11institutetext: School of Mathematics and Computer Science,
Computational Foundry, Swansea University,
Bay Campus, Fabian Way,
Swansea, SA1 8EN, United Kingdom

# The dynamics of belief: continuously monitoring and visualising complex
systems

Edwin J. Beggs  John V. Tucker

###### Abstract

The rise of AI in human contexts places new demands on automated systems to be
transparent and explainable. We examine some anthropomorphic ideas and
principles relevant to such accountablity in order to develop a theoretical
framework for thinking about digital systems in complex human contexts and the
problem of explaining their behaviour. Structurally, systems are made of
modular and hierachical components, which we abstract in a new system model
using notions of modes and mode transitions. A mode is an independent
component of the system with its own objectives, monitoring data, and
algorithms. The behaviour of a mode, including its transitions to other modes,
is determined by functions that interpret each mode’s monitoring data in the
light of its objectives and algorithms. We show how these belief functions can
help explain system behaviour by visualising their evaluation as trajectories
in higher-dimensional geometric spaces. These ideas are formalised
mathematically by abstract and concrete simplicial complexes. We offer three
techniques – a framework for design heuristics, a general system theory based
on modes, and a geometric visualisation – and apply them in three types of
human-centred systems.

Keywords. explainable systems, hierarchical systems, modes, mode transitions,
belief functions, simplicial complex, social systems

##  1 Introduction

> We cannot afford to continue playing catch-up regarding AI – allowing its
> use with limited or no boundaries or oversight and dealing with the almost
> inevitable human rights consequences after the fact.
>
> Michelle Bachelet, UN High Commissioner for Human Rights, 2021[36]

We consider the automation of human-centred systems, a subject made urgent and
controversial by modern AI. Starting with reflections on explainability and
possible design heuristics, we propose a new general system based on new
mathematical models of modes and mode transitions, and having geometric
visualisations of what a system mode ‘believes’ about its monitoring data. We
will address these points in turn and then summarise the contributions.

###  1.1 Automation and the challenge of explainability

Automation aims to reduce human activity and dependency in all sorts of tasks.
To do this, decisions and actions that belong to people must be abstracted,
specified precisely and designed into a relevant system that reduces human
involvement and interventions. Automation in this sense has been expanding its
nature and scope since the 18th Century [14]. Through the new applications and
ambitions of AI, automation is on the rise in the physical world, where it is
long established, and in human affairs, where it is novel and
controversial.111For example, we see it in new ‘smart’ buildings and products,
autonomous vehicles and weapons, staff recruitment and monitoring, customer
services, medical triage, treatment and aftercare, and surveillance and
criminal detection. Under the guise of AI, in so many fields we are seeing a
triumphant march of measurement, data collection, data classification,
computation and autonomous decision making. Wherever there is automation there
is a potential problem understanding the behaviour it produces:

Is the automation functioning as specified or expected? Are its specifications
and expectations what are required or desired or allowed? What parts might
need attention and change?

In the case of automatic physical systems problems almost always can be
discovered, understood and resolved in some way – thanks to centuries of
science, engineering and mathematics. In the case of people the situation is
quite other. Our understanding of automation is poorer, challenged by people’s
individuality, their limitations, expectations, conventions, bias, diverse
intangible experiences; it is challenged by cultural backgrounds and
formalised jurisdictions, and so on. Personal and social contexts refuse to
yield to neat theories with their limited predictive power and many unintended
consequences; personal and social conditions change all the time anyway. We
are still learning how not to manage people in highly structured social
environments, such as companies, as a century of management science, after
Frederick W Taylor, testifies [33, 39]. To the system designers, maintenance
engineers and owners, the specifications are a challenge to formulate, and can
be expected to mutate on deployment for all sorts of reasons.

In this paper, we will address system design for automation in human contexts,
almost from first principles. We will reflect on a few established criteria of
importance to people and suggest some ideas and directions for technical
research that reflect these criteria. Thus, the tone of this theoretical paper
is anthropomorphic, sampling aspects of human thought for technical
inspiration: human thinking has a record of working, and human thought is
explainable to humans. Subsequently, we will introduce high level data
structures and their visualisation, which can be applied, using mathematical
ideas of belief, to monitoring the dynamics of decision making in automation.

###  1.2 System theory as a tool

In human affairs, AI urgently needs to become transparent and explainable
because systems must be trustworthy and ‘easy’ to understand at some level. We
address the problem of transparency, which due to both safety and human rights
considerations can no longer be treated as an after the fact investigation
[36]. Given the likely future economic importance of people and machines
working together [11, 17], it is vitally important that automations can
clearly reveal their state and intentions to human co-workers in realtime.

For us, the key notion is system. Today, the term is everywhere in everyday
speech. It was elevated into a scientific concept by biologist Ludwig Von
Bertalanffy (1901-1972) starting in the late 1920s. Following World War II,
system theories were developed in all sorts of directions notably in the
engineering, management and economic sciences [7, 9]. Many of these theories
were mathematical and had wide scope and grand ambitions [16, 27, 28].
Computer science generated many system models with well-developed theories,
including foundational work in highly abstract mathematical forms that
attempted to bring formal unity.

Indeed, general system theory has identified a remarkable collection of
commonalities.222These include: basic definitions, system thinking, system
topologies, life cycles, system performance, conceptual design, current state
evaluations, solving methods, creative solutions, system synthesis, system
analysis, optimization, solution assessment, virtual optimizing, system
engineering, and evaluation of knowledge in economy and society. In
particular, system theory has long recognised that systems are modular and
hierarchical and the components are a mix of equipment, software and people
[19]. Any of these can – and in practice do – fail. Add to this the threat of
cyberattack, where the objective is to make systems fail. Absolute security or
predictability of a system is impossible, so plans need to be made to contain
and work around damage, the classic problem of resilience.

###  1.3 A new general system model

We introduce an abstract notion of mode of a system designed to describe the
structure and functioning of the system, and a geometric realisation of the
modes to allow visualisation and implement continuity. The description in
terms of modes would be accessible on demand with the visualisation showing
not only which mode the system is in, but whether it is fit-for-purpose, and
to which mode it is likely to transition in the near future [32].

Real world systems are driven by data, which may not be reliable. The
decisions and actions taken by the system are based on judgements about the
data available against known criteria, which we are going to refer to as
beliefs. Thus, as far as mathematical models are concerned, the behaviour of
the system over time is a trajectory in a hugely complex idealised state
space. For this to be transparent we need to reveal a dynamics of belief that
explain the dynamics of system behaviour. In short, this dynamics of belief is
a new trajectory in a space of beliefs about data and perceptions of the
system.

We consider a large range of data sources as simply oracles, which are
providers of information which might, or might not, be reliable. Typically,
these are gathered by sensors and generated by computations external to the
system (e.g., by neural nets or quantum devices). We also include outputs sent
to external devices as oracles (as many actuators talk back). By restricting a
mode’s access to various oracles we have a high-level mechanism for enforcing
restrictions on what a mode can know and do, i.e., a method for enforcing
transparency.

The mode construction and its visualisation using simplicial complexes were
originally designed for complex physical control systems [4]. In this paper we
have deliberately moved away from that to social examples to develop further
the generality of the concepts and methods. Here, for example, we have also
defined the general idea of a frame of belief which has relevance to social
and academic constructs and explainablity. The idea is to present the modes
and their visualisation as an architecture in logical AI.

###  1.4 Contributions and structure of this paper

We present a general approach with an informal characterisation of modes and
mode transitions, algebraic and geometric models, and examples to illustrate
their richness. Main contributions of this paper and their location are:

1\. To consider some aspects of human behaviour informally, before formalising
a model of how the knowledge and beliefs of a system and its responses change
over time (Section 2).

2\. To define concepts and principles for modelling systems using modes and
mode transitions (Section 3).

3\. To address explainability by defining a mathematical model for beliefs and
visualising changes in beliefs about the monitoring data and hence the system,
in modes using simplicial complexes (Sections 4 and 6).

4\. To illustrate the combination of modes and beliefs in three case studies
of human-centred systems (Section 5).

5\. To highlight the role of change in underlying epistemic frameworks of
belief in understanding the design and operation of systems (Section 7).

##  2 Explainability and an anthropic view of systems

We will briefly review some ideas that lie just below the surface of our view
of systems. We will comment on explanations and how people’s performance in a
situation or scenario may be systematised.

###  2.1 What is an explanatory framework?

Our considerations lead to questions about the structure of the system, its
design and deployment, and its operation and performance: What counts as an
explanatory framework? What ways might there be of seeing what is going on? To
develop an informal explanatory framework, consider some general principles,
starting with:

Observation.

An explanation depends upon on data available to do with the system; the data
collected, processed, computed by and about the system.

Categorisation.

An explanation is based on categorisations of possible behaviours that
constitutes a conceptual framework for the human coworker.

Refinement.

An explanation is capable of refinement, revealing more detail, according to
the significance of the information available.

Prediction.

An explanation is capable of making predictions about the system is certain
situations.

Tightly coupled to the explanatory framework is a visualisation framework
ideally satisfying:

Continuity.

