, 30, 17, 24, 50, 46, 28, 37, 49, 85, 16, 86]

  7. 7.

Limited number of modelled layers[63, 64, 56, 65, 66, 87, 88, 83, 84, 89, 67,
68, 47, 81, 51, 69, 82, 70, 71, 48, 72, 59, 61, 73, 74, 75, 76, 90, 91, 92,
93, 77, 78, 49, 85, 86, 79, 80]

  8. 8.

Limited number of aspects in a layer[63, 25, 38, 64, 44, 19, 34, 35, 36, 18,
31, 32, 33, 56, 65, 26, 43, 66, 40, 22, 14, 15, 67, 68, 47, 27, 51, 20, 45,
39, 41, 69, 82, 70, 29, 71, 48, 72, 59, 61, 42, 30, 17, 73, 24, 50, 74, 75,
23, 21, 46, 28, 76, 37, 77, 78, 49, 16, 79, 80]

To demonstrate the consequences of these modeling limitations, the HFGT
text[7] developed a very small, but highly heterogeneous, hypothetical test
case system that exhibited _all eight_ of the modeling limitations identified
by Kivela et. al. Consequently, none of the multi-layer network models
identified by Kivela et. al. would be able to model such a hypothetical test
case. In contrast, a complete HFGT analysis of this hypothetical test case was
demonstrated in the aforementioned text[7].

Figure 1: A Topological Visualizaiton of the Trimetrica Smart City
Infrastructure Test Case[7].

The same text provides the even more complex hypothetical smart city
infrastructure example shown in Fig. 1. It not only includes an electric power
system, water distribution system, and electrified transportation system but
it also makes very fine distinctions in the functionality of its component
elements.

Given the quickly developing _“disparate terminology and the lack of
consensus”_ , Kivela et. al.’s[8] stated goal _“to unify the various disparate
threads and to discern their similarities and differences in as precise a
manner as possible”_ appears imperative. While many may think that the
development of mathematical models is subjective, in reality, ontological
science presents a robust methodological foundation. As briefly explained in
Appendix -A, and as detailed elsewhere[7, 94, 95], the process of developing a
mathematical model of a given (engineering) system is never direct. Rather, a
specific engineering system (which is an instance of a class of systems) has
abstract elements in the mind111It is likely that modeling abstract elements
in the mind is unfamiliar to this journal’s readership. This is purely an
issue of nomenclature. Most physicists and engineers would agree on the
indispensable role that _intuition_ – itself a mental model – has to the
development of mathematical models of systems. For example, the shift from
Newtonian mechanics to Einstein’s relativity constituted first an expansion in
the abstract elements of the mental model and their relationships well before
that mental model could be translated into its associated mathematics.
Similarly, the “disparate terminology and lack of consensus” identified by
Kivela et. al [8] suggests that a reconciliation of this abstract mental model
is required (See Section -A). that constitute an _abstraction_ $\cal A$ (which
is an instance of a domain conceptualization ${\cal C}$). ${\cal C}$ is mapped
to a set of primitive mathematical elements called a language $\cal L$, which
is in turn instantiated to produce a mathematical model ${\cal M}$. The
fidelity of the mathematical model with respect to an abstraction is
determined by the four complementary linguistic properties shown in Figure
2[95]: soundness, completeness, lucidity, and laconicity [96] (See Defns. 24,
25, 26, 27). When all four properties are met, the abstraction and the
mathematical model have an isomorphic (one-to-one) mapping and faithfully
represent each other. For example, the network science and graph theory
literature assume an abstract conceptualization of nodes and edges prior to
defining their 1-to-1 mathematical counterparts[7]. Consequently, as hetero-
functional graph and multi-layer network models of engineering systems are
developed, there is a need to reconcile both the abstraction and the
mathematical model on the basis of the four criteria identified above (See
Appendix -A.).

Fig. 2: Graphical Representation of Four Ontological Properties As Mapping
Between Abstraction and Model: a Soundness, b Completeness, c Lucidity, and d
Laconicity [95].

The ontological strength of hetero-functional graph theory comes from the
“systems thinking” foundations in the model-based systems engineering
literature[97, 7]. In effect, and very briefly, all systems have a “subject +
verb + operand” form where the system form is the subject, the system function
is the verb + operand (i.e. predicate) and the system concept is the mapping
of the two to each other. The key distinguishing feature of HFGT (relative to
multi-layer networks) is its introduction of system function. In that regard,
it is more complete than multi-layer networks if system function is accepted
as part of an engineering system abstraction. Another key distinguishing
feature of HFGT is the differentiation between elements related to
transformation and transportation. In that regard, it takes great care to not
_overload_ mathematical modeling elements and preserve lucidity.

###  I-B Original Contribution

This paper provides a tensor-based formulation of several of the most
important parts of hetero-functional graph theory. More specifically, it
discusses the system concept, the hetero-functional adjacency matrix, and the
hetero-functional incidence tensor. Whereas the hetero-functional graph theory
text[7] is a comprehensive discussion of the subject, the treatment is based
entirely on two-dimensional matrices. The tensor-based formulation described
in this work makes a stronger tie between HFGT and its ontological foundations
in MBSE. Furthermore, the tensor-based treatment developed here reveals
patterns of underlying structure in engineering systems that are less apparent
in a matrix-based treatment. Finally, the tensor-based formulation facilitates
an understanding of the relationships between HFGT and multi-layer networks
(“despite its disparate terminology and lack of consensus”). In so doing, this
tensor-based treatment is likely to advance Kivela et. al’s goal to discern
the similarities and differences between these mathematical models in as
precise a manner as possible.

###  I-C Paper Outline

The rest of the paper is organized as follows. Section II discusses the system
concept as an allocation of system function to system form. Section III
discusses the hetero-functional adjacency matrix emphasizing the relationships
between system capabilities (i.e. structural degrees of freedom as defined
therein). Section IV, then, discusses the hetero-functional incidence tensor
which describes the relationships between system capabilities, operands, and
physical locations in space (i.e. system buffers as defined later). Section V
goes on to discuss this tensor-based formulation from the perspective of
layers and network descriptors. Section VI brings the work to a close and
offers directions for future work. Given the multi-disciplinary nature of this
work, several appendices are provided to support the work with background
material and avoid breaking the logical flow of the main article. Appendix -A
provides the fundamental definitions of ontological science that were used to
motivate this work’s original contribution. Appendix -B describes the notation
conventions used throughout this work. The paper assumes that the reader is
well grounded in graph theory and network science as it is found in any one of
a number of excellent texts[98, 99]. The paper does not assume prior exposure
to hetero-functional graph theory. It’s most critical definitions are tersely
introduced in the body of the work upon first mention. More detailed
classifications of these concepts are compiled in Appendix -C for convenience.
Given the theoretical treatment provided here, the interested reader is
referred to the hetero-functional graph theory text[7] for further explanation
of these well-established concepts and concrete examples. Furthermore, several
recent works have made illustrative comparisons between (formal) graphs and
hetero-functional graphs[100, 101]. Finally, this work makes extensive use of
set, Boolean, matrix, and tensor operations; all of which are defined
unambiguously in Appendices -D, -E, -F, and -G respectively.

##  II The System Concept

At a high-level, the system concept $A_{S}$ describes the allocation of system
function to system form as the central question of engineering design. First,
Subsection II-A provides introductory definitions of system resources,
processes, and knowledge base that serve as prerequisite knowledge for the
remaining subsections. Next, Subsection II-B introduces the transportation
knowledge base as a third-order tensor. Next, Subsection II-C introduces the
refined transportation knowledge base as a fourth-order tensor. The tensor-
based formulations in these two subsection directly support the original
contribution mentioned in Section I-B. Finally, in order to support the
following section, Subsection II-D concludes the section with a discussion of
existence and availability of system capabilities as part of the system
concept.

###  II-A System Resources, Processes, and Knowledge Base

This dichotomy of form and function is repeatedly emphasized in the fields of
engineering design and systems engineering[97, 102, 103, 104]. More
specifically, the allocation of system processes to system resources is
captured in the “design equation”[10, 94]:

$P=J_{S}\odot R$ (1)

where $R$ is set of system resources, $P$ is the set of system processes,
$J_{S}$ is the system knowledge base, and $\odot$ is matrix Boolean
multiplication (Defn. 48).

######  Definition 2 (System Resource):

[4] An asset or object $r_{v}\in R$ that is utilized during the execution of a
process.

######  Definition 3 (System Process[105, 4]):

An activity $p\in P$ that transforms a predefined set of input operands into a
predefined set of outputs.

######  Definition 4 (System Operand):

[4] An asset or object $l_{i}\in L$ that is operated on or consumed during the
execution of a process.

######  Definition 5 (System Knowledge Base[9, 10, 11, 12, 13, 94]):

A binary matrix $J_{S}$ of size $\sigma(P)\times\sigma(R)$ whose element
$J_{S}(w,v)\in\\{0,1\\}$ is equal to one when action $e_{wv}\in{\cal E}_{S}$
(in the SysML sense) exists as a system process $p_{w}\in P$ being executed by
a resource $r_{v}\in R$. The $\sigma()$ notation gives the size of a set.

In other words, the system knowledge base forms a bipartite graph between the
set of system processes and the set of system resources[13].

Hetero-functional graph theory further recognizes that there are inherent
differences within the set of resources as well as within the set of
processes. Therefore, classifications of these sets of resources and sets of
processes are introduced and defined in Appendix -C. $R=M\cup B\cup H$ where
$M$ is the set of transformation resources (Defn. 28), $B$ is the set of
independent buffers (Defn. 29), and $H$ is the set of transportation resources
(Defn. 30). Furthermore, the set of buffers $B_{S}=M\cup B$ (Defn. 31) is
introduced for later discussion. Similarly, $P=P_{\mu}\cup P_{\bar{\eta}}$
where $P_{\mu}$ is the set of transformation processes (Defn. 32) and
$P_{\bar{\eta}}$ is the set of refined transportation processes (Defn. 33).
The latter, in turn, is determined from the Cartesian product (❌) (Defn. 44)
of the set of transportation processes $P_{\eta}$ (Defn. 34) and the set of
holding processes $P_{\gamma}$ (Defn. 35).

$P_{\bar{\eta}}=P_{\gamma}\mbox{{\char 14\relax}}P_{\eta}$ (2)

Fig. 3: The Hetero-functional Graph Theory Meta-Architecture drawn using the
Systems Markup Language (SysML). It consists of three types of resources
$R=M\cup B\cup H$ that are capable of two types of process
$P_{\bar{\eta}}=P_{\gamma}\mbox{{\char 14\relax}}P_{\eta}$[7].

This taxonomy of resources, processes, and their allocation is organized in
the HFGT meta-architecture shown in Figure 3. The taxonomy of resources $R$
and processes $P$ originates from the field of production systems where
transformation processes are viewed as “value-adding”, holding processes
support the design of fixtures, and transportation processes are cost-
minimized. Furthermore, their existence is necessitated by their distinct
roles in the structural relationships found in hetero-functional graphs.
Consequently, subsets of the design equation 1 can be written to emphasize the
relationships between the constitutent classes of processes and resources[9,
10, 11, 12, 13].

$\displaystyle P_{\mu}$ $\displaystyle=J_{M}\odot M$ (3) $\displaystyle
P_{\gamma}$ $\displaystyle=J_{\gamma}\odot R$ (4) $\displaystyle P_{\eta}$
$\displaystyle=J_{H}\odot R$ (5) $\displaystyle P_{\bar{\eta}}$
$\displaystyle=J_{\bar{H}}\odot R$ (6)

where $J_{M}$ is the transformation knowledge base, $J_{\gamma}$ is the
holding knowledge base, $J_{H}$ is the transportation knowledge base, and
$J_{\bar{H}}$ is the refined transportation knowledge base [10, 106, 13, 107,
108, 109]. The original system knowledge base $J_{S}$ is straightforwardly
reconstructed from these smaller knowledge bases[9, 10, 11, 12, 13]:

$J_{S}=\left[\begin{array}[]{ccc}J_{M}&|&\mathbf{0}\\\
\hline\cr&J_{\bar{H}}&\\\ \end{array}\right]$ (7)

###  II-B The Transportation Knowledge Base Tensor

The transportation knowledge base $J_{H}$ is best understood as a _matricized_
3rd-order tensor ${\cal J}_{H}$ where the element ${\cal
J}_{H}(y_{1},y_{2},v)=1$ when the transportation process $p_{u}\in P_{\eta}$
defined by the origin $b_{s_{y_{1}}}\in B_{S}$ and the destination
$b_{s_{y_{2}}}\in B_{S}$ is executed by the resource $r_{v}\in R$.

