# Mathematical Principles of Dynamic Systems and the Foundations of Quantum
Physics

Eric Tesse

###### Abstract

Everybody agrees that quantum physics is strange, and that the world view it
implies is elusive. However, it is rarely considered that the theory might be
opaque because the mathematical language it employs is inarticulate. Perhaps,
if a mathematical language were constructed specifically to handle the
theory’s subject matter, the theory itself would be clarified. This article
explores that possibility. It presents a simple but rigorous language for the
description of dynamics, experiments, and experimental probabilities. This
language is then used to answer a compelling question: What is the set of
allowed experiments? If an experiment is allowed, then the sum of the
probabilities of its outcomes must equal 1. If probabilities are non-additive,
there are necessarily sets of outcomes whose total probability is not equal to
1. Such experiments are therefore not allowed. That being the case, in quantum
physics, which experiments are allowed, and why are the rest disallowed? What
prevents scientists from performing the disallowed experiments? By phrasing
these questions within our mathematical language, we will uncover answers that
are complete, conceptually simple, and clearly correct. This entails no magic
or sleight of hand. To write a rigorous mathematical language, all unnecessary
assumptions must be shed. In this way, the thicket of ad hoc assumptions that
surrounds quantum physics will be cleared. Further, in developing the theory,
the logical consequences of the necessary assumptions will be laid bare.
Therefore, when a question can be phrased in such a language, one can
reasonably expect a clear, simple answer. In this way we will dispel much of
the mystery surrounding quantum measurements, and begin to understand why
quantum probabilities have their peculiar representation as products of
Hilbert space projection operators.

###### Contents

  1. I Introduction
  2. II Dynamics
    1. II.1 Parameters
    2. II.2 Parametrized Functions and Dynamic Sets
    3. II.3 Dynamic Spaces
    4. II.4 Special Sets
    5. II.5 Limits and Closed Sets
  3. III Experiments
    1. III.1 Shells
    2. III.2 E-Automata
      1. III.2.1 Environmental Shells
      2. III.2.2 The Automata Condition
      3. III.2.3 Unbiased Conditions
      4. III.2.4 E-Automata
    3. III.3 Ideal E-Automata
      1. III.3.1 Boolean E-Automata
      2. III.3.2 All-Reet E-Automata
      3. III.3.3 Ideal E-Automata
    4. III.4 Ideal Partitions
    5. III.5 Companionable Sets & Compatible Sets
  4. IV Probabilities
    1. IV.1 Dynamic Probability Spaces
    2. IV.2 T-Algebras and GPS’s
      1. IV.2.1 $\neg t$ and $[t]$
      2. IV.2.2 $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$
      3. IV.2.3 Convergence on a GPS
      4. IV.2.4 $T_{S}$, $T_{S\mathbb{N}}$, and $T_{\mathbb{N}}^{S}$
    3. IV.3 Nearly Compatible Sets
    4. IV.4 Deterministic & Herodotistic Spaces
      1. IV.4.1 DPS’s on Deterministic & Herodotistic Spaces
      2. IV.4.2 DPS’s in Deterministic Universes
      3. IV.4.3 DPS’s in Herodotistic Universes
  5. V Application to Quantum Measurement
    1. V.1 Preliminary Matters
      1. V.1.1 A Note On Paths
      2. V.1.2 Discretely Determined Partitions
      3. V.1.3 Interconnected Dynamic Sets
    2. V.2 Partitions of Unity for Quantum Systems
      1. V.2.1 The Conditional Case
      2. V.2.2 The Non-Conditional Case
    3. V.3 Maximal Quantum Systems
    4. V.4 Terminus & Exordium
  6. A $\Delta$-Additivity
  7. B Conditional Probabilities & Probability Dynamics
  8. C Invariance On Dynamic Sets and DPS’s
    1. C.1 Invariance On Dynamic Sets
    2. C.2 Invariance on DPS’s
  9. D Parameter Theory

##  I Introduction

Our fascination with quantum physics has as much do to with its strangeness as
its success. This strangeness can conjure contradictory responses: on the one
hand, the sense that science has dug so deep as to touch upon profound
metaphysical questions, and on the other, the sense that something is amiss,
as science should strive to uncover simple explanations for seemingly strange
phenomena. Fans of the first response will find little of interest in the
paper, for it explores the second.