The behaviour over time should be understandable from the visualisation.

Faithfulness.

The visualisation provides access to the actual functioning of the system, and
does not conceal important properties of its function.

One way this can be done is to incorporate the explanatory and visualisation
frameworks as an essential part of the design process, allowing us to ‘see’
what the system is ‘thinking’.

Of course, this explanatory framework is impossible without simplification and
information hiding on a grand scale.

However, the frameworks have foundations that are theoretical and empirical
and, indeed, influenced by the cultural, social and political. This is evident
in human centred systems.

Since explainability is defined in terms of human understanding, and humans
have a record of being resilient, we begin by considering how humans might
understand and control a situation systematically. Our purpose is to seek
ideas that can be abstractly formulated and used to shape a conceptual
framework for explainability and our technical proposals.

###  2.2 Designing scenarios

Consider a person responsible for a system and imagine him or her in a
scenario composed of the following:

Objectives.

A person has long term aims which entail shorter term objectives.

Modes.

The system is in a state and they are presented with a choice of possible
actions.

Principles.

There are ‘principles’ – assumptions of various kinds – implying that actions
have certain consequences, good or ill.

Intent.

They intend to take the actions whose consequences they believe will best
further their objectives.

We need to have clear the distinction between between

(a) principles about a real situation

(b) observations of the situation

(c) actions applied to the situation.

In science and engineering these ideas are familiar; for instance, in
ballistics, the objective is to hit a target. Newton’s laws of motion, some
fluid dynamics, would be a source of principles; observation would be
measurements of the wind, humidity, temperature and range of an object to hit
a target; and an action by the gunner would be to set the parameters to aim to
hit the target and fire. Human experience is also a factor providing essential
heuristics, of course. In more complex situations the set of principles is a
large set of assumptions explicit and implicit only some of which will shape
the systematisation.

Oracles.

A person who is uncertain will observe more information about the real
sitiuation.

Beliefs.

Using their principles, they will evaluate the information to form beliefs
about how possible actions best further their objectives.

Choice.

Based on these beliefs they choose actions to perfrom.

Aims.

They may change objectives based on aims and information about reality.

An objective may merely be the legacy of a past decision. For instance, a
general decides to capture a hill, and from then onwards many decisions are
taken using this as an objective. If capturing the hill proves impossible, the
general will likely need to choose another objective. In turn, the aims within
which the general operates may have been set by a dictator deciding to invade
another country.

When humans design or control systems they have an opportunity to specify
objectives which the systems cannot override. For example, the average
electronic heating controller cannot decide to vary the temperature outside a
specified range (unless an error happens). At an extreme of the AI spectrum,
Isaac Asimov’s fictional robots were programmed with three laws of robotics as
immutable objectives which would override any other objectives.

###  2.3 Designing components for actions

> Frustra fit per plura quod potest fieri per pauciora.
>
> [It is futile to do with more things that which can be done with fewer.]
>
> William of Ockham, Summa Totius Logicae, circa 1323 A.D.

Consider two guidelines for people on how to organise the set of possible
actions in a given scenario, one of these addressing simplicity (and, so,
indirectly explainability) and the other addressing resilience, when current
assumptions fail.

Occam’s Razor 1: Actions.

A person will prefer a smaller number of possible actions rather than a larger
number if they believe the smaller number can bring their intent about.

Incompleteness 1: Actions.

If they believe that the possible actions will not bring about their
objectives, they seek more possible actions.

Thus, we reduce the set of possible actions when we can, and expand it when we
have to.

Of course, these two guidelines can also be applied to the principles we use
to design and reason about the system.

Occam’s Razor 2: Principles.

In a given scenario, use the least number of principles required to explain
reality and model the objectives.

Incompleteness 2: Principles.

When reality becomes inconsistent with the principles, seek other principles.

We can also consider Occam’s Razor as a guide to hiding irrelevant data.
Information hiding is an idea central to software engineering, formulated
explicitly as a methodological principle by David Parnas in the 1970s and
brought into mainstream software development in abstract data types and object
oriented programming languages, long before our current data-overloaded
society. In societies, people have specialised roles, often related to data
hiding:

Specialisation.

People only see a minimal amount of information for a role which they do not
perform.

Restricted access.

Only certain people have a right to access certain data to maintain security
or privacy.

Of course, in applying these points to automatic systems we have to make
allowances. The average electronic heating controller has no idea of the laws
of convection or conduction, it only knows the formulae it has been programmed
with. However, even it has some understanding of incompleteness – e.g., when
its actions seem not to work – it uses its fault light and hopes for an
intervention by a technician.

The big problem with incompleteness is that the person’s existing knowledge
may not have a previously prepared method to achieve the objectives – this is
the acquiring of new skills. The person has to be able to make new possible
courses of action, which need to be checked, first purely theoretically and
then more practically.

Innovation.

The generation of new possibilities.

Imagination.

New possibilities are matched against existing principles to predict their
effect.

Testing.

Controlled experiments are used to validate new possibilities.

In the rest of this paper we shall use these ideas to suggest and shape formal
concepts that make up a possible formal complement to this informal framework.
In particular, we consider how to both formalise and visualise scenarios,
using a dynamics generated by beliefs about the basic monitoring data
available to the system. The structure of our thinking is, essentially, to
formalise this critical path:

$\displaystyle\mathrm{MonitoringData}\
\stackrel{{\scriptstyle\mathrm{Evaluate}}}{{\longrightarrow}}\ \mathrm{Belief\
about\ data}\ \stackrel{{\scriptstyle\mathrm{Visualise}}}{{\longrightarrow}}\
\mathrm{Visualisation\ of\ belief}\ \ $ (1)

##  3 Modes and visualisation

> My thesis has been that one path to the construction of a non-trivial theory
> of complex systems is by way of a theory of hierarchy. Empirically, a large
> proportion of the complex systems we observe in nature exhibit hierarchic
> structure. On theoretical grounds we could expect complex systems to be
> hierarchies in a world in which complexity had to evolve from simplicity.
>
> Herbert Simon [19]

Modes are a multi-level categorisation of the objectives and behaviours of
systems. A mode combines data relevant to the objectives and operation of the
system with algorithms governing the behaviour and the external communication
channels and protocols.

###  3.1 What is a mode?

Here is a working definition to begin building the conceptual framework:

###### Definition 1

Consider a system Sys operating in an environment Env. A mode of the system
Sys is defined by these characteristics:

1\. A mode is associated to, or determines, a subset of the possible states of
the system.

2\. A mode is designed to deliver on and meet objectives for the system when
in these states.

3\. A mode consists of

(a) methods to input data from the environment Env and to output data to Env.

(b) data types and algorithms for implementing the objectives.

4\. Methods to evaluate the performance of the mode against its objectives
and, if necessary, choose and transfer to another mode.

Thus, a mode is responsible for specific aspects of the system’s performance,
which may be relevant to the high-level, primary objectives and purposes of
the system, or lower-level, localised technical services. It may be autonomous
or a platform for accessing other modes. Each mode owns its relationship with
the environment.

###  3.2 Principles for designing modes

With the initial intuitions of Definition 1 in mind, we can formulate some
design principles to develop a conceptual framework for thinking about systems
in terms of modes and mode transitions.

Completeness. A set of modes for a system is a classification of the operation
or behaviour of a system. At any time, a system can be in one, or more, modes.

Composition. When a system is in a number of modes then that situation itself
constitutes a mode.

Component. A set of modes for a system consists of (i) a set of basic modes
and (ii) joint modes made by combining other modes.

Localisation. Each mode possesses its own data about its behaviour and
environment. This monitoring data determines the mode’s own state space and
data types.

Thus, our knowledge of the system at any time is localised to the modes active
at that time.

Quantification. If a state of the system is meaningful for a number of modes
then the relevance or suitability of these modes for the state of the system
must be quantified, calibrated and interpreted.

Visualisation. With quantification and calibration comes the possibility of
visualisation via geometric objects drawn to a scale and via derived
qualitative diagrams.

We shall visualise in space the basic modes by vertices, and the joint modes
arising by combining them as lines, triangles, tetrahedra etc. The geometric
objects we build are called simplicial complexes. Simplicial complexes are
made up of geometric pieces called faces.

Modes and faces. In a successful system model using modes, every face of its
simplicial complex is a mode and every mode is represented by a face.

Quantification can take the form of a position in the space representing the
suitability of the mode for the state of the system, in a line or triangle or
tetrahedron, etc. representing the joint mode. This position is computed by
belief functions that from the state of the system calculates measures of the
relevance of other modes to the state.

In the time evolution of the system we need to decide

(i) if, and when, a system should change from one mode to another, and

(ii) which new mode should chosen.

Thresholds. The transition out of one mode into another is governed by the
results of the quantification and calibration. The decision to move to a new
mode may be specified by numerical thresholds. Transition has these stages:

(a) the realisation that the mode is approaching its limitations

(b) the selection of modes that could be more appropriate

(c) triggers to choose and change to a new mode.

The belief functions, in computing relevance, are a means to trigger changes
of mode.