$\displaystyle J_{H}$ $\displaystyle={\cal F}_{M}\left({\cal
J}_{H},[2,1],[3]\right)$ (8) $\displaystyle{\cal J}_{H}$ $\displaystyle={\cal
F}_{M}^{-1}\left(J_{H},[\sigma(B_{S}),\sigma(B_{S}),\sigma(R)],[2,1],[3]\right)$
(9)

where ${\cal F}_{M}$ and ${\cal F}_{M}^{-1}$ are the matricization and
tensorization functions (Defns. 56 and 57) respectively. Here, ${\cal
F}_{M}()$ serves to vectorize the dimensions of the origin and destination
buffers into the single dimension of transportation processes. Appendix -G,
more generally, introduces the reader to tensor-based operations.

The ${\cal J}_{H}$ tensor reveals that the transportation knowledge base is
closely tied to the classical understanding of a graph $A_{B_{S}}$ where point
elements of form called nodes, herein taken to be the set of buffers $B_{S}$,
are connected by line elements of form called edges. Such a graph in hetero-
functional graph theory (and model-based systems engineering) is called a
formal graph[97] because all of its elements describe the system form and any
statement of function is entirely _implicit_.

$\displaystyle A_{B_{S}}(y_{1},y_{2})$
$\displaystyle=\bigvee_{v}^{\sigma(R)}{\cal J}_{H}(y_{1},y_{2},v)$
$\displaystyle=\bigvee_{v}^{\sigma(R)}J_{H}(u,v)\qquad\forall
y_{1},y_{2}\in\\{1,\ldots,\sigma(B_{S})\\},u=\sigma(B_{S})(y_{1}-1)+y_{2},v\in\\{1,\ldots,\sigma(R)\\}$
(10) $\displaystyle A_{B_{S}}$ $\displaystyle={\cal
J}_{H}\odot_{3}\mathds{1}^{\sigma(R)}$
$\displaystyle=vec^{-1}\left(J_{H}\odot\mathds{1}^{\sigma(R)},[\sigma(B_{S}),\sigma(B_{S})]\right)^{T}$
(11) $\displaystyle A_{B_{S}}^{TV}$ $\displaystyle=\left({\cal
J}_{H}\odot_{3}\mathds{1}^{\sigma(R)}\right)^{TV}$
$\displaystyle=J_{H}\odot\mathds{1}^{\sigma(R)}$ (12)

where the $\bigvee$ notation is the Boolean analogue of the $\sum$ notation
(Defn. 45), $\odot_{n}$ is the n-mode Boolean matrix product (Defn. 62),
$vec^{-1}()$ is inverse vectorization (Defn. 59) and $()^{V}$ is shorthand for
vectorization (Defn. 58). Appendix -E, more generally, introduces the reader
to Boolean operations. Furthermore, the notation $\mathds{1}^{n}$ is used to
indicate a ones-vector of length n. The transportation system knowledge base
$J_{H}$ replaces the edges of the formal graph $A_{B_{S}}$ with an _explicit_
description of function in the transportation processes $P_{\eta}$. The multi-
column nature of the transportation knowledge base $J_{H}$ contains more
information than the formal graph $A_{B_{S}}$ and allows potentially many
resources to execute any given transportation process. Consequently, the OR
operation across the rows of $J_{H}$ (or the third dimension of ${\cal
J}_{H}$) is sufficient to reconstruct the formal graph $A_{B_{S}}$. In short,
a single column transportation knowledge base is mathematically equivalent to
a vectorized formal graph $A_{B_{S}}$.

###  II-C The Refined Transportation Knowledge Base Tensor

Similarly, the refined transportation knowledge base is best understood as a
matricized 4th order tensor ${\cal J}_{\bar{H}}$ where the element ${\cal
J}_{H}(g,y_{1},y_{2},v)=1$ when the refined transportation process
$p_{\varphi}\in P_{\bar{\eta}}$ defined by the holding process $p_{\gamma
g}\in P_{\gamma}$, the origin $b_{s_{y_{1}}}\in B_{S}$ and the destination
$b_{s_{y_{2}}}\in B_{S}$ is executed by the resource $r_{v}\in R$.

$\displaystyle J_{\bar{H}}$ $\displaystyle={\cal F}_{M}\left({\cal
J}_{\bar{H}},[3,2,1],[4]\right)$ (13) $\displaystyle{\cal J}_{\bar{H}}$
$\displaystyle={\cal
F}_{M}^{-1}\left(J_{\bar{H}},[\sigma(P_{\gamma}),\sigma(B_{S}),\sigma(B_{S}),\sigma(R)],[3,2,1],[4]\right)$
(14)

The $J_{\bar{H}}$ tensor reveals that the refined transportation knowledge
base is closely tied to the classical understanding of a multi-commodity flow
network ${\cal A}_{LB_{S}}$[110, 111, 112]. Mathematically, it is a 3rd-order
tensor whose element ${\cal A}_{LB_{S}}(i,y_{1},y_{2})=1$ when operand
$l_{i}\in L$ is transported from buffer $b_{s_{y_{1}}}$ to $b_{s_{y_{2}}}$.
Again, the multi-commodity flow network ${\cal A}_{LB_{S}}$ is purely a
description of system form and any statement of function is entirely
_implicit_. In the special case222By Defn. 35, holding processes are
distinguished by three criteria: 1.) different operands, 2.) how they hold
those operands, and 3.) if they change the state of the operand. The special
case mentioned above is restricted to only the first of these three
conditions. of a system where the set of operands $L$ maps 1-to-1 the set of
holding processes $P_{\gamma}$ (i.e. $i=g$):

$\displaystyle{\cal A}_{LB_{S}}(i,y_{1},y_{2})$
$\displaystyle=\bigvee_{v}^{\sigma(R)}{\cal J}_{\bar{H}}(g,y_{1},y_{2},v)$
$\displaystyle\quad\forall
g\in\\{1,\ldots,\sigma(P_{\gamma})\\},y_{1},y_{2}\in\\{1,\ldots,\sigma(B_{S})\\},v\in\\{1,\ldots,\sigma(R)\\}$
(15) $\displaystyle=\bigvee_{v}^{\sigma(R)}J_{\bar{H}}(\varphi,v)$
$\displaystyle\quad\forall\varphi=\sigma^{2}(B_{S})(g-1)+\sigma(B_{S})(y_{1}-1)+y_{2},v\in\\{1,\ldots,\sigma(R)\\}$
(16) $\displaystyle{\cal A}_{LB_{S}}$ $\displaystyle={\cal
J}_{\bar{H}}\odot_{4}\mathds{1}^{\sigma(R)}$
$\displaystyle=vec^{-1}\left(J_{\bar{H}}\odot\mathds{1}^{\sigma(R)},\left[\sigma(B_{S}),\sigma(B_{S}),\sigma(P_{\gamma})\right]\right)^{T}$
(17) $\displaystyle{\cal A}_{LB_{S}}^{TV}$ $\displaystyle=\left({\cal
J}_{\bar{H}}\odot_{4}\mathds{1}^{\sigma(R)}\right)^{TV}$
$\displaystyle=J_{\bar{H}}\odot\mathds{1}^{\sigma(R)}$ (18)

The refined transportation system knowledge base $J_{\bar{H}}$ replaces the
operands and edges of the multi-commodity flow network $A_{MP}$ with an
_explicit_ description of function in the holding processes $P_{\gamma}$ and
transportation processes $P_{\eta}$. The multi-column nature of the refined
transportation knowledge base $J_{\bar{H}}$ contains more information than the
multi-commodity flow network ${\cal A}_{LB_{S}}$ and allows potentially many
resources to execute any given refined transportation process. Consequently,
the OR operation across the rows of $J_{\bar{H}}$ (or the fourth dimension of
${\cal J}_{\bar{H}}$) is sufficient to reconstruct the multi-commodity flow
network ${\cal A}_{LB_{S}}$. In short, a single column of the refined
transportation knowledge base is mathematically equivalent to a vectorized
multi-commodity flow network.

The transformation, holding, transportation and refined transportation
knowledge bases ($J_{M}$, $J_{\gamma}$, $J_{H}$ and $J_{\bar{H}}$) readily
serve to reconstruct the system knowledge base $J_{S}$. First, the refined
transportation knowledge base is the Khatri-Rao product of the holding and
transportation knowledge bases.

$\displaystyle J_{\bar{H}}(\sigma(P_{\eta})(g-1)+u,v)$
$\displaystyle=J_{\gamma}(g,v)\cdot J_{H}(u,v)\qquad\forall
g\in\\{1,\ldots,\sigma(P_{\gamma})\\},u=\sigma(B_{S})(y_{1}-1)+y_{2},v\in\\{1,\ldots,\sigma(R)\\}$
(19) $\displaystyle J_{\bar{H}}$ $\displaystyle=J_{\gamma}\circledast J_{H}$
(20)
$\displaystyle=\left[J_{\gamma}\otimes\mathds{1}^{\sigma(P_{\eta})}\right]\cdot\left[\mathds{1}^{\sigma(P_{\gamma})}\otimes
J_{H}\right]$ (21)

where $\cdot$ is the Hadamard (or scalar) product (Defn. 51), $\circledast$ is
the Khatri-Rao product (Defn. 54) and $\otimes$ is the Kronecker product
(Defn. 53).

###  II-D Existence, Availability, and Concept of System Capabilities

Hetero-functional graph theory also differentiates between the _existence_ and
the _availability_ of physical capabilities in the system[10, 107]. While the
former is described by the system knowledge base the latter is captured by the
system constraints matrix (which is assumed to evolve in time).

######  Definition 6 (System Constraints Matrix[9, 10, 11, 12, 13, 94]):

A binary matrix $K_{S}$ of size $\sigma(P)\times\sigma(R)$ whose element
$K_{S}(w,v)\in\\{0,1\\}$ is equal to one when a constraint eliminates event
$e_{wv}$ from the event set.

The system constraints matrix is constructed analogously to the system
knowledge base[9, 10, 11, 12, 13].

$K_{S}=\left[\begin{array}[]{ccc}K_{M}&|&\mathbf{0}\\\
\hline\cr&K_{\bar{H}}&\\\ \end{array}\right]$ (22)

In this regard, the system constraints matrix has a similar meaning to graph
percolation [113, 114] and temporal networks [115].

Once the system knowledge base $J_{S}$ and the system constraints matrix
$K_{S}$ have been constructed, the system concept $A_{S}$ follows
straightforwardly.

######  Definition 7 (System Concept[9, 10, 11, 12, 13, 94]):

A binary matrix $A_{S}$ of size $\sigma(P)\times\sigma(R)$ whose element
$A_{S}(w,v)\in\\{0,1\\}$ is equal to one when action $e_{wv}\in{\cal E}_{S}$
(in the SysML sense) is available as a system process $p_{w}\in P$ being
executed by a resource $r_{v}\in R$.

$A_{S}=J_{S}\ominus K_{S}=J_{S}\cdot\bar{K}_{S}$ (23)

where $\ominus$ is Boolean subtraction (Defn. 49) and
$\overline{K}_{S}=NOT(K_{S})$.

Every filled element of the system concept indicates a _system capability_
(Defn. 36) of the form: “Resource $r_{v}$ does process $p_{w}$”. The system
constraints matrix limits the availability of capabilities in the system
knowledge base to create the system concept $A_{S}$. The system capabilities
are quantified by the structural degrees of freedom.

######  Definition 8 (Structural Degrees of Freedom[9, 10, 11, 12, 13, 94]):

The set of independent actions ${\cal E}_{S}$ that completely defines the
instantiated processes in a large flexible engineering system. Their number is
given by:

$\displaystyle DOF_{S}=\sigma({\cal E}_{S})$
$\displaystyle=\sum_{w}^{\sigma(P)}\sum_{v}^{\sigma(R)}\left[J_{S}\ominus
K_{S}\right](w,v)$ (24)
$\displaystyle=\sum_{w}^{\sigma(P)}\sum_{v}^{\sigma(R)}A_{S}(w,v)$ (25)
$\displaystyle=\langle J_{S},\bar{K}_{S}\rangle_{F}$ (26)

As has been discussed extensively in prior publications, the term structural
degrees of freedom is best viewed as a generalization of kinematic degrees of
freedom (or generalized coordinates)[9, 10, 11, 12, 13, 116]. Note that the
transformation degrees of freedom $DOF_{M}$ and the refined transportation
degrees of freedom $DOF_{H}$ are calculated similarly[9, 10, 11, 12]:

$\displaystyle DOF_{M}$
$\displaystyle=\sum_{j}^{\sigma(P_{\mu})}\sum_{k}^{\sigma(M)}\left[J_{M}\ominus
K_{M}\right](j,k)$ (27) $\displaystyle DOF_{H}$
$\displaystyle=\sum_{\varphi}^{\sigma(P_{\bar{\eta}})}\sum_{v}^{\sigma(R)}\left[J_{\bar{H}}\ominus
K_{\bar{H}}\right](u,v)$ (28)

##  III Hetero-functional Adjacency Matrix

This section serves provides a tensor-based formulation of the hetero-
functional adjacency matrix. First, Subsection III-A introduces this matrix as
pairwise sequences of system capabilities. Next, Subsection III-B provides a
tensor-based formulation of the system sequence knowledge base. Next,
Subsection III-C provides a tensor-based formulation of the system sequence
constraints. Both of these subsections directly support the paper’s original
contribution. Finally, Subsection III-D concludes the section with a
discussion of sequence-dependent degrees of freedom.