Let’s start by noting that the mathematical language employed by quantum
mechanics was not developed to investigate the types of problems that are of
interest in that field. Hilbert spaces, for example, were developed to
investigate analogies between certain function spaces & Euclidean spaces; they
were only later adopted by physicists to describe quantum systems. This is in
sharp contrast with classical mechanics; the development of differential
calculus was, to a large extent, driven by the desire to describe the observed
motion of bodies - the very question with which classical mechanics is
concerned. In consequence, it is no exaggeration to say that if a question is
well defined within classical mechanics, it can be described using calculus.

In quantum mechanics, the mathematical language is far less articulate. For
example, it leaves unclear what empirical properties a system must posses in
order for the quantum description to apply. It’s also unclear how, and even
whether, the language quantum mechanics can be used to describe the
experiments employed to test the theory. It is similarly unclear whether
and/or how quantum mechanics can be used to describe the world of our direct
experience.

Such considerations lead to a simple question: To what extent are the
difficulties of quantum theory due to limitations in our ability to phrase
relevant questions in the theory’s mathematical language? If such limitations
do play a role, it would not represent a unique state of affairs. As most
everyone knows, Zeno’s paradoxes seemed to challenge some of our most basic
notions of time and motion, until calculus resolved the paradoxes by creating
a clear understanding of the continuum. Somewhat more remotely, the drawing up
of annual calendars (and other activities founded on cyclic heavenly activity)
was once imbued with a mystery well beyond our current awe of quantum physics.
With the slow advance of the theory of numbers the mystery waned, until now
such activity requires nothing more mysterious than straightforward
arithmetic.

In this article, the question of whether a similar situation exists for
quantum mechanics will be investigated. Three simple mathematical theories
will be created, each addressing a basic aspect of dynamic systems. The
resulting mathematical language will then be used to analyze quantum systems.
If the analysis yields core characteristics of quantum theory, some portion of
the theory’s underpinnings will necessarily be revealed, and a measure of
insight into previously mentioned questions ought to be gained.

The mathematics will be constructed to speak to two of the most basic
differences between experimental results in quantum mechanics and those in
classical mechanics: in quantum mechanics experimental outcomes are non-
deterministic & the experimental probabilities are not additive. Non-
additivity in turn puts limitations on the kinds of experiments that can be
performed. To see this, note that if an experiment has outcomes of $\\{X,\neg
X\\}$, and another has outcomes of $\\{X_{1},X_{2},\neg X\\}$ then
$P(X)+P(\neg X)=1=P(X_{1})+P(X_{2})+P(\neg X)$, and so
$P(X)=P(X_{1})+P(X_{2})$; if $P(X)\neq P(X_{1})+P(X_{2})$ then one of these
experiments can not be performed (the only other possibility is that the
probability of a given outcome depends on the make-up of the experiment as a
whole, but this is not the case in quantum physics). Turning this around, if
there is no set $Y$, s.t. $\\{X\\}\bigcup Y$ and $\\{X_{1},X_{2}\\}\bigcup Y$
are both sets of experimental outcomes, one may wonder if we can expect $P(X)$
to equal $P(X_{1})+P(X_{2})$.

This line of reasoning provides some of the central questions to be answered
in this article: What sorts of experiments can be performed? What limits
scientists to only being able to perform these experiments? What does the set
of allowed experiments imply about the nature of the experimental
probabilities? How do these experimental probabilities correspond to quantum
probabilities? We will seek simple, readily understandable answers to these
questions.