Explanation. The conceptual system of modes, mode evaluation functions,
thresholds and mode transition functions, and their visual representation in
simplicial complexes, can serve as an explanatory framework for the dynamical
behaviour of automatic systems.

###  3.3 What is visualised? Some simple examples

We consider some simple examples to begin to shape our informal ideas of a
system of modes, the evaluation of beliefs and how they might be visualised.

judgement of monitoring
datainterventionincomingcallspoliceambulancefiretriggeringan actiontriage
foremergency services Figure 1: Visualising a mode transition for a trigger
mechanism and triage for emergency services

######  Example 1 (A simple trigger)

In Figure 1, the left diagram represents visually a trigger for action that
has three modes: two vertices and one line. Consider the line between a mode
that makes judgements of some attribute using data monitoring the environment
(the point at the top of the interval) and the mode that performs an
intervention (the point at the bottom). Unlike a usual state transition
graphs, where lines merely represent transitions or jumps, here the line
itself represents a (joint) mode where the decision to intervene unfolds
according to its monitoring data. These observations are formed into a
numerical representation of beliefs about the state of the system as points
along the line, beginning at the top and moving to the bottom. Threshold
points on the line begin to warn of, and then trigger, a change of mode; on
reaching the bottom, the intervention begins.

Witnesses can note the position along the line to see how things were going,
and perhaps be reassured that the action was progressing correctly. Such a
point on a line indicator – a progress bar – is quite a common visualisation
of estimated time taken for a task to be completed. Here, however, the line
indicator is displaying the information (= an interpretation of the data) that
the control system is actually using to make decisions. If the point is near
the top, then a witness knows that the control system has no intention of
intervening because they are looking directly at the system’s own assessment
of the situation. ∎

In this example, we see that a key to transparency is continuity. We do not
discontinuously move between vertices on a graph, but have a continuous motion
on a geometric visualisation, which allows an estimate of what the system is
about to do and when. Another key to transparency is reliability, that what we
see is really related to the state of the system. This means that the geometry
of the visualisation is intimately related to how the system actually makes
decisions.

######  Example 2 (A simple emergency)

In the second diagram of Figure 1, we consider a notification warning of a
possible incident – such as personal attack, social disturbance, or collision
of vehicles. In general, the incident could involve some or all of the primary
first responders of police, ambulance and fire brigade. These are represented
by vertices in the diagram. As more information becomes available over
(hopefully a short) time period, relevant choices of services can be made and
deployed to the scene. In particular, between the separate actions of say
deploying police and ambulance we have a combined action of doing both
simultaneously, and that is represented in the diagram by a line between the
two vertices.

Altogether the 15 faces of the diagram represent 15 possible modes of the
system! The filled-in tetrahedron between the four vertices is the mode which
gathers all available information and assesses the response. When beliefs
about the state of the system (represented by a point in this tetrahedron or
3-simplex) reaches the triangle at the end then full deployment is instigated.
If the incident in some way is misrepresented then the point moves, possibly
returning to the warning vertex, and then to the rest of the system (not
shown). The state when an action is actually taken should be one of the three
action vertices two of the lines between them. ∎

Systems frequently have to take account of several factors or carry out
several tasks, and we take account of this by allowing combinations of modes
to also form modes, which translates into edges and triangles, etc.
representing modes. In general, we shall represent beliefs about the state of
a system as a point in a simplicial complex.

What exactly is a simplicial complex? A 0-simplex is just a point, and a
1-simplex is a line between some of the 0-simplices. Thus, a simplicial
complex consisting of only 0 and 1-simplices is just a graph with edges the
1-simplices and vertices the 0-simplices. We shall not assume that this graph
is directed – if there are arrows they are either imposed by principles (e.g.,
not being able to deploy a parachute which we have already jettisoned) or by
the current objectives. Increasing the dimension, a basic 2-simplex is a
filled in triangle and a basic 3-simplex is a solid tetrahedron, and so on.

The belief about the state of the system is visualised as a point in the
geometric simplicial object, showing which mode the system is in and which
mode is likely to come next. The position is given by the belief generated by
algorithms on the basis of evidence. Technically, there is mathematical belief
theory that is a generalisation of probability theory and, again, we refer to
Section 6 for a discussion.

As the modes are linked to the behaviour of the system, the visualisation of
the modes gives the current behaviour of the system – at least its intentions
and its capabilities.

Since each face of simplicial complex is a mode, the choice of a simplicial
complex for a system can reveal the existence of possible modes that may be
difficult to anticipate and account for (cf. Example 2).

##  4 Mathematical Tools 1: Simplicial complexes

The three-stage critical path (1), from monitoring data to interpretation and
belief to a visualisation of belief for our general conception of modes, can
be modelled mathematically using abstract and concrete simplicial complexes,
which we describe in this and the next section.

###  4.1 Simplicial complexes

###### Definition 2

An abstract simplicial complex consists of a collection $\mathcal{C}$ of
finite subsets of a set $\mathcal{M}$ with the property that if $Y\subset
X\in\mathcal{C}$ then $Y\in\mathcal{C}$.

In line with the Components Principle in Section 3, we can take $\mathcal{M}$
to be the set of basic modes of the system, and $\mathcal{C}$ to be the set of
joint modes. Thus, a basic mode becomes a singleton set $\\{\alpha\\}$ and a
joint mode is a set $\\{\alpha_{1},\ldots,\alpha_{k}\\}$. The rule means that
any subset of a mode is a mode.

We visualise this abstract simplicial complex as a geometric simplicial
complex made up of parts called simplicies. This is done by taking each basic
mode $\alpha\in\mathcal{M}$ to be a point (= a 0-simplex). Then the subsets of
size two $\\{\alpha,\beta\\}\in\mathcal{C}$ are visualised as a line (= a
1-simplex) connecting the points $\alpha$ and $\beta$. So far, the resulting
structure is an undirected graph, but now we continue in higher dimensions:
for example, the subsets of size three
$\\{\alpha,\beta,\gamma\\}\in\mathcal{C}$ are visualised as a filled in
triangle (= a 2-simplex) connecting the points $\alpha$, $\beta$ and $\gamma$,
and so on.

###### Proposition 1

To every abstract simplicial complex (as in Definition 2) is associated its
standard realisation $\Delta_{\mathcal{C}}$, a simplicial complex as follows.
Form a vector space $\mathbb{R}^{\mathcal{M}}$ with basis $e_{\alpha}$ for
$\alpha\in\mathcal{M}$. The simplex spanned by $X\in\mathcal{C}$ is

$\Delta_{X}=\Big{\\{}\sum_{x\in X}\lambda_{x}\,e_{x}:\lambda_{x}\in[0,1],\
\sum_{x\in X}\lambda_{x}=1\Big{\\}}.$

The simplex $\Delta_{X}$ is a $(|X|-1)$-simplex where $|X|$ is the size of
$X$, and if $Y\subset X$ then $\Delta_{Y}$ is a face of $\Delta_{X}$.

###### Example 3

Suppose there are three basic modes $\alpha,\beta,\gamma$. In general, the set
of basic and joint modes in the 2-simplex (filled in triangle) shown in Fig. 2
is

$\\{\alpha,\beta,\gamma,\alpha\beta,\alpha\gamma,\beta\gamma,\alpha\beta\gamma\\}.$

$\alpha$$\beta$$\gamma$ Figure 2: Triangle for Example 3

It is usually possible to re-draw the geometric simplicial complex in a
smaller number of dimensions, the very large number in the Proposition simply
gives an existence result.

###  4.2 Evaluating states

Consider the system from the viewpoint of its states. In line with the
Quantification Principle in Section 3, we would like to associate to a state
of the system a point in the geometric simplicial complex. To do this it is
easiest to consider a space $\mathbb{S}$ of possible states of the system. The
basic mode $\alpha$ will apply when the system state is in a certain subset
$U_{\alpha}\subset\mathbb{S}$ of all the possible states. If we want every
possible state of the system to be assigned to a mode then the union of all
the $U_{\alpha}\subset\mathbb{S}$ is all of $\mathbb{S}$, i.e., we have a
cover of $\mathbb{S}$.

###### Definition 3

Let $\mathcal{M}$ be a set of basic modes. We localise the state space to the
states for the joint modes by defining $\mathbb{S}_{X}=\cap_{\alpha\in
X}U_{\alpha}$ for $X\subset\mathcal{M}$ for which $\mathbb{S}_{X}$ is not
empty. Thus $\mathbb{S}_{\\{\alpha,\beta\\}}=U_{\alpha}\cap
U_{\beta}\subset\mathbb{S}$.

Now define

$\mathcal{C}=\big{\\{}X\subset\mathcal{M}:\cap_{\alpha\in
X}U_{\alpha}\neq\emptyset\big{\\}},$

and it is immediate that $\mathcal{C}$ is itself an abstract simplicial
complex.

###### Definition 4

A partition of unity for the cover $U_{\alpha}\subset\mathbb{S}$ is a function
$\phi_{\alpha}:\mathbb{S}\to[0,1]$ for every $\alpha\in\mathcal{M}$ such that

(1) if $\phi_{\alpha}(s)\neq 0$ then $s\in U_{\alpha}$

(2) $\sum_{\alpha\in\mathcal{M}}\phi_{\alpha}(s)=1$ for all $s\in\mathbb{S}$.