###  III-A Pairwise Sequences of System Capabilities

Once the system’s physical capabilities (or structural degrees of freedom have
been defined), the hetero-functional adjacency matrix $A_{\rho}$ is introduced
to represent their pairwise sequences [13, 117, 118, 108, 119].

######  Definition 9 (Hetero-functional Adjacency Matrix[13, 117, 118, 108,
119]):

A square binary matrix $A_{\rho}$ of size
$\sigma(R)\sigma(P)\times\sigma(R)\sigma(P)$ whose element
$J_{\rho}(\chi_{1},\chi_{2})\in\\{0,1\\}$ is equal to one when string
$z_{\chi_{1},\chi_{2}}=e_{w_{1}v_{1}}e_{w_{2}v_{2}}\in{\cal Z}$ is available
and exists, where index $\chi_{i}\in\left[1,\dots,\sigma(R)\sigma(P)\right]$.

In other words, the hetero-functional adjacency matrix corresponds to a
hetero-functional graph $G=\\{{\cal E}_{S},{\cal Z}\\}$ with structural
degrees of freedom (i.e. capabilities) ${\cal E}_{S}$ as nodes and feasible
sequences ${\cal Z}$ as edges.

Much like the system concept $A_{S}$, the hetero-functional adjacency matrix
$A_{\rho}$ arises from a Boolean difference[13, 117, 118, 108, 119].

$A_{\rho}=J_{\rho}\ominus K_{\rho}$ (29)

where $J_{\rho}$ is the system sequence knowledge base and $K_{\rho}$ is the
system sequence constraints matrix.

######  Definition 10 (System Sequence Knowledge Base [13, 117, 118, 108,
119]):

A square binary matrix $J_{\rho}$ of size
$\sigma(R)\sigma(P)\times\sigma(R)\sigma(P)$ whose element
$J_{\rho}(\chi_{1},\chi_{2})\in\\{0,1\\}$ is equal to one when string
$z_{\chi_{1},\chi_{2}}=e_{w_{1}v_{1}}e_{w_{2}v_{2}}\in{\cal Z}$ exists, where
index $\chi_{i}\in\left[1,\dots,\sigma(R)\sigma(P)\right]$.

######  Definition 11 (System Sequence Constraints Matrix[13, 117, 118, 108,
119]):

A square binary constraints matrix $K_{\rho}$ of size
$\sigma(R)\sigma(P)\times\sigma(R)\sigma(P)$ whose elements
$K(\chi_{1},\chi_{2})\in\\{0,1\\}$ are equal to one when string
$z_{\chi_{1}\chi_{2}}=e_{w_{1}v_{1}}e_{w_{2}v_{2}}\in{\cal Z}$ is eliminated.

The definitions of the system sequence knowledge base $J_{\rho}$ and the
system sequence constraints matrix $K_{\rho}$ feature a translation of indices
from $e_{w_{1}v_{1}}e_{w_{2}v_{2}}$ to $z_{\chi_{1}\chi_{2}}$. This fact
suggests that these matrices have their associated 4th order tensors ${\cal
J}_{\rho}$, ${\cal K}_{\rho}$ and ${\cal A}_{\rho}$.

$\displaystyle J_{\rho}$ $\displaystyle={\cal F}_{M}\left({\cal
J}_{\rho},[1,2],[3,4]\right)$ (30) $\displaystyle K_{\rho}$
$\displaystyle={\cal F}_{M}\left({\cal K}_{\rho},[1,2],[3,4]\right)$ (31)
$\displaystyle A_{\rho}$ $\displaystyle={\cal F}_{M}\left({\cal
A}_{\rho},[1,2],[3,4]\right)$ (32)

###  III-B The System Sequence Knowledge Base Tensor

The system sequence knowledge base $J_{\rho}$ and its tensor-equivalent ${\cal
J}_{\rho}$ create all the _potential_ sequences of the capabilities in
$A_{S}$.

$\displaystyle{\cal J}_{\rho}(w_{1},v_{1},w_{2},v_{2})$
$\displaystyle=A_{S}(w_{1},v_{1})\cdot A_{S}(w_{2},v_{2})\quad\forall
w_{1},w_{2}\in\\{1\ldots\sigma(P)\\},v_{1},v_{2}\in\\{1\ldots\sigma(R)\\}$
(33) $\displaystyle{J}_{\rho}(\chi_{1},\chi_{2})$
$\displaystyle=A_{S}^{V}(\chi_{1})\cdot
A^{V}_{S}(\chi_{2})\qquad\quad\;\;\forall\chi_{1},\chi_{2}\in\\{1\ldots\sigma(R)\sigma(P)\\}$
(34) $\displaystyle J_{\rho}$ $\displaystyle=A_{S}^{V}A_{S}^{VT}$ (35)
$\displaystyle J_{\rho}$
$\displaystyle=\left[J_{S}\cdot\bar{K}_{S}\right]^{V}\left[J_{S}\cdot\bar{K}_{S}\right]^{VT}$
(36) $\displaystyle{\cal J}_{\rho}$ $\displaystyle={\cal
F}_{M}^{-1}\left(J_{\rho},[\sigma(P),\sigma(R),\sigma(P),\sigma(R)],[1,2],[3,4]\right)$
(37)

###  III-C The System Sequence Constraints Tensor

Of these potential sequences of capabilities, the system sequence constraints
matrix $K_{\rho}$ serves to eliminate the infeasible pairs. The feasibility
arises from five types of constraints:

  1. I:

$P_{\mu}P_{\mu}$. Two transformation processes that follow each other must
occur at the same transformation resource. $m_{1}=m_{2}$.

  2. II:

$P_{\mu}P_{\bar{\eta}}$. A refined transportation process that follows a
transformation process must have an origin equivalent to the transformation
resource at which the transformation process was executed.
$m_{1}-1=(u_{1}-1)/\sigma(B_{S})$ where $/$ indicates integer division.

  3. III:

$P_{\bar{\eta}}P_{\mu}$. A refined transportation process that precedes a
transformation process must have a destination equivalent to the
transformation resource at which the transformation process was executed.
$m_{2}-1=(u_{1}-1)\%\sigma(B_{S})$ where $\%$ indicates the modulus.

  4. IV:

$P_{\bar{\eta}}P_{\bar{\eta}}$. A refined transportation process that follows
another must have an origin equivalent to the destination of the other.
$(u_{1}-1)\%\sigma(B_{S})=(u_{2}-1)/\sigma(B_{S})$

  5. V:

$PP$. The type of operand of one process must be equivalent to the type of
output of another process. In other words, the ordered pair of processes
$P_{w_{1}}P_{w_{2}}$ is feasible if and only if $A_{P}(w_{1},w_{2})=1$ where
$A_{P}$ is the adjacency matrix that corresponds to a _functional graph_ in
which pairs of system processes are connected.

In previous hetero-functional graph theory works, the system sequence
constraints matrix $K_{\rho}$ was calculated straightforwardly using for FOR
loops to loop over the indices $\chi_{1}$ and $\chi_{2}$ and checking the
presence of the five feasibility constraints identified above.

Here, an alternate approach based upon tensors is provided for insight into
the underlying mathematical structure. For convenience,
$\overline{K}_{\rho}=NOT(K_{\rho})$ captures the set of all _feasibility_
conditions that pertain to valid sequences of system capabilities. This set
requires that _any_ of the first four constraints above _and_ the last
constraint be satisfied.

$\overline{K}_{\rho}=\left(\overline{K}_{\rho I}\oplus\overline{K}_{\rho
II}\oplus\overline{K}_{\rho III}\oplus\overline{K}_{\rho
IV}\right)\cdot\overline{K}_{\rho V}$ (38)

where $\oplus$ is Boolean addition (Defn. 46) and $\overline{K}_{\rho
I},\overline{K}_{\rho II},\overline{K}_{\rho III},\overline{K}_{\rho
IV},\overline{K}_{\rho V}$ are the matrix implementations of the five types of
feasibility constraints identified above. Their calculation is most readily
achieved through their associated 4th-order tensors.

Type I Constraints: For the Type I constraint, $\overline{\cal K}_{\rho I}$ is
constructed from a sum of 4th-order outer products (Defn. 55) of elementary
basis vectors.

$\displaystyle\overline{\cal K}_{\rho I}$
$\displaystyle=\sum_{w_{1}=1}^{\sigma(P_{\mu})}\sum_{v_{1}=1}^{\sigma(M)}\sum_{w_{2}=1}^{\sigma(P_{\mu})}\sum_{v_{2}=v_{1}}^{v_{1}}e_{w_{1}}^{\sigma(P)}\circ
e_{v_{1}}^{\sigma(R)}\circ e_{w_{2}}^{\sigma(P)}\circ e_{v_{2}}^{\sigma(R)}$
(39)

where the $e_{i}^{n}$ notation places the value 1 on the $i^{th}$ element of a
vector of length $n$. $\overline{K}_{\rho I}$ is calculated straightforwardly
by matricizing both sides and evaluating the sums.

$\displaystyle\overline{K}_{\rho I}$
$\displaystyle=\sum_{w_{1}=1}^{\sigma(P_{\mu})}\sum_{v_{1}=1}^{\sigma(M)}\sum_{w_{2}=1}^{\sigma(P_{\mu})}\sum_{v_{2}=v_{1}}^{v_{1}}\left(e_{v_{1}}^{\sigma(R)}\otimes
e_{w_{1}}^{\sigma(P)}\right)\otimes\left(e_{v_{2}}^{\sigma(R)}\otimes
e_{w_{2}}^{\sigma(P)}\right)^{T}$ (40) $\displaystyle\overline{K}_{\rho I}$
$\displaystyle=\sum_{v_{1}=1}^{\sigma(M)}\left(e_{v_{1}}^{\sigma(R)}\otimes\begin{bmatrix}\mathds{1}^{\sigma(P_{\mu})}\\\
\mathbf{0}^{\sigma(P_{\bar{\eta}})}\end{bmatrix}\right)\left(e_{v_{1}}^{\sigma(R)}\otimes\begin{bmatrix}\mathds{1}^{\sigma(P_{\mu})}\\\
\mathbf{0}^{\sigma(P_{\bar{\eta}})}\end{bmatrix}\right)^{T}$ (41)

Type II Constraints: Similarly, for the Type II constraint:

$\displaystyle\overline{\cal K}_{\rho II}$
$\displaystyle=\sum_{w_{1}=1}^{\sigma(P_{\mu})}\sum_{v_{1}=1}^{\sigma(M)}\sum_{y_{1}=v_{1}}^{v_{1}}\sum_{v_{2}=1}^{\sigma(R)}e_{w_{1}}^{\sigma(P)}\circ
e_{v_{1}}^{\sigma(R)}\circ\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{1}}^{\sigma(P_{\bar{\eta}})}\end{bmatrix}\circ e_{v_{2}}^{\sigma(R)}$
(42)

Here, the $X_{y_{1}}^{\sigma(P_{\bar{H}})}$ vector has a value of 1 wherever a
refined transportation process $p_{w_{2}}$ originates at the transformation
resource $m_{v_{1}}$. Drawing on the discussion of the 3rd-order tensor ${\cal
J}_{\bar{H}}$ in Section II, $X_{y_{1}}^{\sigma(P_{\bar{H}})}$, itself, is
expressed as a vectorized sum of 3rd-order outer products.