For specific circumstances we can impose extra conditions on $\phi_{\alpha}$,
e.g., continuity or computability.

###### Definition 5

We now define a function $\underline{\phi}:\mathbb{S}\to\Delta_{\mathcal{C}}$
that classifies and visualises beliefs about the the states of a system using
the simplicial complex $\Delta_{\mathcal{C}}$ by

$\underline{\phi}(s)=\sum_{\alpha\in\mathcal{M}}\phi_{\alpha}(s)\,e_{\alpha}\
.$

###### Example 4

Figure 3 visualises a simplicial complex and partition of unity for a cover by
four sets. Note that the triangle in Figure 3 is shaded to form a 2-simplex
precisely because $U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$ is not empty.

$\alpha$$\beta$$\gamma$$\delta$$v_{\alpha}$$v_{\beta}$$v_{\gamma}$$v_{\delta}$$\phi_{\alpha}$$\phi_{\beta}$$\phi_{\gamma}$$\phi_{\delta}$
Figure 3: A partition of unity for a cover by four sets

###  4.3 Computations within modes

$\Delta_{\mathcal{C}}$$\mathbb{S}$$S_{X}$$S_{Y}$queries$\phi$$\phi_{X}$$\phi_{Y}$
Figure 4: Localisation allowing computation of $\phi$

In general we may not know the state $s\in\mathbb{S}$ of the system. We may
need to perform observations or measurements to do this. If we suppose that we
are in mode $X\subset\mathcal{C}$ then oracles bundled with $X$ are queried
for information, and we need to compute with this data.

###### Definition 6

For the mode $X$ we have the associated datatype $S_{X}$ containing, amongst
other things, the data from queries (e.g. information from sensors). Then
$S_{X}$ is our computable model of the state space $\mathbb{S}$ given that we
believe that we are in mode $X$. We use a computable function
$\phi_{X}:S_{X}\to\Delta_{\mathcal{C}}$ to approximate our idealised function
$\underline{\phi}:\mathbb{S}\to\Delta_{\mathcal{C}}$. Fig. 4 illustrates the
actual computation ot this approximation to
$\underline{\phi}:\mathbb{S}\to\Delta_{\mathcal{C}}$.

The function $\phi_{X}:S_{X}\to\Delta_{\mathcal{C}}$ is used to see when we
need to change mode from $X$ to another mode. If there is uncertainty about
the current mode (as opposed to a normal mode transfer), then we could
simultaneously run queries for other modes (e.g. $Y\subset\mathcal{C}$ in Fig.
4).

Decisions to prepare for a change of mode or to make a change of mode are
based on positions in the simplicial complex. When in mode $X$ with state $s$
the interpretation of $X$’s monitoring data, represented by
$\phi_{X}(s)\in\Delta_{\mathcal{C}}$, can trigger actions according to where
$\phi_{X}(s)$ lies in appropriately defined subsets or zones. In particular, a
change of mode is triggered by some exit zones. Recall the definition of
thresholds in Section 3.2.

##  5 Three socially-centred examples

So far we have mentioned physical control systems to illustrate the ideas
surrounding modes. Here we give some social situations which can be modelled
similarly. To relate them to the first part of the paper we need to consider
the function $\underline{\phi}:\mathbb{S}\to\Delta_{\mathcal{C}}$ given after
Definition 4. We assess the idealised state space, the data which allows us to
localise our position in that state space, and how $\phi$ depends on that
data.

###  5.1 An offender monitoring problem

Scenario and its visualisation. Consider an offender who has been sentenced
for a crime committed under the influence of alcohol and who is being
monitored for blood alcohol concentration and also tagged to limit how far
they may go from home. For both alcohol and position we can assign a number,
$x_{\mathrm{alc}}$ and $x_{\mathrm{tag}}$ respectively, in $[0,1]$ where $0$
is clearly OK and $1$ is clearly a problem. Given the context of this
monitoring problem, the system state is an ordered pair
$(x_{\mathrm{alc}},x_{\mathrm{tag}})\in[0,1]^{2}=\mathbb{S}$. This context
deliberately omits much information, such as whether the offender is reading
or sleeping, which if it was available is simply not recorded. The simplicity
of assigning $\mathbb{S}=[0,1]^{2}$ is justified by the simplicity of the
legal restrictions and the privacy of the offender.

If at any time the following conditions are satisfied, interventions are
triggered:

1) If the position is out of the allowed area, the offender is asked to see
their probation officer.

2) If alcohol is higher than a trigger amount, the offender is asked to see
their counsellor.

3) If both the above, the police are called as an incident may occur.

For both $x_{\mathrm{alc}}$ and $x_{\mathrm{tag}}$ we assign a region of
values which are not problematic, and a region of values which are a problem,
say $[0,\frac{1}{4}]$ and $[\frac{3}{4},1]$ respectively. The fact that there
is a gap between these regions is important:

a) It allows for error in a device or a difference in alcohol metabolism.

b) It allows for a warning to be sent to the offender, so that action can be
taken to avoid an intervention (e.g. going home or stopping drinking).

Evidence and quantification. In Figure 5 we give a mode diagram for the
actions, where we have used ovals to emphasise the points and lines where
interventions occur. The filled in triangle (2-simplex) mode is responsible
for monitoring the situation and issuing a warning to the offender if
necessary. The point and line modes at the bottom are responsible for alerting
the relevant services.

alcohol problemtagging problemboth problemsOKwarning line Figure 5: Diagram
describing interventions in the alcohol-tagging problem

We now consider a cover of the state space and its relation to the function
$\underline{\phi}$ as discussed in Definition 4 and pictured in Figure 3. We
use the square state space $[0,1]^{2}$ and the cover given in Figure 6. There
the first shaded region indicates an alcohol problem, the second a tagging
problem, and the third the absence of a problem. The function $\phi$ is
pictured in Figure 7.

alcoholtag011 Figure 6: A cover of $[0,1]^{2}$ for the alcohol-tagging problem
$({\mathrm{alcOK}},{\mathrm{tagOK}})$$({\mathrm{alcProb}},{\mathrm{tagOK}})$warning
line$({\mathrm{alcOK}},{\mathrm{tagProb}})$alcoholtag011 Figure 7: The
function $\underline{\phi}$ mapping $[0,1]^{2}$ to a 2-simplex for the
alcohol-tagging problem

System | Model
---|---
Algorithm | Automated system alerts professionals
Data | Subject’s records, limited time memory
Objectives | Prevent incident, promote treatment
Oracles | Electronic tag, blood alcohol device
Visualisation of belief | Supporting decision in court

###  5.2 Hospital admission

wardsurgeryleave hospitaladmitdischargetreatment ondischargetriagearrive
inhospitaltake ambulancetake other
transport$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 8: Visualising unplanned admissions to hospital

Scenario and its visualisation. In Figure 8 we consider a person arriving at a
hospital and entering an initial assessment process or triage. In the mode ‘in
ambulance’, the line between ‘take ambulance’ and ‘arrive’, the patient will
be in the care of paramedics who may inform the hospital of an imminent
arrival.

On arrival the patient enters the ‘triage’ mode, the filled in triangle with
vertices ‘arrive’, ‘admit’ and ‘discharge’. The purpose of this mode is to
sort patients into those who have to stay in hospital and those who can go
home. As evidence is collected the state of the patient will move along the
triangle until it reaches one of the vertices ‘admit’ or ‘discharge’. If the
patient is known to be critical (e.g. by a report from the paramedics) then
they may proceed along the line ‘emergency admission’ connecting ‘arrive’ and
‘admit’. If the hospital is closed to admissions then the ‘compulsory
discharge’ mode connecting ‘arrive’ and ‘discharge’ will be used.

If the decision is made to discharge the patient may still receive treatment
before leaving, indicated by the line leading to ‘leave’. Similarly being sent
to surgery or a ward requires preparation.

We will concentrate on the triage process itself, the filled in triangle in
Figure 8. In contrast to the simple state space in the last example, we have
$\mathbb{S}$ consisting of the possible states of health of a person, an
enormously complicated space. Fortunately, we only have to make a decision to
admit or discharge, and it is that path of likelihood of decisions as a
function of time during the triage process that is plotted on the diagram.

Evidence and quantification. To gather evidence we use a query tree. At each
node in the tree a question is asked or an observation is made, and the next
node is chosen depending on the result. Each note will have its own score for
the three coordinates

$\phi(node)=\big{(}\mathrm{begin\ triage}\ ,\ \mathrm{discharge}\ ,\
\mathrm{admit}\big{)}\ .$

This is a triple of positive numbers adding to one, and specifies a point in
the triangle. For the later vertices, we might also get a list of possible
conditions which will be added to the patients notes. The list of coordinates
gives the function $\phi$, and this function is updated as the examination
proceeds. Early in the process the ‘begin triage’ value will be close to 1,
and it will decrease as the examination continues.

Thresholds and mode transitions. The threshold values for decisions may depend
on other factors. For example, if the hospital is not busy a patient might be
admitted as soon as the ‘admit’ coordinate reaches $0.8$, whereas on a very
busy day it might have to reach $0.95$.

System | Model
---|---
Algorithm | Formulae as helpers only, staff make decisions
Data | Medical records,
Objectives | Triage to a set reliability in a set average time
Oracles | Observations by medical staff, tests, patients narrative
Visualisation of belief | Expert system, medical staff have final say

###### Remark 1

This example can be seen from several different points of view. The account
above is from the point of view of the hospital. There are other agents at
work, most obviously the machines; the ambulance and the medical equipment;
and the people, the patient and the medical staff. All of these have their own
state spaces and their own reasons for acting and communicating as they do.
(E.g. a blood pressure monitor may not have been properly maintained.)

###  5.3 A judicial process

releasedjailcustodial sentencesuspended sentencebegin courtprobation Figure 9:
A simplified judicial process

Scenario and its visualisation. Consider someone who is accused of a crime
appearing in court. In Figure 9 they enter on the leftmost vertex. In our
simplified picture the court can take one of three options: innocent
(released), guilty (custodial sentence– go to jail) or guilty (suspended or
deferred sentence – allowed to leave under the supervision of the probation
service). This can be represented by a 3-simplex (solid tetrahedron) with the
weights of the vertices being the belief in the court in the three possible
decisions and a weight indicating whether the court believes that it is time
to reach a decision, i.e. whether the evidence has been considered. In terms
of the previous subsets, the 3-simplex corresponds to the vertices

$\\{\mathrm{begin\ court,custodial,release,suspended}\\}\in\mathcal{C}$

Initially we start at the begin court vertex, as no evidence has been
presented, and (ideally) the court should not have a prior belief about the
verdict. As time passes and the evidence is presented the beliefs change and
we flow away from the court vertex, towards the 2-simplex giving the
judgements (this is indicated by the arrows giving a control direction to the
system).

The system then enters the 2-simplex on the right. It would be possible to
move from jail to probation (time served and good behaviour) or the other way
for breaking conditions. The flow lines here indicate the progress towards the
eventual end of the sentence (possibly reduced for good behaviour).

Evidence and quantification. The court process (the solid tetrahedron in Fig.
9) has some similarities to the medical triage we discussed earlier, in that
it depends on expert opinions. Instead, we look at a more straightforward part
of the proceedings. We choose to look at the triangle in Figure 9 formed by
the vertices released, jail and probation in detail. The state space
$\mathbb{S}$ will be the behaviour of the offender up to the current time $t$.
(We measure time $t\in[0,1]$ as a fraction of the original sentence.) For
simplicity we summarise the behaviour of the offender into a single good
behaviour function $g(t)\in[0,1]$. We take $g(t)=0$ to be bad at time $t$, and
$g(t)=1$ to be good. The function $g(t)$ could be given by a start value (say
$g(0)=1$) at the beginning of the sentence. Subsequently every recorded
incident of bad behaviour subtracts an amount from $g(t)$, but this deduction
decays with time.

We now consider the 2-simplex in Figure 13 and the visualisation of the
trajectory $\phi$ of the offender in the system as a function of time. At any
time the offender is in one of the three states Released ($r$), Jail ($j$) or
Probation ($p$). we will consider transitions out of these states, and fill in
the $(r,j,p)$ values afterwards. We only specify these values on the boundary
of the given domains, and fill in the other values by interpolation. (We could
take a much more precise approach, but it would only obscure what is going
on.)

Thresholds and mode transitions. As regards transferring between the three
possibilities, we do not have a single threshold value. This might result in
someone being moved between jail and probation every single day if they were
near that value. Instead we assume that we have overlapping stable regions.
There are many ways in which we might choose to model this, but we shall
choose catastrophe theory. This was invented to give a theory of continuous
changes giving rise to discontinuous effects (see [2] for theory and [40] for
applications).

probationjail$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$g=0$$g=a$$g=b$$g=1$$P$
Figure 10: Moving from jail to and from probation

In Fig. 10 we model the transition from jail to probation, assuming that the
transitions are time independent so the result only depends on good behaviour
$g$. Beginning at point $P$, with $g=0$ in jail, a character decides to reform
and increases their $g$ value. They move to the right along the curve until
they reach $g=b$, at which point they ‘fall off’ the jail surface onto the
probation surface. If they behave more badly they will not be moved back to
jail until they move left along the probation curve to the point $g=a$, at
which point the possibility of remaining on probation disappears and they end
back in jail. Seen from above, we would have a stable jail region from $g=0$
to $g=b$ and a stable jail region from $g=a$ to $g=1$. If a person is in one
of these regions, they stay in it until they cross the boundary.

Now we consider how to visualise this in a simplicial complex, in this
simplified case an interval with end points ‘jail’ and ‘probation’. In Fig. 10
for $g<a$ there is no possibility other than jail, and likewise for $g>b$ we
have probation. The interesting intermediate case is when $a\leq g\leq b$,
when the person has a possibility of changing status, depending on how they
behave. We give a suitable function from the state space (in this case just
the $g\in[0,1]$ values) to the simplicial complex in Fig. 11.

probationjail$\bullet$$\bullet$$\bullet$$\bullet$$g=0$$g=1$$g=a$$g=b$ Figure
11: The map into the 1-simplex for jail and probation

Now we consider adding time $t\in[0,1]$ as a fraction of the original sentence
to the state space, and being released to the simplicial complex.

$\bullet$JailProbationRelease$\bullet$$\bullet$$j=1$$j=1$$p=1$$p=1$$r=1$$j=1$$r=1$$r=1$$r=1$$\bullet$$a$$a$$b$$b$$c$$c$$\bullet$$\bullet$$d$$d$$\bullet$$\bullet$$\bullet$$e$$\bullet$$f$$g$$(0,0)$$(1,0)$$(0,1)$$(1,1)$$\bullet$$\bullet$$\bullet$$\bullet$

Good behaviour

Time Figure 12: Domains for jail, probation and early release

In Figure 12 we the stable domains for jail, probation and early release and
the corresponding maps into the triangle with vertices $j,p,r$ respectively.
As previously mentioned, a convict will have a path through the square of good
behaviour as a function of time (as a proportion of sentence).
By stable domains we mean that if a convict is in the shaded domain ‘Jail’ in
the left square of Figure 12, then they will remain in jail unless their path
leaves the shaded region. If their path leaves the Jail region on the interval
$ac$ then they are transferred to probation. If it leaves on the interval $cd$
then they are given early release.
Correspondingly, a convict in the domain ‘Probation’ can exit to early release
along the lines $bce$ and to jail along $gf$. (They cannot exit by crossing
$ag$ unless they have a time machine.)

ReleaseProbationJail$\bullet$$\bullet$$\bullet$$g,f$$e,c,b$$a$ Figure 13: Map
to the 2-simplex for the Probation domain

To show the map to the simplex for Figure 12 we take the most complicated case
of the probation domain shown in Figure 13. The map is interpolated between
the values on the boundary to give values on the whole probation domain in a
continuous manner.

System | Model
---|---
Algorithm | Given by domains in Fig. 12.
Data | Time $t$, good behaviour function $g(t)$.
Objectives | Decide on release or parole.
Oracles | Reports on behaviour from warders, police, probation offlcers.
Visualisation of belief | Show how close the offender is to a transition.

##  6 Mathematical Tools 2: Belief and visualisation

In complex systems beliefs change. When a plane approaches within, say, 100
miles of its target airport the pilot will believe that it should prepare to
land. If it is then found to be running out of fuel then the belief in landing
will rise to absolute necessity, even overriding bad weather warnings. Thus,
we emphasise that we are using beliefs as dynamic quantities, they are not
merely assessing the likelihood of static conditions.

We shall use the Dempster-Shafer theory of belief functions [21] to quantify
belief on a scale of 0 (no evidence) to 1 (convincing evidence). There is a
large literature on this subject, but it shares the idea of a measure for
belief which can be updated on the presentation of additional evidence. Here
we are not concerned with any particular mechanism of how this update is
performed.

###  6.1 Dempster-Shafer belief

It is important to distinguish between belief in something and a probability
that that thing occurs. On being told by a completely reliable source that a
person has a car which has a single colour, we could form two statements,

$B=$‘the car is blue’ and $NB=$‘the car is not blue’.

The combination $B$ or $NB$ we would consider to have a belief of 1, but in
the absence of any other information we would have no evidence base to assign
a non-zero belief value to the separate statement $B$ or to the statement
$NB$. We could randomly assign belief values to each which added up to 1, but
this would only undermine the idea that belief ought be assigned according to
evidence. Sensibly our belief $\mathrm{Bel}(\\{B,NB\\})$ in the statement ($B$
or $NB$) is strictly greater than the sum of $\mathrm{Bel}(\\{B\\})$ and
$\mathrm{Bel}(\\{NB\\})$. Such a conclusion means that we are not dealing with
a probability distribution.