$\displaystyle
X_{y_{1}}^{\sigma(P_{\bar{H}})}=\sum_{g=1}^{\sigma(P_{\gamma})}\sum_{y_{2}=1}^{\sigma(B_{S})}\left(e_{g}^{\sigma(P_{\gamma})}\circ
e_{y_{1}}^{\sigma(B_{S})}\circ
e_{y_{2}}^{\sigma(B_{S})}\right)^{TV}=\left(\mathds{1}^{\sigma(P_{\gamma})}\otimes
e_{y_{1}}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\right)$ (43)

$\overline{K}_{\rho II}$ is then calculated straightforwardly by matricizing
both sides of Eq. 42 and evaluating the sums.

$\displaystyle\overline{K}_{\rho II}$
$\displaystyle=\sum_{w_{1}=1}^{\sigma(P_{\mu})}\sum_{v_{1}=1}^{\sigma(M)}\sum_{y_{1}=v_{1}}^{v_{1}}\sum_{v_{2}=1}^{\sigma(R)}\left(e_{v_{1}}^{\sigma(R)}\otimes
e_{w_{1}}^{\sigma(P)}\right)\otimes\left(e_{v_{2}}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{1}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\right)^{T}$ (44)
$\displaystyle\overline{K}_{\rho II}$
$\displaystyle=\sum_{v_{1}=1}^{\sigma(M)}\left(e_{v_{1}}^{\sigma(R)}\otimes\begin{bmatrix}\mathds{1}^{\sigma(P_{\mu})}\\\
\mathbf{0}^{P_{\bar{\eta}}}\end{bmatrix}\right)\left(\mathds{1}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
\mathds{1}^{\sigma(P_{\gamma})}\otimes
e_{v_{1}}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\end{bmatrix}\right)^{T}$
(45)

Type III Constraints: Similarly, for the Type III constraint:

$\displaystyle\overline{\cal K}_{\rho III}$
$\displaystyle=\sum_{y_{2}=v_{2}}^{v_{2}}\sum_{v_{1}=1}^{\sigma(R)}\sum_{w_{2}=1}^{\sigma(P_{\mu})}\sum_{v_{2}=1}^{\sigma(M)}\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{2}}^{\sigma(P_{\bar{\eta}})}\end{bmatrix}\circ
e_{v_{1}}^{\sigma(R)}\circ e_{w_{2}}^{\sigma(P)}\circ e_{v_{2}}^{\sigma(R)}$
(46)

Here, the $X_{y_{2}}^{\sigma(P_{\bar{H}})}$ vector has a value of 1 wherever a
refined transportation process $p_{w_{1}}$ terminates at the transformation
resource $m_{v_{2}}$. $X_{y_{2}}^{\sigma(P_{\bar{H}})}$, itself, is expressed
as a vectorized sum of 3rd-order outer products.

$\displaystyle
X_{y_{2}}^{\sigma(P_{\bar{H}})}=\sum_{g=1}^{\sigma(P_{\gamma})}\sum_{y_{1}=1}^{\sigma(B_{S})}\left(e_{g}^{\sigma(P_{\gamma})}\circ
e_{y_{1}}^{\sigma(B_{S})}\circ
e_{y_{2}}^{\sigma(B_{S})}\right)^{TV}=\left(\mathds{1}^{\sigma(P_{\gamma})}\otimes\mathds{1}^{\sigma(B_{S})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\right)$ (47)

$\overline{K}_{\rho III}$ is then calculated straightforwardly by matricizing
both sides of Eq. 46 and evaluating the sums.

$\displaystyle\bar{K}_{\rho III}$
$\displaystyle=\sum_{y_{2}=v_{2}}^{v_{2}}\sum_{v_{1}=1}^{\sigma(R)}\sum_{w_{2}=1}^{\sigma(P_{\mu})}\sum_{v_{2}=1}^{\sigma(M)}\left(e_{v_{1}}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{2}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\right)\otimes\left(e_{v_{2}}^{\sigma(R)}\otimes
e_{w_{2}}^{\sigma(P)}\right)^{T}$ (48) $\displaystyle\overline{K}_{\rho III}$
$\displaystyle=\sum_{v_{2}=1}^{\sigma(M)}\left(\mathds{1}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
\mathds{1}^{\sigma(P_{\gamma})}\otimes\mathds{1}^{\sigma(B_{S})}\otimes
e_{v_{2}}^{\sigma(B_{S})}\end{bmatrix}\right)\left(e_{v_{2}}^{\sigma(R)}\otimes\begin{bmatrix}\mathds{1}^{\sigma(P_{\mu})}\\\
\mathbf{0}^{P_{\bar{\eta}}}\end{bmatrix}\right)^{T}$ (49)

Type IV Constraints: Similarly, for the Type IV constraint:

$\displaystyle\overline{\cal K}_{\rho IV}$
$\displaystyle=\sum_{y_{2}=1}^{\sigma{(B_{S})}}\sum_{v_{1}=1}^{\sigma(R)}\sum_{y_{1}=y_{2}}^{y_{2}}\sum_{v_{2}=1}^{\sigma(R)}\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{2}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\circ
e_{v_{1}}^{\sigma(R)}\circ\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{1}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\circ e_{v_{2}}^{\sigma(R)}$ (50)

$\overline{K}_{\rho IV}$ is then calculated straightforwardly by matricizing
both sides of Eq. 50 and evaluating the sums.

$\displaystyle\overline{K}_{\rho IV}$
$\displaystyle=\sum_{y_{2}=1}^{\sigma{(B_{S})}}\sum_{v_{1}=1}^{\sigma(R)}\sum_{y_{1}=y_{2}}^{y_{2}}\sum_{v_{2}=1}^{\sigma(R)}\left(e_{v_{1}}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{2}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\right)\otimes\left(e_{v_{2}}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
X_{y_{1}}^{\sigma(P_{\bar{H}})}\end{bmatrix}\right)^{T}$ (51)
$\displaystyle\overline{K}_{\rho IV}$
$\displaystyle=\sum_{y_{2}=1}^{\sigma{(B_{S})}}\left(\mathds{1}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
\mathds{1}^{\sigma(P_{\gamma})}\otimes\mathds{1}^{\sigma(B_{S})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\end{bmatrix}\right)\left(\mathds{1}^{\sigma(R)}\otimes\begin{bmatrix}\mathbf{0}^{\sigma(P_{\mu})}\\\
\mathds{1}^{\sigma(P_{\gamma})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\end{bmatrix}\right)^{T}$
(52)

Type V Constraints: The Type V constraint must make use of the functional
graph adjacency matrix $A_{P}$. Consequently, the fourth-order tensor ${\cal
K}_{\rho V}$ is calculated first on a scalar basis using the Kronecker delta
function $\delta_{i}$ (Defn. 50) and then is matricized to $K_{\rho V}$.

$\displaystyle{\cal\overline{K}}_{\rho V}(w_{1},v_{1},w_{2},v_{2})$
$\displaystyle=\delta_{v_{1}v_{1}}\cdot\delta_{v_{2}v_{2}}\cdot
A_{P}(w_{1},w_{2})\qquad\forall
w_{1},w_{2}\in\\{1,\ldots,\sigma(P)\\},\;v_{1},v_{2}\in\\{1,\ldots,\sigma(R)\\}$
(53) $\displaystyle\overline{K}_{\rho V}$
$\displaystyle=\sum_{v_{1}=1}^{\sigma(R)}\sum_{v_{2}=1}^{\sigma(R)}\left(e_{v_{1}}^{\sigma(R)}\otimes
e_{v_{2}}^{\sigma(R)T}\right)\otimes A_{P}$ (54)
$\displaystyle\overline{K}_{\rho V}$
$\displaystyle=\left(\mathds{1}^{\sigma(R)}\otimes\mathds{1}^{\sigma(R)T}\right)\otimes
A_{P}$ (55)

###  III-D Sequence-Dependent Degrees of Freedom

Once the system sequence knowledge base and constraints matrix have been
calculated, the number of sequence-dependent degrees of freedom follow
straightforwardly.

######  Definition 12 (Sequence-Dependent Degrees of Freedom [13, 117, 118,
108, 119]):

The set of independent pairs of actions
$z_{\chi_{1}\chi_{2}}=e_{w_{1}v_{1}}e_{w_{2}v_{2}}\in{\cal Z}$ of length 2
that completely describe the system language. The number is given by:

$\displaystyle DOF_{\rho}=\sigma({\cal Z})$
$\displaystyle=\sum_{\chi_{1}}^{\sigma(R)\sigma(P)}\sum_{\chi_{2}}^{\sigma(R)\sigma(P)}[J_{\rho}\ominus
K_{\rho}](\chi_{1},\chi_{2})$ (56)
$\displaystyle=\sum_{\chi_{1}}^{\sigma(R)\sigma(P)}\sum_{\chi_{2}}^{\sigma(R)\sigma(P)}[A_{\rho}](\chi_{1},\chi_{2})$
(57)

For systems of substantial size, the size of the hetero-functional adjacency
matrix may be challenging to process computationally. However, the matrix is
generally very sparse. Therefore, projection operators are used to eliminate
the sparsity by projecting the matrix onto a one’s vector[108, 119]. This is
demonstrated below for $J_{S}^{V}$ and $A_{\rho}$:

$\displaystyle\mathds{P}_{S}J_{S}^{V}$ $\displaystyle=\mathds{1}^{\sigma({\cal
E}_{S})}$ (58) $\displaystyle\mathds{P}_{S}A_{\rho}\mathds{P}_{S}^{T}$
$\displaystyle=\widetilde{A}_{\rho}$ (59)

where $\mathds{P}_{S}$ is a (non-unique) projection matrix for the vectorized
system knowledge base and the hetero-functional adjacency matrix [108, 119].
Note that the number of sequence dependent degrees of freedom for the
projected hetero-functional adjacency matrix can be calculated as:

$DOF_{\rho}=\sigma({\cal Z})=\sum_{\psi_{1}}^{\sigma({\cal
E}_{S})}\sum_{\psi_{2}}^{\sigma({\cal
E}_{S})}[\widetilde{A}_{\rho}](\psi_{1},\psi_{2})$ (60)

where $\psi\in\left[1,\dots,\sigma({\cal E}_{S})\right]$.

##  IV Hetero-functional Incidence Tensor

This section serves to introduce the hetero-functional incidence tensor as
part of the paper’s original contribution. Subsection IV-A describes the
tensor in third-order form. Subsection IV-B then elaborates why it sometimes
useful to present this tensor in fourth-order form. Finally, Subsection IV-C
shows how matricizing the heter-functional incidence tensor (into second-order
form) can serve to reconstruct the hetero-functional adjacency matrix.

###  IV-A Third Order Form

To complement the concept of a hetero-functional adjacency matrix $A_{\rho}$
and its associated tensor ${\cal A}_{\rho}$, the hetero-functional incidence
tensor $\widetilde{\cal M}_{\rho}$ describes the structural relationships
between the physical capabilities (i.e. structural degrees of freedom) ${\cal
E}_{S}$, the system operands $L$, and the system buffers $B_{S}$.

$\widetilde{\cal M}_{\rho}=\widetilde{\cal M}_{\rho}^{+}-\widetilde{\cal
M}_{\rho}^{-}$ (61)

######  Definition 13 (The Negative 3rd Order Hetero-functional Incidence
Tensor $\widetilde{\cal M}_{\rho}^{-}$):

The negative hetero-functional incidence tensor $\widetilde{\cal
M_{\rho}}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(B_{S})\times\sigma({\cal
E}_{S})}$ is a third-order tensor whose element $\widetilde{\cal
M}_{\rho}^{-}(i,y,\psi)=1$ when the system capability
${\epsilon}_{\psi}\in{\cal E}_{S}$ pulls operand $l_{i}\in L$ from buffer
$b_{s_{y}}\in B_{S}$.

######  Definition 14 (The Positive 3rd Order Hetero-functional Incidence
Tensor $\widetilde{\cal M}_{\rho}^{+}$):

The positive hetero-functional incidence tensor $\widetilde{\cal
M}_{\rho}^{+}\in\\{0,1\\}^{\sigma(L)\times\sigma(B_{S})\times\sigma({\cal
E}_{S})}$ is a third-order tensor whose element $\widetilde{\cal
M}_{\rho}^{+}(i,y,\psi)=1$ when the system capability
${\epsilon}_{\psi}\in{\cal E}_{S}$ injects operand $l_{i}\in L$ into buffer
$b_{s_{y}}\in B_{S}$.

The calculation of these two tensors depends on the definition of two more
matrices which further depend on the hetero-functional graph theory
definitions in Appendix -C.