###### Definition 7

We consider beliefs in a finite set $X$ of statements. A generalised belief
function on $X$ is a function

$\mathrm{Bel}:P(X)\to[0,1],$

where $P(X)$ is the set of subsets of $X$. Furthermore, for the empty set
$\mathrm{Bel}(\emptyset)=0$ and the sets satisfy the ‘super-additivity’
property

$\displaystyle\mathrm{Bel}(Y\cup Z)+\mathrm{Bel}(Y\cap
Z)\geq\mathrm{Bel}(Y)+\mathrm{Bel}(Z)\quad\mathrm{for\ all}\ Y,Z\subset X.$
(2)

We denote by $\mathcal{B}(X)$ the set of belief functions on $X$.

We shall say that a belief function is normalised if $\mathrm{Bel}(X)=1$. It
is important to note that we do not impose this normalisation condition –
hence our word generalised above. Thus, we reserve the right to believe that
the set of presented alternatives $X$ may be incomplete. The ‘super-
additivity’ property is simply justified in that, in addition to those people
who believe in $Y$ and those who believe in $Z$, there may be people who just
believe that at least one is true.

Dual to the belief function is this:

###### Definition 8

The plausibility function

$\mathrm{Pla}:P(X)\to[0,1]$

is defined by

$\mathrm{Pla}(Y)=\mathrm{Bel}(X)-\mathrm{Bel}(X\setminus Y),$

where $X\setminus Y$ is the complement of $Y$ in $X$.

The plausibility of a subset of statements may be taken as the extent to which
people think that it might possibly be true.

###### Proposition 2

$\mathrm{Bel}(Y)\leq\mathrm{Pla}(Y)$ for all $Y\subset X.$

###### Proof

Note that

$\mathrm{Bel}(Y)\leq\mathrm{Bel}(Y\cup(X\setminus Y))-\mathrm{Bel}(X\setminus
Y)=\mathrm{Bel}(X)-\mathrm{Bel}(X\setminus Y)=\mathrm{Pla}(Y).$

∎

Also, we can deduce that the plausibility function is sub-additive, i.e.,

$\mathrm{Pla}(Y\cup Z)+\mathrm{Pla}(Y\cap
Z)\leq\mathrm{Pla}(Y)+\mathrm{Pla}(Z)\quad\mathrm{for\ all}\ Y,Z\subset X.\ $

###  6.2 Visualisation of a belief function

In principle, belief functions give a large amount of data which is difficult
to understand – for a set $X$ of size $n$ we have $2^{n}$ values.333For a
probability measure we would only need the $n$ point measures by additivity,
but belief functions contain much more information. However, as long as we are
clear about what information we actually want, it is possible to visualise the
results in a much lower dimension.

So, in practice, what do we need to know about beliefs?

(1) We need to know when to add to our list of things $X$ to believe – when
our existing tool collection is incomplete, as described in subsection 2.3.

(2) We need to subtract things from $X$ which we no longer think are
necessary, in accordance with Occam’s razor, as described in subsection 2.3.

Now (1) has a simple answer, even though in practice that answer might lead to
many more questions.

###### Definition 9

Let $\mathcal{B}(X)$ be the set of belief functions on $X$. The first piece of
the visualisation of a belief function $\mathrm{Bel}\in\mathcal{B}(X)$ is the
belief in the whole of $X$, that is $\mathrm{Bel}(X)\in[0,1]$.

For (2), how can we visualise a belief in subsets of $X$ so that it is obvious
which objects in $X$ are no longer needed? Answer: we visualise the belief as
a point in the simplex $\Delta_{X}$ spanned by $X$, as given in Proposition 1.

###### Definition 10

The second piece of the visualisation of a belief function
$\mathrm{Bel}\in\mathcal{B}(X)$ is the point in the simplex

$\frac{\sum_{x\in X}\mathrm{Pla}(\\{x\\})\,e_{x}}{\sum_{x\in
X}\mathrm{Pla}(\\{x\\}))}\in\Delta_{X}\ .$

The first thing to note is that the behaviour of this fraction is, shall we
say, erratic if the denominator is small. In our case, by sub-additivity, the
denominator is

$\sum_{x\in X}\mathrm{Pla}(\\{x\\})\geq\mathrm{Pla}(X)=\mathrm{Bel}(X).$

Thus, if $\mathrm{Bel}(X)\in[0,1]$ is close to 1, it is reasonable to take the
fraction. If it is close to zero we may have problems, but in that case our
belief in the set $X$ is so small that all conclusions will be considered
unreliable.

The second thing to note is that this function in Definition 10 really does
solve requirement (2). To make this precise we need the idea of a face of a
simplex.

###### Definition 11

For $Y\subset X$ the face $\Delta_{Y}\subset\Delta_{X}$ of $\Delta_{X}$ is

$\Delta_{Y}=\Big{\\{}\sum_{y\in Y}\lambda_{y}\,e_{y}:\lambda_{y}\geq 0,\
\sum_{y\in Y}\lambda_{y}=1\Big{\\}}\subset\Delta_{X}\ .$

###### Proposition 3

For a belief function on $X$ with $\mathrm{Bel}(X)\neq 0$ and $Y\subset X$

$\mathrm{Bel}(Y)=\mathrm{Bel}(X)\ \mathrm{if,and\ only,if}\ \frac{\sum_{x\in
X}\mathrm{Pla}(\\{x\\})\,e_{x}}{\sum_{x\in
X}\mathrm{Pla}(\\{x\\}))}\in\Delta_{Y}\ .$

Proof: First, we assume the RHS. Then, for $x\in X\setminus Y$ we have
$\mathrm{Pla}(\\{x\\})=0$, and, by sub-additivity, $\mathrm{Pla}(X\setminus
Y)=0$; by definition of $\mathrm{Pla}$, we get
$\mathrm{Bel}(Y)=\mathrm{Bel}(X)$.

Second, we assume the LHS. Then for $x\in X\setminus Y$ we have
$\mathrm{Bel}(X)\geq\mathrm{Bel}(X\setminus\\{x\\})\geq\mathrm{Bel}(Y)=\mathrm{Bel}(X)$
so $\mathrm{Bel}(X\setminus\\{x\\})=\mathrm{Bel}(X)$. We deduce
$\mathrm{Pla}(\\{x\\})=0$ and so we get the RHS. $\square$

###### Definition 12

For a belief function on $X$ with $\mathrm{Bel}(X)\neq 0$ we define the
visualisation map

$\mathrm{Vis}:\mathcal{B}(X)\to[0,1]\times\Delta_{X}$

by

$\mathrm{Vis}(\mathrm{Bel})=\Big{(}\mathrm{Bel}(X),\frac{\sum_{x\in
X}\mathrm{Pla}(\\{x\\})\,e_{x}}{\sum_{x\in X}\mathrm{Pla}(\\{x\\}))}\Big{)}\
.$

###  6.3 The meanings and visualisation of belief functions

We shall explain the use of the visualisation function in Definition 12 with
the set $X=\\{a,b,c\\}$. The 2-simplex $\Delta_{X}$ is the filled in triangle
in Figure 14, and the face $\Delta_{Y}$ for $Y=\\{a,b\\}\subset X$ is shown by
the thickened line.

The first component of the visualisation function can be interpreted as the
reliability of the whole procedure. For example, before designers embark on a
system, they may evaluate how much confidence they have in the methods and
resources they are asked to use. This confidence level
$\mathrm{Bel}(X)\in[0,1]$ affects the interpretation of everything which
follows. It would be common to use a colour code where green would denote
$\mathrm{Bel}(X)\cong 1$, orange an intermediate value and red for a low value
of $\mathrm{Bel}(X)$. An alternative would be to attach a numerical value to
the point in the simplex given by the second component of the visualisation.
Because of problems with colours and to give a more exact value we use both
methods in Figure 14, where coloured dots give the positions in the simplex
and in addition the value of $\mathrm{Bel}(X)$ is given as a percentage.

$e_{a}$$e_{b}$$e_{c}$$x$95% $\bullet$$w$95% $\bullet$$y$95% $\bullet$$z$85%
$\bullet$$u$65% $\bullet$ Figure 14: Values of the visualisation map for the
set $X=\\{a,b,c\\}$

First, we consider the three green dots in Figure 14 with a belief value of
$\mathrm{Bel}(X)=0\cdot 95$. We suppose that this value is sufficiently high
that we have reasonable confidence in the rest of the values. The point $w$ is
on the face $\Delta_{\\{b,c\\}}$ so, according to Proposition 3, in this case

$\mathrm{Bel}(\\{b,c\\})=\mathrm{Bel}(X)\ \textrm{and}\
\mathrm{Bel}(\\{a\\})=0.$

The point $x$ is on the face $\Delta_{\\{c\\}}$ so, according to Proposition
3, in this case

$\mathrm{Bel}(\\{c\\})=\mathrm{Bel}(X)\ \textrm{and}\
\mathrm{Bel}(\\{a,b\\})=0.$

The point $y$ is close to the face $\Delta_{\\{b,c\\}}$ so
$\mathrm{Bel}(\\{a\\})$ is small. In practice, we could decide that $a\in X$
was not needed if $\mathrm{Bel}(\\{a\\})$ was sufficiently small, it would be
almost impossible to obtain a value which was exactly zero.

The orange point $z$ corresponds to a belief function for which
$\mathrm{Bel}(X)=0\cdot 85$. At this level we suppose that there is
significant concern that the model based on $X$ is not sufficient, and may not
be performing optimally. (Of course the threshold levels for concern would be
set at different values for different models and problems.) Still, according
to the position being not close to the faces, we see that there is no evidence
that we can ignore any elements of $X$.

The red point $u$ corresponds to a belief function for which
$\mathrm{Bel}(X)=0\cdot 65$. At this level we suppose that there is great
concern that the model based on $X$ is not sufficient, and may be giving
dubious values.

##  7 Changing frames of belief and theories

So far, we have been considering systems in which we suppose the modes are
chosen and fixed in the process of designing the system. The focus is on how
we move between these modes according to prescribed conditions and algorithms
that interpret monitoring data. Recalling our anthropomorphic guidelines in
Section 2, and the simple principles about the ideas of innovation,
imagination and testing, we can ask: How might modes be modified or even
created in response to unforeseen situations? This is an open problem but one
worthy of some light speculation, given its relevance for our approach. The
light speculation soon encounters ancient and deep philosophical issues,
however, especially in the epistemologies of the sciences, physical and
social.

###  7.1 Frames of belief as modes

> In so far as a scientific statement speaks about reality, it must be
> falsifiable: and in so far as it is not falsifiable, it does not speak about
> reality.
>
> Karl R. Popper, The Logic of Scientific Discovery [20]

Consider how the components of a mode in a system (see Definition 1)
correspond with models of a real situation, broadly conceived and interpreted,
in the table:

System | Model
---|---
Algorithm | Theories and formal descriptions to process
| results and predict observations
Data | Data, determined by the theories,
| from previous observations and computations
| e.g., constants of nature, parameters, …
Objectives | Predict observations, …
Oracles | Experimental apparatus, instruments, tools
| for making observations, and sampling…
Visualisation of the | Means to evaluate how well the model predicts observations.
belief function. |

The visualisation of belief is one of the primary ideas of this paper (Section
6.3), and is focussed on understanding, accounting for and explaining a
system; it is also valuable in thinking about design, e.g., could a simpler
model work? The question now is:

On what foundations do the models and the beliefs depend?

We will use the terminology of a frame of belief to denote whatever
‘knowledge’ the models and belief rest upon. Here is a working definition:

###### Definition 13

Suppose that $\mathcal{M}$ is a list of theories, laws, principles, axioms,
theorems, models, heuristics and guidelines, etc. that may be relevant to a
situation; we call these theoretical elements. A subset $X\subset\mathcal{M}$
we will call a frame of belief, where to the description of each theoretical
element we add methods of interaction between them. For modelling a particular
problem we likely have to consider only a (small) subset $Y\subset X$ of the
contents of $\mathcal{M}$.

In modelling, and especially in modelling social contexts, there is a frame of
belief upon which any particular model is built. In science and engineering,
we expect the frames of belief to be clearly identifiable and indeed known, if
not always forming a concensus.

In social contexts, the frames of belief we expect to be harder to identify
and in many cases contested. In particular, in social affairs, commonly there
is a cocktail of social theory, social and cultural orthodoxies, religious
traditions, formal regulation, received opinion, political influence,
prejudice, bias, etc.

What can be said about a frame of belief in general?

Here there is something of a cliff edge for authors of a humble technical
paper on systems. The intellectual ocean that is the philosophy of knowledge
is in sight with its schools of thought, philosophical theories, and profound
debates, which cover so many physical, social and individual concerns.

From our point view, we see and propose an abstraction:

Frame of Belief 1: Form.

A frame of belief can be considered as a mode

Frame of Belief 2: Change.

A change of frame of belief can be considered as a change of mode.

This means that the set of principles for modes we introduced earlier, in
Section 3, may be relevant to shaping and developing the notion, e.g., there
are basic frames of belief, joint frames of belief, and frames of belief have
hierarchical structure.

###  7.2 Forms of frames of belief in science

However, in our context of thinking about human scenarios, there is something
to gained initially for reflecting on changes in our scientific understanding
of the physical world. In science and engineering it is easy to see ideas of
innovation, imagination and testing as drivers of problem solving. So, for
inspiration, let us turn to the history and philosophy of science and
engineering, where we may find some suggestive bits and pieces to illustrate
what ‘knowledge’ underpins the making of a mode.

The situation in physical science is philosophically simpler than in human
science:

###### Example 5

Simplicial complex. Suppose that $\mathcal{M}$ is a list of scientific
theories, old and new, e.g., general relativity, quantum mechanics, atomic
theory, Newtonian mechanics, thermodynamics, the luminiferous aether, the
fluid theory of heat, the kinetic theory of heat, etc. At some period in
history, we can say that human knowledge has a frame of belief $X$ – i.e., is
in mode $X\subset\mathcal{M}$ – if the science in $X$ is commonly applied to
our understanding of (an aspect of) the natural world. We can take the
abstract simplicial complex $\mathcal{C}$ to be the set of subsets $X$ which
do not contain immediate contradictions (e.g., Flat Earth and Round Earth
theories do not both belong to an element of $\mathcal{C}$). ∎

A mode or model formed of several elements of $\mathcal{M}$ is not just the
sum of the different components.

###### Example 6

Joint modes. Consider combining Maxwell’s equations of electromagnetism with
Newtonian gravity. We need to decide how the theories affect each other. For
example, we could decide that electromagnetic fields are not sources of
gravity, and that they are in turn not affected by gravitational fields. Of
course, this would run into problems because it could not predict
gravitational lensing.

###### Example 7

Hierarchy of modes. We can illustrate frames of belief by opening up a mode
‘atomic theory’ into various submodes, corresponding to different ideas of
atomic structure [1, 15]. From ideas by the classical Greeks (including the
name atom), John Dalton proposed an idea of atomic theory, but without any
internal structure of the atoms. In 1867, William Thomson proposed that atoms
were knots in the aether. This was followed in 1878 by Alfred Mayer’s
proposition of atoms as magnets arranging themselves into molecules. After the
discovery of the electron William Thomson and J.J. Thompson (the discoverer of
the electron) proposed the ‘plum pudding’ model for the atom. Rutherford
proposed a nucleus surrounded by electrons. Niels Bohr introduced the
quantisation of the orbits of the electrons, and Schrodinger gave the first
modern treatment of atoms using wave functions, which has since been modified
by the introduction of relativity and spin (the Klein-Gordon and Dirac
equations). ∎

Scientific examples, old and new, are readily available and can be quite
specific, with the theoretical elements being precise laws or equations. A
major contemporary example would be the competing theories for explaining dark
matter [13].

However, for us it is important to note that Definition 13 needs to be
explored in terms of human-centred theoretical elements: sociological
theories, social categories, economic models, cultural practices, regulations
and laws, priorities and bias, social media tags, etc. The epistemic space of
social science is much more complicated, of course, as many working notions
are imprecise, debated and are culture and country specific. Yet, there is no
shortage of government regulations, formal procedures, and notions of ‘best
practice’ that require precision and agreement at some level. Indeed it is
these that invite automation.

For our general speculative discussion here, the social examples would contain
too many distractions.

However, the attempt to define frames of belief for human and social contexts
is directly relevant to our research agenda here, e.g., consider frames of
belief as envelopes for the examples in Section 5.

###  7.3 Changing frames of belief in science

Now we need to look at how frames of belief change. Interpreted as modes,
change is facilitated by transition functions, and there is an interesting
link between transition functions and information hiding. Two types of change
have been highlighted in Section 2.3.

We consider moving from a mode $X$ to a subset mode $Y\subset X$ when we no
longer believe that its complement $X\setminus Y$ in $X$ is relevant and may
be hidden or discarded. In agreement with Ockham’s Razor 2 in Section 2.3, we
established a principle to move to a simpler (subset) mode where possible.

###### Example 8

Phlogiston was a substance whose release was supposed to cause combustion
[24]. Over time the existence of Phlogiston became simply not necessary to
explain phenomena. In the notation above from mode $X$ we have $X\setminus
Y=\\{\mathrm{Phlogiston}\\}$. ∎

Alternately, consider moving from mode $X$ to a superset mode $X\subset Z$.
This is more difficult as we are moving into an unknown country where there
are factors in mode $Z$ that mode $X$ was never designed to understand.
Firstly, we would not want to move from mode $X$ if we believe that it is
doing well, there has to be a motivation to move, such as problems. In
science, change has come about from ‘a state of growing crisis’:

> Thermodynamics was born from the collision of two existing nineteenth-
> century physical theories, and quantum mechanics from a variety of
> difficulties surrounding black-body radiation, specific heats, and the
> photoelectric effect. Furthermore, in all these cases except that of Newton
> the awareness of anomaly had lasted so long and penetrated so deep that one
> can appropriately describe the fields affected by it as in a state of
> growing crisis.
>
> Thomas S. Kuhn, The Structure of Scientific Revolutions [25]

However, not so obviously, there is a problem when this crisis is too large.
It is not that we cannot change mode in a time of great crisis (indeed there
may be little alternative), it is rather that we would be so far into an
unknown country that the result of doing so would not be predictable and beset
with risk.