######  Definition 15 (The Negative Process-Operand Incidence Matrix
$M_{LP}^{-}$):

A binary incidence matrix $M_{LP}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(P)}$
whose element $M_{LP}^{-}(i,w)=1$ when the system process $p_{w}\in P$ pulls
operand $l_{i}\in L$ as an input. It is further decomposed into the negative
transformation process-operand incidence matrix $M_{LP_{\mu}}^{-}$ (Defn. 37)
and the negative refined transformation process-operand incidence matrix
$M_{LP_{\bar{\eta}}}^{-}$ (Defn. 38) which by definition is in turn calculated
from the negative holding process-operand incidence matrix
$M_{LP_{\gamma}}^{-}$ (Defn. 39).

$M_{LP}^{-}=\begin{bmatrix}M_{LP_{\mu}}^{-}&M_{LP_{\bar{\eta}}}^{-}\end{bmatrix}=\begin{bmatrix}M_{LP_{\mu}}^{-}&M_{LP_{\gamma}}^{-}\otimes\mathds{1}^{\sigma(P_{\eta})T}\end{bmatrix}$
(62)

######  Definition 16 (The Positive Process-Operand Incidence Matrix
$M_{LP}^{+}$):

A binary incidence matrix $M_{LP}^{+}\in\\{0,1\\}^{\sigma(L)\times\sigma(P)}$
whose element $M_{LP}^{+}(i,w)=1$ when the system process $p_{w}\in P$ injects
operand $l_{i}\in L$ as an output. It is further decomposed into the positive
transformation process-operand incidence matrix $M_{LP_{\mu}}^{+}$ (Defn. 40)
and the positive refined transformation process-operand incidence matrix
$M_{LP_{\bar{\eta}}}^{+}$ (Defn. 41) which, by definition, is, in turn,
calculated from the positive holding process-operand incidence matrix
$M_{LP_{\gamma}}^{+}$ (Defn. 42)

$M_{LP}^{+}=\begin{bmatrix}M_{LP_{\mu}}^{+}&M_{LP_{\bar{\eta}}}^{+}\end{bmatrix}=\begin{bmatrix}M_{LP_{\mu}}^{+}&M_{LP_{\gamma}}^{+}\otimes\mathds{1}^{\sigma(P_{\eta})T}\end{bmatrix}$
(63)

With the definitions of these incidence matrices in place, the calculation of
the negative and positive hetero-functional incidence tensors $\widetilde{\cal
M}_{\rho}^{-}$ and $\widetilde{\cal M}_{\rho}^{+}$ follows straightforwardly
as a third-order outer product. For $\widetilde{\cal M}_{\rho}^{-}$:

$\displaystyle\widetilde{\cal
M}_{\rho}^{-}=\sum_{i=1}^{\sigma(L)}\sum_{y_{1}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{1}}^{\sigma(B_{S})}\circ\mathds{P}_{S}\left(\left(X^{-}_{iy_{1}}\right)^{V}\right)$
(64)

where

$X^{-}_{iy_{1}}=\left[\begin{array}[]{c}M_{LP_{\mu}}^{-T}e_{i}^{\sigma(L)}e_{y_{1}}^{\sigma(M)T}\quad|\quad\mathbf{0}\\\
\hline\cr
M_{LP_{\gamma}}^{-T}e_{i}^{\sigma(L)}\otimes\left(e_{y_{1}}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\right)\otimes\mathds{1}^{\sigma(R)T}\\\
\end{array}\right]$ (65)

The $X^{-}_{iy_{1}}$ matrix is equivalent in size to the system concept
$A_{S}$. It has a value of one in all elements where the associated process
both withdraws input operand $l_{i}$ and originates at the buffer
$b_{s_{y_{1}}}$. Consequently, when $X^{-}_{iy_{1}}$ is vectorized and then
projected with $\mathds{P}_{S}$, the result is a vector with a value of one
only where the associated system capabilities meet these criteria.

For $\widetilde{\cal M}_{\rho}^{+}$:

$\displaystyle\widetilde{\cal
M}_{\rho}^{+}=\sum_{i=1}^{\sigma(L)}\sum_{y_{2}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{2}}^{\sigma(B_{S})}\circ\mathds{P}_{S}\left(\left(X^{+}_{iy_{2}}\right)^{V}\right)$
(66)

where

$X^{+}_{iy_{2}}=\left[\begin{array}[]{c}M_{LP_{\mu}}^{+T}e_{i}^{\sigma(L)}e_{y_{2}}^{\sigma(M)T}\quad|\quad\mathbf{0}\\\
\hline\cr
M_{LP_{\gamma}}^{+T}e_{i}^{\sigma(L)}\otimes\left(\mathds{1}^{\sigma(B_{S})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\right)\otimes\mathds{1}^{\sigma(R)T}\\\
\end{array}\right]$ (67)

The $X^{+}_{iy_{2}}$ matrix is equivalent in size to the system concept
$A_{S}$. It also has a value of one in all elements where the associated
process both injects output operand $l_{i}$ and terminates at the buffer
$b_{s_{y_{2}}}$. Consequently, when $X^{+}_{iy_{2}}$ is vectorized and then
projected with $\mathds{P}_{S}$, the result is a vector with a value of one
only where the associated system capabilities meet these criteria.

It is important to note that the definitions of the 3rd order hetero-
functional incidence tensors $\widetilde{\cal M}_{\rho}^{-}$, and
$\widetilde{\cal M}_{\rho}^{+}$ are provided in projected form as indicated by
the presence of the projection operator $\mathds{P}_{S}$ in Equations 64 and
66 respectively. It is often useful to use the un-projected form of these
tensors.

$\displaystyle{\cal
M}_{\rho}^{-}=\sum_{i=1}^{\sigma(L)}\sum_{y_{1}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{1}}^{\sigma(B_{S})}\circ\left(X^{-}_{iy_{1}}\right)^{V}$ (68)
$\displaystyle{\cal
M}_{\rho}^{+}=\sum_{i=1}^{\sigma(L)}\sum_{y_{2}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{2}}^{\sigma(B_{S})}\circ\left(X^{+}_{iy_{2}}\right)^{V}$ (69)

###  IV-B Fourth Order Form

The third dimension of these unprojected 3rd order hetero-functional incidence
tensors can then be split into two dimensions to create 4th order hetero-
functional incidence tensors.

$\displaystyle{\cal M}_{PR}^{+}$ $\displaystyle=vec^{-1}\left({\cal
M}_{\rho}^{+},[\sigma(P),\sigma(R)],3\right)$ (70) $\displaystyle{\cal
M}_{PR}^{-}$ $\displaystyle=vec^{-1}\left({\cal
M}_{\rho}^{-},[\sigma(P),\sigma(R)],3\right)$ (71)

These fourth order tensors describe the structural relationships between the
system processes $P$, the physical resources $R$ that realize them, the system
operands $L$ that are consumed and injected in the process, and the system
buffers $B_{S}$ from which these are operands are sent and the system buffers
$B_{S}$ to which these operands are received. They are used in the following
section as part of the discussion on layers.

${\cal M}_{PR}={\cal M}_{PR}^{+}-{\cal M}_{PR}^{-}$ (72)

######  Definition 17 (The Negative 4th Order Hetero-functional Incidence
Tensor ${\cal M}_{PR}^{-}$):

The negative 4th Order hetero-functional incidence tensor ${\cal
M}_{PR}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(B_{S})\times\sigma(P)\times\sigma(R)}$
has element ${\cal M}_{PR}^{-}(i,y,w,v)=1$ when the system process $p_{w}\in
P$ realized by resource $r_{v}\in R$ pulls operand $l_{i}\in L$ from buffer
$b_{s_{y}}\in B_{S}$.

######  Definition 18 (The Positive 4th Order Hetero-functional Incidence
Tensor ${\cal M}_{PR}^{-}$):

The positive 4th Order hetero-functional incidence tensor ${\cal
M}_{PR}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(B_{S})\times\sigma(P)\times\sigma(R)}$
has element ${\cal M}_{PR}^{-}(i,y,w,v)=1$ when the system process $p_{w}\in
P$ realized by resource $r_{v}\in R$ injects operand $l_{i}\in L$ into buffer
$b_{s_{y}}\in B_{S}$.

Furthermore, the negative and positive 4th hetero-functional incidence tensors
can be used to demonstrate a direct relationship to the system concept
$A_{S}$.

$\displaystyle{\cal M}_{PR}^{-}(i,y,w,v)=$ $\displaystyle
X^{-}_{iy_{1}}(w,v)\cdot A_{S}(w,v)$ (73) $\displaystyle{\cal M}_{PR}^{-}=$
$\displaystyle\sum_{i}^{\sigma(L)}\sum_{y_{1}}^{\sigma(B_{S})}X^{-}_{iy_{1}}\cdot
A_{S}$ (74) $\displaystyle{\cal M}_{PR}^{+}(i,y,w,v)=$ $\displaystyle
X^{+}_{iy_{2}}(w,v)\cdot A_{S}(w,v)$ (75) $\displaystyle{\cal M}_{PR}^{+}=$
$\displaystyle\sum_{i}^{\sigma(L)}\sum_{y_{2}}^{\sigma(B_{S})}X^{+}_{iy_{2}}\cdot
A_{S}$ (76)

Equations 74 and 76 show that the 4th order hetero-functional incidence
tensors contain three types of information:

  1. 1.

the mapping of system processes to system resources in the system concept
$A_{S}$,

  2. 2.

the mapping of processes to their operands in $M_{LP}^{-}$ and $M_{LP}^{+}$,

  3. 3.

the implicit knowledge that by definition transformation processes occur at a
stationary buffer, and that transportation processes are defined by their
origin and destination buffers.

In other words, the hetero-functional incidence tensor is a complete descption
of a system’s allocated architecture.

###  IV-C Second Order Form

Returning back to the third-order hetero-functional incidence tensor
$\widetilde{M}_{\rho}$, it and and its positive and negative components
$\widetilde{M}_{\rho}^{+},\widetilde{M}_{\rho}^{-}$, can also be easily
matricized.

$\displaystyle M_{\rho}$ $\displaystyle={\cal F}_{M}\left({\cal
M}_{\rho},[1,2],[3]\right)$ (77) $\displaystyle M_{\rho}^{-}$
$\displaystyle={\cal F}_{M}\left({\cal M}_{\rho}^{-},[1,2],[3]\right)$ (78)
$\displaystyle M_{\rho}^{+}$ $\displaystyle={\cal F}_{M}\left({\cal
M}_{\rho}^{+},[1,2],[3]\right)$ (79)

The resulting matrices have a size of
$\sigma(L)\sigma(B_{S})\times\sigma(\cal{E_{S}})$ which have a corresponding
physical intuition. Each buffer $b_{s_{y}}$ has $\sigma(L)$ copies to reflect
a place (i.e. bin) for each operand at that buffer. Each of these places then
forms a bipartite graph with the system’s physical capabilities. Consequently,
and as expected, the hetero-functional adajacency matrix $A_{\rho}$ can be
calculated as a matrix product of the positive and negative hetero-functional
incidence matrices $M_{\rho}^{+}$ and $M_{\rho}^{+}$.

$A_{\rho}=M_{\rho}^{+T}\odot M_{\rho}^{-}=M_{\rho}^{+T}M_{\rho}^{-}$ (80)

Such a product systematically enforces all five of the feasibility constraints
identified in Section III. Furthermore, the Boolean and real matrix products
are interchangeable because each process is associated with exactly one
origin-destination pair.

##  V Discussion

Given the discussion on multi-layer networks in the introduction, it is
worthwhile reconciling the gap in terminology between multi-layer networks and
hetero-functional graph theory. First, the concept of layers in hetero-
functional graphs is discussed. Second, an ontological comparison of layers in
hetero-functional graphs and multi-layer networks is provided. Third, a
discussion of network descriptors in the context of layers is provided. Given
the “disparate terminology and the lack of consensus” in the multi-layer
network literature, the discussion uses the multi-layer description provided
De Dominico et. al[14].

###  V-A Layers in Hetero-functional Graphs

######  Definition 19 (Layer):

[120] A layer $G_{\lambda}=\\{{\cal E}_{S\lambda},Z_{S\lambda}\\}$ of a
hetero-functional graph $G=\\{{\cal E}_{S},Z_{S}\\}$ is a subset of a hetero-
functional graph, $G_{\lambda}\subseteq G$, for which a predefined layer
selection (or classification) criterion applies. A set of layers in a hetero-
functional graph adhere to a classification scheme composed of a number of
selection criteria.

Note that this definition of a layer is particularly flexible because it
depends on the nature of the classification scheme and its associated
selection criteria. Nevertheless, and as discussed later, it is important to
choose a classification scheme that leads to a set of mutually exclusive
layers that are also collectively exhaustive of the hetero-functional graph as
a whole.

To select out specific subsets of capabilities (or structural degrees of
freedom), HFGT has used the concept of “selector matrices” of various types[7,
121]. Here a layer selector matrix is defined.

######  Definition 20:

[120] Layer Selector Matrix: A binary matrix $\Lambda_{\lambda}$ of size
$\sigma(P)\times\sigma(R)$ whose element $\Lambda_{\lambda}(w,v)=1$ when the
capability $e_{wv}\subset E_{S\lambda}$.

From this definition, the calculation of a hetero-functional graph layer
follows straightforwardly. First, a layer projection operator
$\mathds{P}_{\lambda}$ is calculated[120]:

$\displaystyle\mathds{P}_{\lambda}\Lambda_{\lambda}^{V}$
$\displaystyle=\mathds{1}^{\sigma({\cal E}_{S})}$ (81)

Next, the negative and positive hetero-functional incidence tensors
$\widetilde{\cal M}_{\rho\lambda}^{-}$ and $\widetilde{\cal
M}_{\rho\lambda}^{+}$ for a given layer $\lambda$ are calculated
straightforwardly[120].

$\displaystyle\widetilde{\cal
M}_{\rho\lambda}^{-}=\sum_{i=1}^{\sigma(L)}\sum_{y_{1}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{1}}^{\sigma(B_{S})}\circ\mathds{P}_{\lambda}\left(\left(X^{-}_{iy_{1}}\right)^{V}\right)=\widetilde{\cal
M}_{\rho}^{-}\odot_{3}\mathds{P}_{\lambda}$ (82) $\displaystyle\widetilde{\cal
M}_{\rho\lambda}^{+}=\sum_{i=1}^{\sigma(L)}\sum_{y_{2}=1}^{\sigma(B_{S})}e_{i}^{\sigma(L)}\circ
e_{y_{2}}^{\sigma(B_{S})}\circ\mathds{P}_{S}\left(\left(X^{+}_{iy_{2}}\right)^{V}\right)=\widetilde{\cal
M}_{\rho}^{+}\odot_{3}\mathds{P}_{\lambda}$ (83)

From there, the positive and negative hetero-functional incidence tensors for
a given layer can be matricized and the adjacency matrix of the associated
layer $\widetilde{A}_{\rho\lambda}$ follows straightforwardly[120].

$\displaystyle\widetilde{M}_{\rho\lambda}^{+}$ $\displaystyle={\cal
F}_{M}(\widetilde{\cal M}_{\rho\lambda}^{+},[1,2],[3])$ (84)
$\displaystyle\widetilde{M}_{\rho\lambda}^{-}$ $\displaystyle={\cal
F}_{M}(\widetilde{\cal M}_{\rho\lambda}^{-},[1,2],[3])$ (85)
$\displaystyle\widetilde{A}_{\rho\lambda}$
$\displaystyle=\widetilde{M}_{\rho\lambda}^{+T}\odot\widetilde{M}_{\rho\lambda}^{-}$
(86)

This approach of separating a hetero-functional graph into its constituent
layers is quite generic because the layer selector matrix $\Lambda_{\lambda}$
can admit a wide variety of classification schemes. Three classification
schemes are discussed here:

  1. 1.

An Input Operand Set Layer

  2. 2.

An Output Operand Set Layer

  3. 3.

A Dynamic Device Model Layer

Fig. 4: The Trimetric Smart City Infrastructure Test Case Visualized as Five
Layers Defined by Input Operand Sets: The Potable Water Topology, The
Electrified Potable Water Topology, the Electric Power Topology, the Charging
Topology, and the Transportation Topology

######  Definition 21:

Input Operand Set Layer: A hetero-functional graph layer for which all of the
node-capabilities have a common set of input operands $L_{\lambda}\subseteq
L$.

This definition of an Operand Set Layer was used in the HFGT text[7] to
partition the Trimetrica test case (first mentioned in Figure 1) into the
multi-layer depiction in Figure 4. In this classification scheme, any system
contains up to 2σ(L) possible layers. For completeness, an index
$\lambda_{D}\in\\{1,\dots,2^{\sigma(L)}\\}$ is used to denote a given layer.
In reality, however, the vast majority of physical systems exhibit far fewer
than 2σ(L) layers. Consequently, it is often useful to simply assign an index
$\lambda$ to each layer and create a 1-1 mapping function (i.e. lookup table)
$f_{\lambda}$ back to the $\lambda_{D}$ index.

$f_{\lambda}:\lambda\rightarrow\lambda_{D}$ (87)

The utility of the $\lambda_{d}$ index (stated as a base 10 number) becomes
apparent when it is converted into a binary (base 2) number
$\lambda_{v}\in\\{0,1\\}^{\sigma(L)}$ which may be used equivalently as a
binary vector of the same length.

$\lambda_{v}=bin(\lambda_{D})$ (88)

The resulting binary vector $\lambda_{v}$ has the useful property that
$\lambda_{v}(l_{i})=1$, iff operand $l_{i}\in L_{\lambda}$. Consequently, a
given value of $\lambda_{v}$ serves to select from $L$ the operands that
pertain to layer $\lambda$. The associated layer selector matrix follows
straightforwardly:

$\Lambda_{\lambda}(w,v)=\left\\{\begin{array}[]{cc}1&\mbox{if}\quad\lambda_{v}=M_{LP}^{-}(:,w)\quad\forall
r_{v}\in R\\\ 0&\mbox{otherwise}\end{array}\right.$ (89)

It is also worth noting that the layer selector matrix $\Lambda$ above is
effectively a third order tensor whose value $\Lambda(\lambda,w,v)=1$ when the
capability $e_{wv}$ is part of layer $\lambda$.

One advantage of a classification scheme based on _sets_ of input operands is
that they lead to the generation of a mutually exclusive and collectively
exhaustive set of layers. Because no process (and consequently capability) has
two sets of input operands, it can only exist in a single layer (mutual
exclusivity). In the meantime, the presence of $2^{\sigma(L)}$ assures that
all capabilities fall into (exactly) one layer (exhaustivity). It is worth
noting that a classification scheme based on _individual_ operands would not
yield these properties. For example, a water pump consumes electricity and
water as input operands. Consequently, it would have a problematic existence
in both the “water layer” as well as the “electricity layer”. In contrast, a
classification scheme based on operand sets creates an “electricity-water”
layer.

Analogously to Defn. 21, an output-operand set layer can be defined and its
associated layer selector matrix calculated.

######  Definition 22:

Output Operand Set Layer: A hetero-functional graph layer for which all of the
node-capabilities have a common set of output operands $L_{\lambda}\subseteq
L$.

$\Lambda_{\lambda}(w,v)=\left\\{\begin{array}[]{cc}1&\mbox{if}\quad\lambda_{v}=M_{LP}^{+}(:,w)\quad\forall
r_{v}\in R\\\ 0&\mbox{otherwise}\end{array}\right.$ (90)

The third classification scheme is required when developing dynamic equations
of motion from the structural information of a hetero-functional graph[117,
122]. Every process is said to have a “dynamic device model” that is usually
described as a set of differential, algebraic, or differential-algebraic
equations[117, 122]. The simplest of these are the constitutive laws of basic
dynamic system elements (e.g. resistors, capacitors, and inductors). Some
processes, although distinct, may have device models with the same functional
form. For example, two resistors at different places in an electrical system
have the same constitutive (Ohm’s) law, but have different transportation
processes because their origin and destinations are different. Consequently,
layers that distinguish on the basis of dynamic device model (i.e.
constitutive law) are necessary.

######  Definition 23:

Dynamic Device Model Layer: A hetero-functional graph layer for which all of
the node-capabilities have a dynamic device model with the same functional
form.

In such a case, the layer selector matrix $\Lambda_{\lambda}$
straightforwardly maps capabilities to their layer and dynamic device model
interchangeably. A sufficient number of layers need to be created to account
for all of the different types of dynamic device models in the system. This
classification scheme may be viewed as a generalization of the well-known
literature on “linear-graphs”[123] and “bond graphs”[124].

###  V-B Finding Commonality between Multilayer Networks and Hetero-functional
Graphs

The above discussion of layers in a hetero-functional graph inspires a
comparison with multi-layer networks. The multi-layer adjacency tensor (${\cal
A}_{MLN}$) defined by De Dominico et. al.[14] is chosen to facilitate the
discussion. This fourth order tensor has elements ${\cal
A}_{MLN}(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ where the indices
$\alpha_{1},\alpha_{2}$ denote “vertices” and $\beta_{1},\beta_{2}$ denote
“layers”. De Dominico et. al write that this multilayer adjacency tensor is a
[14]: _“…very general object that can be used to represent a wealth of
complicated relationships among nodes.”_ The challenge in reconciling the
multi-layer adjacency tensor ${\cal A}_{MLN}$ and the hetero-functional
adjacency tensor ${\cal A}_{\rho}$ is an ontological one. Referring back to
the ontological discussion in the introduction and more specifically Figure 2
reveals that the underlying abstract conceptual elements (in the mind) to
which these two mathematical models refer may not be the same.

Consider the following interpretation of ${\cal
A}_{MLN}(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})={\cal
A}_{B_{S}l_{1}}(y_{1},y_{2},i_{1},i_{2})$ where the multi-layer network’s
vertices are equated to the buffers $B_{S}$ and the layers are equated to the
operands $L$. This interpretation would well describe the departure of an
operand $l_{i_{1}}$ from buffer $b_{sy_{1}}$ and arriving as $l_{i_{2}}$ at
$b_{sy_{2}}$. The equivalence of vertices to buffers is effectively a
consensus view in the literature. In contrast, the concept of a “layer” in a
multi-layer network (as motivated in the introduction) remains relatively
unclear. The equivalence of layers to operands warrants further attention.

######  Theorem 1:

The mathematical model $A_{B_{S}l_{1}}$ is neither lucid nor complete with
respect to the system processes $P$ (as an abstraction).

######  Proof 1:

By contradiction. Assume that ${\cal A}_{B_{S}l_{1}}$ is both lucid and
complete network model with respect to system processes $P$. Consider an
operand $l_{1}$ that departs $b_{s1}$, undergoes process $p_{1}$, and arrives
as $l_{1}$ at $b_{s2}$. Now consider the same operand $l_{1}$ that departs
$b_{s1}$, undergoes process $p_{2}$, and arrives as $l_{1}$ at $b_{s2}$. Both
of these scenarios would be denoted by ${\cal A}_{B_{S}l_{1}}(1,2,1,1)=1$.
Consequently, this modeling element is overloaded and as such violates the
ontological property of lucidity. Furthermore, because ${\cal A}_{B_{S}l_{1}}$
makes no mention of the concept of system processes, then it violates the
completeness property as well.

The counter-example provided in the proof above is not simply a theoretical
abstraction but rather quite practical. For several decades, the field of
mechanical engineering has used “linear graphs”[123] to derive the equations
of motion of dynamic systems with multi-domain physics. Consider the RLC
circuit shown in Figure 5 and its associated linear graph. As parallel
elements, the inductor and capacitor both transfer electrical power (as an
operand) between the same pair of nodes. However, the constitutive law (as a
physical process) of a capacitor is distinct from that of the inductor.
Consequently, the interpretation ${\cal A}_{B_{S}l_{1}}$ of a multi-layer
network is inadequate even for this very simple counter-example333Although
electric power systems and circuits have served as a rich application domain
for graph theory and network science, these approaches usually parameterize
the circuit components homogeneously as a fixed-value impedance/admittance at
constant frequency. When the constant frequency assumption is relaxed, the
diversity of constitutive laws for resistors, capacitors, and inductors must
be explicitly considered..

Fig. 5: A Simple RLC Circuit shown as a circuit diagram on left and as a
linear graph model on right. Each resistor, capacitor and inductor can be said
to be part of its own layer by virtue of their distinct constitutive laws.

Another possible interpretation of a multi-layer network is ${\cal
A}_{MLN}(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ = ${\cal
A}_{B_{S}l_{2}}(y_{1},y_{2},w_{1},w_{2})$ where the multi-layer network’s
vertices are equated to the buffers $B_{S}$ and the layers are equated to the
processes $P$. This interpretation would well describe the execution of a
process $p_{w_{1}}$ that is realized by buffer $b_{sy_{1}}$ followed by a
process $p_{w_{2}}$ that is realized by buffer $b_{sy_{2}}$. The equivalence
of layers to processes warrants further attention as well.

######  Theorem 2:

The mathematical model $A_{B_{S}l_{2}}$ is neither lucid nor complete with
respect to the system’s transportation resources $H$ (as an abstraction).