Finally, recall Example 6 with visualisation in mind:

###### Example 9

For a philosophical example, we take $\mathcal{M}$ to be the theories of the
universe. A set of theories $X\subset\mathcal{M}$ will be a simplex if those
theories could possibly co-exist simultaneously, i.e., are not contradictory.
Introducing the geometry of simplicies, this means there is no edge between
the vertices ‘flat Earth’ and ‘round Earth’, but there is an edge between
‘special relativity’ and ‘thermodynamics’. Further, being on that edge would
allow more theory and predictions of observations than either theory could in
isolation. Evidence from observations and experiments would be used to place
our actual universe (from a hypothetical ‘state space’ of universes) at some
point in the simplicial complex, or to show that our theories are faulty by
awkwardly pointing at a state not compatible with our construction. ∎

###### Example 10

In particle physics we have modes consisting of field theories, so Quantum
Electrodynamics would be a mode and the Standard Model would be another mode.
We begin in such a mode (currently the Standard Model), and experiments are
proposed and modelled (typically by Feynman diagrams). Then data from actual
experiments is compared to the calculated values. If the experimental results
differ from the calculated values by $3\sigma$ ($\sigma$ denotes one standard
deviation in terms of experimental and statistical error) then we are
concerned that the model may not be correct. If the experimental results
differ from the calculated values by $5\sigma$ then we believe that there is a
real problem and seek a solution by modifying the model. This would typically
be by proposing a new particle or field to add to the model.

##  8 Concluding remarks

###  8.1 Reflections

Our paper is about automation in the context of human-centred systems. With
automation in mind, we have introduced

1\. a new general system model based on a notion of modes and mode
transitions;

2\. formalised the model mathematically both algebraically and geometrically;

3\. advocated an anthropomorphic approach that focuses on the explication of
systems, based on intuitions about conceptual frameworks and frames of belief;

4\. proposed human-centred case studies to illustrate/test the model; and

5\. connected our concept of ‘belief about data’ with the established
quantitative theory of belief pioneered by Dempster and Shafer.

With purely technical ideas in mind, the contributions are (1), (2) and (5),
introducing concepts and data structures for modelling modes in which the
position of the system in the geometric visualisation of the modes represents
quantified judgements or beliefs about the state of the system. Thus, our
system model focuses on the dynamics of changing interpretations of monitoring
data and offers some insight into reasons for decisions taken to change the
behaviour of the system.

With human-centred applications in mind, the contributions are (3) and (4) in
which we align the system model with simple working assumptions and principles
of human reasoning and scenarios for automation. To guide the design of
changes in the system we considered, for example, Incompleteness and Occam’s
Razor (as described in Section 2), which motivate a move to a simpler system
when possible, and to a more complicated system only when forced to do so. We
noted how changes in the ‘frames of belief’ that underly the design of modes
are also a factor, and we speculated that they can have the same structure as
modes.

###  8.2 Questions and answers

Here are some questions that come to mind that may help to clarify a few
points:

Q: Does the modes model apply only to the highest level of the organisation of
the system?

A: No, we can have a hierarchy where each of the modes in the top level
becomes, on closer examination, a simplicial complex of modes (called sub-
modes, and having the same organising principles). From the system design
point of view, this is simply a form of information hiding (e.g., a single
mode labelled ‘court’ for a police officer opens into a multitude of submodes
for those organising court proceedings). From a mathematical point of view,
the relevant idea to preserve the structure is a simplicial map between
simplicial complexes.

Q: How do modes relate to the environment?

A: For dealing mathematically with the real world we have in mind the
formalism of physical oracles [3] that both gather information and affect the
real world, and allows for errors and delays. Each mode has a set of such
oracles with which it can communicate.

Q: How does the geometry specify the algorithms?

A: There is a huge amount of information hiding in which the geometry of the
modes specify which modes neighbour the current mode, and the algorithm only
knows about those neighbouring modes.

Q: Can a system of modes be retro-fitted to explain some system?

A: Perhaps: because modes combine objectives, specifications and
implementations, this combination can identify the scope and limits of
components and refactor a complex system.

Modes have a number of connections with general techniques and themes in
computing:

Q: How do modes relate to learning algorithms such as neural nets?

A: The oracle formalism can also be used to import output from a potentially
non-explainable algorithm (such as a neural net), where the possible error can
quantified and considered along with other information. Actually, the output
of a classification procedure, such as a neural net, can be taken as an
element of a simplex spanned by the elements of the set of possible
classifications, and thus can fit into a simplicial approach.

Q: How do modes relate to general themes such as security, safety and privacy?

A: Modes can be classified by allocated a label for security clearance,
potential harm or sensitivity. This can be done via a simplicial map to a
simplicial complex of appropriate classifications and this can determine what
information (via the oracles) each mode has access to. Only certain modes
would have access to oracles allowing some possibly problematic behaviour –
other modes simply cannot access these oracles. For instance, simplicial
complexes stand comparison with lattices [12] for a formal study of security.
Oracles cover all external computers and databases and sensors and actuators .
Judgements can be made as to how reliable these oracles are.

###  8.3 On systems in AI and current controversies

Considering the list of five contributions of this paper above, numbers (1),
(2) and (5) are about general system theories, which are relevant to the
design of many kinds of computing system. The problem of controlling
autonomous systems in unpredictable environments is central to the remit of
AI, which has benefitted from new system models with different motivations,
features and mathematical theories. Since the 1960s many new system models
have been developed such as deterministic and nondeterministic networks of
computing units and concurrent systems shaped by theories of automata,
probability and physical systems as in control systems [16] and hybrid systems
[29]. Our ‘first principles approach’ to the technical material aligns with
more fundamental conceptual approaches to systems, such as Carl Hewitt’s
thinking about concurrency and his actor model, which finds a home in AI
through the modern and conceptually strong theory of agents [37, 38, 23]. The
usefulness of our technical work to aspects of multiagent modelling is
something worth investgating. Perhaps agents can be factorised into modes or
simply be modes.

Contributions numbered (3) and (4) are relevant to human-centred AI
applications, which are the cause of current AI controversies. Issues such as
transparency, explainability, etc. are more prominent, significant and
difficult to tackle for human-centred AI applications. In this paper, we
consider automation of human-centred tasks. We also try to consider human
reasoning and innovation, and their links with modes through examples.

One underlying theme is an idea of belief, and we use this and simplicial
geometry both to make decisions and to visualise beliefs about the state of
the system, both these being important topics in AI. The simplicial model is a
higher dimensional data structure whose compactess and natural heirachical
structure improves on planar graphs. Graphs of various kinds can be found in
most system theories, and play a role in explainability, e.g., in machine
learning [34].

The frameworks of belief underlying human-centred systems are hard to analyse
and codify even for our simplified examples and seem destined to become a
topic in their own right. Our refections are shaped by the role of models and
theories in the history of science, which are convenient, general and better
understood than examples of frameworks of belief that are highly contextual,
made from from social theories, regulations, cultural norms, or best practice.

Thus, we see our paper as offering some conceptual and mathematical tools for
thinking about AI systems to better understand and make them more more easily
accountable, which are essential as AI systems deal with people.

Any discussion of methods for understanding automation may be relevant to
currently growing AI controversies. For example, among a number of headline
grabbing actions an open letter “Pause Giant AI Experiments” has been signed
by many leading industrialists and academics.444See:
https://futureoflife.org/open-letter/pause-giant-ai-experiments. In the space
of a few months, this was followed by further warnings, international meetings
about regulation, widespread debates on AI tools in many sectors. Our
connection to these ‘growing crises’ is through our interest in explainable
automations, most urgently in decision making in human affairs.

Legacy liabilities. We are proposing an explanatory framework for logical AI
based on human reasoning, which would be used to contain and limit machine
learning (e.g., by imposing Asimov’s laws and other social norms).
Sufficiently extensive interconnected logical AIs, developed over decades with
many technologies, are bound to pose problems in the years to come. If legacy
AIs were also granted permission to innovate and mutate, the problem could be
intractable – given our problems with current legacy software. It is rather
essential that any means by which we can understand, impose limits on, reason
with and keep track of, etc, an automation is explored. And it helps to know
what we might be looking for as we research the future.

Decision making. At the centre of our thinking about modes is decision making.
Decision making permeates computing applications. Monitoring and managing
systems involves decisions to change modes of operation based on judgements
about available monitoring data. In human-centred systems, classifying
people’s data involves choices about the characteristics that define
typologies, from recommender systems in shopping to identifying and tagging
objects and faces. In the professional and social realm, decisions based on
the information available about people have automatically filtered and
processed applications and requests, from creditworthy loans to job
applications. Commonly, decisions in applications involve weighing up data,
arguments, evidence, etc. and making a judgement. Probability theories have
helped analyse decision making since the 18th Century, in the birth of the
mathematics of insurance. More recently, belief theories [18, 21], many valued
logics [10], and sundry machine learning techniques [26], have provided
mathematical theories with which to explore inexact reasoning and making
decisions. Underwritten by the guiding formalisation of an abstract simplicial
complex, the scope of the modes could become much wider.

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