######  Proof 2:

By contradiction. Assume that ${\cal A}_{B_{S}l_{2}}$ is both a lucid and
complete network model with respect to system’s transportation resources $H$.
Consider transportation process $p$ between a buffer $b_{s1}$ and a distinct
buffer $b_{s2}$. If such a transportation process were realized by any buffer
$b_{s}\in B_{S}$, then by definition it would no longer be a buffer but rather
a transportation resource. Consequently, ${\cal A}_{B_{S}l_{2}}$ is not
complete with respect the system’s transportation resources $H$. Now consider
a process $p_{1}$ that is realized by buffer $b_{s1}$ followed by a process
$p_{2}$ that is realized by a distinct buffer $b_{s2}$. This is denoted by
${\cal A}_{B_{S}l_{2}}(1,2,1,2)=1$. Given the distinctness of $b_{s1}$ and
$b_{s2}$, a transportation process must have happened in between $p_{1}$ and
$p_{2}$ although it is not explicitly stated by the mathematical statement
${\cal A}_{B_{S}l_{2}}(1,2,1,2)=1$. Such a transportation process, although
well-defined by its origin and destination could have been realized by any one
of a number of transportation resources. Consequently, the modeling element is
overloaded and as such violates the property of lucidity. The lack of an
explicit description of transportation processes or resources limits the
utility of this type of multi-layer network model.

It is worth noting that the first multi-layer network interpretation ${\cal
A}_{B_{S}l_{1}}$ can be derived directly from the positive and negative
hetero-functional incidence matrices[120].

$\displaystyle{\cal A}_{B_{S}L_{1}}(y_{1},y_{2},i_{1},i_{2})=$
$\displaystyle\bigvee_{\psi}{\cal M}_{\rho}^{-}(i,y_{1},\psi)\cdot{\cal
M}_{\rho}^{+}(i,y_{2},\psi)$ (91) $\displaystyle{\cal A}_{B_{S}l_{1}}=$
$\displaystyle{\cal F}_{M}^{-1}\left(M_{\rho}^{-T}\odot
M_{\rho}^{+},[\sigma(B_{S}),\sigma(B_{S}),\sigma(L),\sigma(L)],[1,3],[2,4]\right)$
(92)

When $M_{\rho}^{-T}$ and $M_{\rho}^{+}$ are multiplied so that the
capabilities ${\cal E}_{s}$ are the inner dimension, the result is an
adjacency matrix that when tensorized becomes ${\cal A}_{B_{S}l_{1}}$. In
effect, ${A}_{B_{S}l_{1}}$ (in matricized form) is the dual adjacency
matrix[125] of the hetero-functional adjacency matrix $A_{\rho}$. The presence
of this matrix multiplication obfuscates (i.e. creates a lack of lucidity) as
to whether one capability or another occurred when expressing the adjacency
tensor element ${\cal A}_{B_{S}l_{1}}(y_{1},y_{2},i_{1},i_{2})$. In contrast,
the matrix multiplication in Eq. 80 does not cause the same problem. When two
capabilities succeed one another, the information associated with their
physical feasibility in terms of intermediate buffers and their functional
feasibility in terms of intermediate operands remains intact. In other words,
given the sequence of capabilities $e_{w_{1}v_{1}}e_{w_{2}v_{2}}$, one can
immediately deduce the exchanged operands in $L_{\lambda}\subseteq L$ and the
intermediate buffer $b_{s}$. In the case of the exchanged operands, one simply
needs to intersect the output-operand set of the first process with the input
operand set of the second process. In the case of the intermediate buffer, one
checks if either or both of the resources are buffers. If not, then two
transportation processes followed one another and the intermediate buffer is
deduced by Eq. 95. In short, the hetero-functional adjacency matrix (or
tensor) unambigously describes the sequence of two subject+verb+operand
sentences whereas neither of the above interpretations of a multi-layer
network do.

###  V-C Network Descriptors

In light of the commonalities and differences between hetero-functional graphs
and (formal) multilayer networks, this section discusses the meaning of
network descriptors in the context of hetero-functional graphs. In this
regard, the hetero-functional adjacency matrix is an adjacency matrix like any
other. Consequently, network descriptors can be calculated straightforwardly.
Furthermore, network descriptors can be applied to subsets of the graph so as
to conduct a layer-by-layer analysis. Nevertheless, that the nodes in a
hetero-functional graph represent whole-sentence-capabilities means that
network descriptors have the potential to provide new found meanings over
formal graphs based on exclusively formal elements.

####  V-C1 Degree Centrality

Degree centrality measures the number of edges attached to a vertex. Since a
hetero-functional graph is a directed graph, there is a need to distinguish
between the in-degree centrality, which measures the number of edges going
into vertex, and the out-degree centrality, which measures the number of edges
going out of a vertex [100]. In the context of hetero-functional graph theory,
the in-degree centrality of a vertex calculates the number of capabilities
that potentially proceed the capability related to the vertex. The out-degree
centrality calculates the number of capabilities that potentially succeed the
vertex’s capability. The higher the degree centrality of a capability, the
more connected that capability is to the other capabilities in the hetero-
functional graph. It is important to recognize that because transportation
capabilities receive nodes in a hetero-functional graph, they can become the
most central node. In contrast, the degree centrality of a formal graph could
not reach such a conclusion because the function of transportation is tied to
formal edges rather than formal nodes.

####  V-C2 Closeness Centrality

Closeness centrality measures the average shortest path from one vertex to
every other reachable vertex in the graph. In a hetero-functional graph, the
meaning of closeness centrality shows how a disruption has the potential to
propagate through the graph across all different types of operands [100]. This
metric is especially valuable for the resilience studies of interdependent
systems, where the propagation of disruption across multiple disciplines is
often poorly understood.

####  V-C3 Eigenvector Centrality

Eigenvector centrality calculates the importance of a node relative to the
other nodes in the network [126]. It also includes the eigenvector centrality
of the node’s direct neighbors [14]. The eigenvector centrality is
specifically designed for the weighting of the in-degree of nodes in a
directed network. The _Katz centrality_ , on the other hand, provides an
approach to study the relative importance of nodes based on the out-degree
[14].

####  V-C4 Clustering Coefficients

Clustering coefficients describe how strongly the nodes of a network cluster
together. This is performed by searching for “triangles” or “circles” of nodes
in a network. In a directed network, these circles can appear in multiple
distinct combinations of directed connections. Each of these combinations
needs to be measured and counted differently. Fagiolo discussed this taxonomy
and accompanying clustering coefficients [127]. These clustering coefficients
for directed networks can be directly applied to hetero-functional graphs and
show which capabilities are strongly clustered together. The definition of
layers in hetero-functional graphs allows for a consistent definition and
calculation of clustering coefficients within and across layers for different
types of systems. When investigating a system, the clustering coefficient may
show clusters of capabilities that were not yet recognized as heavily
interdependent. Such information can be used to revise control structures such
that clusters of capabilities are controlled by the same entity for
efficiency.

####  V-C5 Modularity

Modularity serves as a measure to study if a network can be decomposed in
disjoint sets. In the hetero-functional graph theory literature, much has been
published about modularity as it was a prime motivation towards the inception
of the theory [10, 128]. Hetero-functional graph theory introduces the concept
of the _Degree-of-Freedom-based Design Structure Matrix_ (or: the capability
DSM) that does not only encompass the hetero-functional adjacency matrix, but
extends the concept to the other elements of hetero-functional graph theory:
the service model and the control model. The hetero-functional graph design
structure matrix has the ability to visualize the couplings between the
subsystems of an engineering system and to classify those interfaces. Note
that the capability DSM can also be applied to just the hetero-functional
adjacency matrix. Furthermore, the capability DSM applies to the concept of
layers in a hetero-functional graph. To study the interfaces between layers,
the capability DSM can adopt layers as subsystems and classify the interfaces
between the layers as mentioned previously. In conclusion, hetero-functional
graphs are described by flat adjacency matrices, regardless of the number of
layers in the analysis. Consequently, conventional graph theoretic network
descriptors can be applied. The main difference in definition between the
conventional graph theoretic application and the hetero-functional graph
theoretic application is the result of the difference in the _definition of
the fundamental modeling elements_ , the nodes and edges, in a hetero-
functional graph.

##  VI Conclusions and Future Work

This paper has provided a tensor-based formulation of several of the most
important parts of hetero-functional graph theory. More specifically, it
discussed the system concept showing it as a generalization of formal graphs
and multi-commodity networks. It also discussed the hetero-functional
adjacency matrix and its tensor-based closed form calculation. It also
discussed the hetero-functional incidence tensor and related it back to the
hetero-functional adjacency matrix. The tensor-based formulation described in
this work makes a stronger tie between HFGT and its ontological foundations in
MBSE. Finally, the tensor-based formulation facilitates an understanding of
the relationships between HFGT and multi-layer networks “despite its disparate
terminology and lack of consensus”. In so doing, this tensor-based treatment
is likely to advance Kivela et. al’s goal to discern the similarities and
differences between these mathematical models in as precise a manner as
possible.

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###  -A Ontological Science Definitions

The formal definitions of soundness, completeness, lucidity, and laconicity
rely on “Ullman’s Triangle” in Figure 6.

Fig. 6: Two Versions of Ullman’s Triangle. On the left is the relationship
between reality, the understanding of reality, and the description of reality.
On the right the instantiated version of the ontological definition[95].

######  Definition 24 (Soundness[96]):

A language ${\cal L}$ is sound w.r.t. a domain conceptualization ${\cal C}$
iff every modeling primitive in the language (${\cal M}$) has an
interpretation in the domain abstraction ${\cal A}$. (The absence of soundness
results in the excess of modeling primitives w.r.t. the domain abstractions as
shown in Figure 2.c on lucidity.)

######  Definition 25 (Completeness[96]):

A language ${\cal L}$ is complete w.r.t. a domain conceptualization ${\cal C}$
iff every concept in the domain abstraction ${\cal A}$ of that domain is
represented in a modeling primitive of that language. (The absence of
completeness results in one or more concepts in the domain abstraction not
being represented by a modeling primitive, as shown in Figure 2.d on
laconicity.)

######  Definition 26 (Lucidity[96]):

A language ${\cal L}$ is lucid w.r.t. a domain conceptualization ${\cal C}$
iff every modeling primitive in the language represents at most one domain
concept in abstraction ${\cal A}$. (The absence of lucidity results in the
overload of a modeling primitive w.r.t. two or more domain concepts as shown
in Figure 2.a on soundness.)

######  Definition 27 (Laconicity[96]):

A language ${\cal L}$ is laconic w.r.t. a domain conceptualization ${\cal C}$
iff every concept in the abstraction ${\cal A}$ of that domain is represented
at most once in the model of that language. (The absence of laconicity results
in the redundancy of modeling primitives w.r.t the domain abstractions as
shown in Figure 2.b on completeness.)

###  -B Notation Conventions

Several notation conventions are used throughout this work:

  * •

All sets are indicated by a capital letter. e.g. P – the set of processes..

  * •

All elements within a set are indicated by a lower case letter. e.g. $p\in P$.

  * •

A subscript number indicates the position in an ordered set. e.g. $p_{i}\in
P$.

  * •

The $i^{th}$ elementary basis vector of size n is denoted by $e_{i}^{n}$.

  * •

A matrix of ones of size $m\times n$ is denoted by $\mathds{1}^{m\times n}$.

  * •

A matrix of zeros of size $m\times n$ is denoted by $\mathbf{0}^{m\times n}$.

  * •

With the exception of elementary basis vectors, all vectors and matrices are
indicated with a capital letter. e.g. $J_{H}$.

  * •

All tensors are indicated with capital letters in calligraphic script. e.g.
$\cal{J}_{H}$.

  * •

All elements in vectors, matrices, and tensors are indicated with indices
within parentheses. e.g. $J_{S}(w,v)$.

  * •

A(:,i) denotes the $i^{th}$ column of A or equivalently the $i^{th}$ mode-1
fiber. The : indicates all elements of the vector.

  * •

A(i,:) denotes the $i^{th}$ row of A or equivalently the $i^{th}$ mode-2
fiber.

  * •

A(i,j,:) denotes the $i,j$ mode-3 fiber of $\cal A$.

  * •

Given the presence of Booleans, real numbers and their operators, this work
refrains from the use of Einstein’s (shorthand) tensor notation where the
sigma-notation $\sum$ is eliminated.

###  -C Hetero-functional Graph Theory Definitions

######  Definition 28 (Transformation Resource[7]):

A resource $r\in R$ is a transformation resource $m\in M$ iff it is capable of
one or more transformation processes on one or more operands and it exists at
a unique location in space.

######  Definition 29 (Independent Buffer[7]):

A resource $r\in R$ is an independent buffer $b\in B$ iff it is capable of
storing one or more operands and is not able to transform them or transport
them to another location and it exists at a unique location in space.

######  Definition 30 (Transportation Resource[7]):

A resource $r\in R$ is a transportation resource $h\in H$ iff it is capable of
transporting one or more operands between an origin and a distinct
destination, without transforming these operands.

######  Definition 31 (Buffer[7]):

A resource $r\in R$ is a buffer $b_{s}\in B_{S}$ iff it is capable of storing
one or more operands at a unique location in space. $B_{S}=M\cup B$.

######  Definition 32 (Transformation Process[7]):

A process is a transformation process $p_{\mu j}\in P_{\mu}$ iff it is capable
of transforming one or more properties of a set of operands into a distinct
set of output properties in place. It’s syntax is:

$\\{\mbox{transitive verb, operands}\\}\rightarrow\\{\mbox{outputs}\\}$ (93)

######  Definition 33 (Refined Transportation Process[7]):

A process is a refined transportation process $p_{\bar{\eta}\varphi}\in
P_{\bar{\eta}}$ iff it is capable of transporting one or more operands between
an origin buffer $b_{sy_{1}}\in B_{S}$ to a destination buffer $b_{sy_{2}}\in
B_{S}$ while it is realizing holding process $p_{\gamma g}\in P_{\gamma}$.
It’s syntax is:

$\displaystyle\\{\mbox{transport, operands, origin, destination,while
transitive verb}\\}\rightarrow$ $\displaystyle\\{\mbox{outputs,
destination}\\}$ (94)

######  Definition 34 (Transportation Process[7]):

A process is a transportation process $p_{\eta u}\in P_{\eta}$ iff it is
capable of transporting one or more operands between an origin buffer
$b_{sy_{1}}\in B_{S}$ to a destination buffer $b_{sy_{2}}\in B_{S}$ according
to the following convention of indices[9, 10, 11, 12, 13]444Note that a
“storage process” is merely a transportation process with the same origin and
destination.:

$u=\sigma(B_{S})(y_{1}-1)+y_{2}$ (95)

It’s syntax is:

$\displaystyle\\{\mbox{transport, operands,}$
$\displaystyle\mbox{origin,destination}\\}\rightarrow\\{\mbox{outputs,destination}\\}$
(96)

######  Definition 35 (Holding Process[7]):

A process is a holding process $p_{\gamma g}\in P_{\gamma}$ iff it holds one
or more operands during the transportation from one buffer to another. In
order to maintain the independence axiom and the mutual exclusivity of the
system processes (Theorem 3), holding processes are specified so as to
distinguish between transportation processes that:

  * •

Have _different_ operands,

  * •

Hold a given operand in a _given way_ , or

  * •

Change the _state_ of the operand.

######  Theorem 3 (Mutual Exclusivity of System Processes[7]):

A lucid representation of system processes as a domain conceptualization
distinguishes between two system processes as modeling primitives with
different sets of inputs and outputs.

######  Definition 36 (Capability[9, 10, 11, 12, 13, 94]):

An action $e_{wv}\in{\cal E}_{S}$ (in the SysML sense) defined by a system
process $p_{w}\in P$ being executed by a resource $r_{v}\in R$. It constitutes
a subject + verb + operand sentence of the form: “Resource $r_{v}$ does
process $p_{w}$”.

######  Definition 37 (The Negative Transformation Process-Operand Incidence
Matrix $M_{LP_{\mu}}^{-}$):

A binary incidence matrix
$M_{LP_{\mu}}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\mu})}$ whose element
$M_{LP}^{-}(i,j)=1$ when the transformation system process $p_{\mu_{j}}\in P$
pulls operand $l_{i}\in L$ as an input.

######  Definition 38 (The Negative Refined Transportation Process-Operand
Incidence Matrix $M_{LP_{\bar{\eta}}}^{-}$):

A binary incidence matrix
$M_{LP_{\bar{\eta}}}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\bar{\eta}})}$
whose element $M_{LP}^{-}(i,\varphi)=1$ when the refined transportation
process $p_{\varphi}\in P_{\bar{\eta}}$ pulls operand $l_{i}\in L$ as an
input. It is calculated directly from the negative holding process-operand
incidence matrix $M_{LP_{\gamma}}^{-}$.

$\displaystyle M_{LP_{\bar{\eta}}}^{-}(i,\varphi)$
$\displaystyle=\sum_{u=1}^{\sigma(P_{\eta})}M_{LP_{\gamma}}^{-}(i,g)\cdot\delta_{u}\qquad\forall
i\in\\{1,\ldots\sigma(L)\\},\;g\in\\{1,\ldots,\sigma(P_{\gamma})\\},u\in\\{1,\ldots,\sigma(P_{\eta})\\},\varphi=\sigma(P_{\eta})(g-1)+u$
(97) $\displaystyle M_{LP_{\bar{\eta}}}^{-}(i,\varphi)$
$\displaystyle=\sum_{y_{1}=1}^{\sigma(B_{S})}\sum_{y_{2}=1}^{\sigma(B_{S})}M_{LP_{\gamma}}^{-}(i,g)\cdot\delta_{y_{1}}\cdot\delta_{y_{2}}\quad\forall
y_{1},y_{2}\in\\{1,\ldots,\sigma(B_{S})\\},\varphi=\sigma^{2}(B_{S})(g-1)+\sigma(B_{S})(y_{1}-1)+y_{2}$
(98) $\displaystyle M_{LP_{\bar{\eta}}}^{-}$
$\displaystyle=\sum_{u=1}^{\sigma(P_{\eta})}M_{LP_{\gamma}}^{-}\otimes
e_{u}^{\sigma(P_{\eta})T}\quad\;\;=\sum_{y_{1}=1}^{\sigma(B_{S})}\sum_{y_{2}=1}^{\sigma(B_{S})}M_{LP_{\gamma}}^{-}\otimes\left(e_{y_{1}}^{\sigma(B_{S})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\right)^{T}$ (99)
$\displaystyle=M_{LP_{\gamma}}^{-}\otimes\mathds{1}^{\sigma(P_{\eta})T}\qquad\quad=M_{LP_{\gamma}}^{-}\otimes\left(\mathds{1}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\right)^{T}$
(100)

######  Definition 39 (The Negative Holding Process-Operand Incidence Matrix
$M_{LP_{\gamma}}^{-}$):

A binary incidence matrix
$M_{LP_{\gamma}}^{-}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\gamma})}$ whose
element $M_{LP}^{-}(i,g)=1$ when the holding process $p_{g}\in P_{\gamma}$
pulls operand $l_{i}\in L$ as an input.

######  Definition 40 (The Positive Transformation Process-Operand Incidence
Matrix $M_{LP_{\mu}}^{+}$):

A binary incidence matrix
$M_{LP_{\mu}}^{+}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\mu})}$ whose element
$M_{LP}^{+}(i,j)=1$ when the transformation system process $p_{\mu_{j}}\in P$
ejects operand $l_{i}\in L$ as an output.

######  Definition 41 (The Positive Refined Transportation Process-Operand
Incidence Matrix $M_{LP_{\bar{\eta}}}^{+}$):

A binary incidence matrix
$M_{LP_{\bar{\eta}}}^{+}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\bar{\eta}})}$
whose element $M_{LP}^{+}(i,\varphi)=1$ when the refined transportation
process $p_{\varphi}\in P_{\bar{\eta}}$ ejects operand $l_{i}\in L$ as an
output. It is calculated directly from the negative holding process-operand
incidence matrix $M_{LP_{\gamma}}^{+}$.

$\displaystyle M_{LP_{\bar{\eta}}}^{+}(i,\varphi)$
$\displaystyle=\sum_{u=1}^{\sigma(P_{\eta})}M_{LP_{\gamma}}^{+}(i,g)\cdot\delta_{u}\qquad\forall
i\in\\{1,\ldots\sigma(L)\\},\;g\in\\{1,\ldots,\sigma(P_{\gamma})\\},u\in\\{1,\ldots,\sigma(P_{\eta})\\},\varphi=\sigma(P_{\eta})(g-1)+u$
(101) $\displaystyle M_{LP_{\bar{\eta}}}^{+}(i,\varphi)$
$\displaystyle=\sum_{y_{1}=1}^{\sigma(B_{S})}\sum_{y_{2}=1}^{\sigma(B_{S})}M_{LP_{\gamma}}^{+}(i,g)\cdot\delta_{y_{1}}\cdot\delta_{y_{2}}\quad\forall
y_{1},y_{2}\in\\{1,\ldots,\sigma(B_{S})\\},\varphi=\sigma^{2}(B_{S})(g-1)+\sigma(B_{S})(y_{1}-1)+y_{2}$
(102) $\displaystyle M_{LP_{\bar{\eta}}}^{+}$
$\displaystyle=\sum_{u=1}^{\sigma(P_{\eta})}M_{LP_{\gamma}}^{+}\otimes
e_{u}^{\sigma(P_{\eta})T}\quad\;\;=\sum_{y_{1}=1}^{\sigma(B_{S})}\sum_{y_{2}=1}^{\sigma(B_{S})}M_{LP_{\gamma}}^{+}\otimes\left(e_{y_{1}}^{\sigma(B_{S})}\otimes
e_{y_{2}}^{\sigma(B_{S})}\right)^{T}$ (103)
$\displaystyle=M_{LP_{\gamma}}^{+}\otimes\mathds{1}^{\sigma(P_{\eta})T}\qquad\quad=M_{LP_{\gamma}}^{+}\otimes\left(\mathds{1}^{\sigma(B_{S})}\otimes\mathds{1}^{\sigma(B_{S})}\right)^{T}$
(104)

######  Definition 42 (The Positive Holding Process-Operand Incidence Matrix
$M_{LP_{\gamma}}^{+}$):

A binary incidence matrix
$M_{LP_{\gamma}}^{+}\in\\{0,1\\}^{\sigma(L)\times\sigma(P_{\gamma})}$ whose
element $M_{LP}^{+}(i,g)=1$ when the holding process $p_{g}\in P_{\gamma}$
ejects operand $l_{i}\in L$ as an output.

###  -D Definitions of Set Operations

######  Definition 43 ($\sigma$() Notation [7]):

returns the size of the set. Given a set S with n elements, $n=\sigma(S)$.

######  Definition 44 (Cartesian Product ❌[129]):

Given three sets, A, B, and C,

$A\mbox{{\char 14\relax}}B=\\{(a,b)\in C\quad\forall a\in A\;\mbox{and}\;b\in
B\\}$ (105)

###  -E Definitions of Boolean Operations

The conventional symbols of $\wedge$, $\vee$, and $\lnot$ are used to indicate
the AND, OR, and NOT operations respectively.

######  Definition 45 ($\bigvee$ Notation):

$\bigvee$ notation indicates a Boolean OR over multiple binary elements
$a_{i}$.

$\bigvee_{i}^{n}a_{i}=a_{1}\vee a_{2}\vee\ldots\vee a_{n}$ (106)

######  Definition 46 (Matrix Boolean Addition $\oplus$):

Given Boolean matrices $A,B,C\in\\{0,1\\}^{m\times n}$, $C=A\oplus B$ is
equivalent to

$C(i,j)=A(i,j)\vee B(i,j)\qquad\forall i\in\\{1\ldots m\\},j\in\\{1\ldots
n\\}$ (107)

######  Definition 47 (Matrix Boolean Scalar Multiplication $(\cdot$)):

Given Boolean matrices $A,B,C\in\\{0,1\\}^{m\times n}$, $C=A\cdot B$ is
equivalent to

$C(i,j)=A(i,j)\wedge B(i,j)=A(i,j)\cdot B(i,j)\qquad\forall i\in\\{1\ldots
m\\},j\in\\{1\ldots n\\}$ (108)

######  Definition 48 (Matrix Boolean Multiplication $\odot$[10, 130]):

Given matrices $A\in\\{0,1\\}^{m\times n}$, $B\in\\{0,1\\}^{n\times p}$, and
$C\in\\{0,1\\}^{m\times p}$, $C=A\times B=AB$ is equivalent to

$C(i,k)=\bigvee_{i=1}^{n}A(i,j)\wedge B(j,k)=\bigvee_{i=1}^{n}A(i,j)\cdot
B(j,k)\qquad\forall i\in\\{1\ldots m\\},k\in\\{1\ldots p\\}$ (109)

######  Definition 49 (Matrix Boolean Subtraction):