Of the three mathematical theories to be constructed, the first will be a
simple theory of dynamic systems that encompasses both deterministic and non-
deterministic dynamics. Determinism refers to the particular case in which a
complete knowledge of the present grants complete knowledge of the future; all
other cases represent types of non-determinism. The theory of dynamic systems
will then be utilized to construct a theory of experiments; dynamic systems
will be used to describe both experiments as a whole, and the (sub)systems
whose natures the experiments probe. The analysis will be somewhat analogous
to that found in automata theory: when a (sub)system path is “read into” an
experimental set-up, the set-up determines which outcome the path belongs to.
By understanding how such processes can be constructed, we can obtain an
understanding of what types of experiments are performable, reproducible, and
have well defined outcomes.111It should be pointed that, while the analogy
between experiments and automata is useful for creating a quick sketch of the
theory, it is no more than an analogy. In the automata studied by computer
scientists, time is assumed to be discrete and the number of automata states
is assumed to be finite. These assumptions must be made in order to assert
that the automata are performing “calculations”, and they have significant
impact in deriving classes of calculable functions, but such assumptions would
be out of place when discussing experiments.

Finally, building on this understanding, a probability theory will be given
for collections of experiments by assuming that the usual rules of probability
and statistics hold on the set of outcomes for any individual experiment &
that if any two experiments in the collection share an outcome, then they
agree on that outcome’s probability.

These constructs will then be applied to quantum measurements, which are
commonly described in terms of projection operators in a Hilbert space. It
will be found that the nature and structure of these measurements can indeed
be reduced to a clear, simple, rational understanding. At no point will there
be any need to invoke anything the least bit strange, spooky, or beyond the
realm of human understanding, nor any need to rely on any procedures that are
utilized because they work even though we don’t understand why.

Because this paper only considers a fraction of all quantum phenomenon, no
claim can be made that all (or even most) quantum phenomenon can yield to some
simple, rational understanding. What we seek to establish is more modest -
that at least some of our bafflement in the face of quantum physics is due to
the manner with which we address the phenomenon, rather than the nature of the
phenomena themselves.

### On the Question of Interpretations

Though quantum interpretations is a large topic, in this article it will play
a small role. However, because a theory can not be fully understood without
some notion of its possible interpretations, a word or two are in order before
proceeding.

The concept of an "interpretation" will be taken here as being more or less
equivalent to the mathematical concept of a "model".222The mathematical notion
of a “model” is a basic concept from the field of mathematical logic. Since
mathematical logic is not generally a part of the scientific curriculum,
here’s a brief description of model theory. A mathematical theory is a set of
formal statements, generally taken to be closed under logical implication. A
model for the theory is a “world” in which all the statements are true. For
example, group theory starts with three statements involving a binary
function, $\cdot$, and a constant, $I$. They are: For all $x,y,z$, $(x\cdot
y)\cdot z=x\cdot(y\cdot z)$; For all $x$, $x\cdot I=x$; For all $x$ there
exists a $y$ s.t. $x\cdot y=I$. One model for this theory is the set of
integers, with “$\cdot$” meaning $+$ and “$I$” meaning $0$. Another is the set
of non-zero real numbers, with “$\cdot$” meaning $\times$ and “$I$” meaning
$1$. It is generally the case that a theory will have a more than one model. A
model may be thought of as a reality that underlies the theory (in which case
the fact that a theory has many different models simply means that it holds
under many different circumstances). It is this sense of models referring to a
theory’s underlying reality that leads to the correspondence between
scientific interpretations and mathematical models. The theories in this
article will have many models, and there will be no attempt to single out any
one as being preferred. In this section, informal sample models will be given
for the two properties of central interest: non-deterministic dynamics and
non-additive probabilities. These should help provide a background
understanding for the theories.

#### Two Models for Non-Determinism

As noted previously, non-determinism simply means that a complete knowledge of
a system’s current state does not imply a complete knowledge of all the
system’s future states. In the simplest model of non-determinism, at any given
time the system is in a single state, but that state doesn’t contain enough
information to be able to deduce what all the future states will be. This will
be referred to as _type-i_ non-determinism (the “i” standing for “individual”,
because the system is always in an individual state). Turning towards the past
rather than the future, this is akin to the situations found in archeology and
paleontology, in which greater knowledge of the present state does yield
greater knowledge of the past, but you would not expect any amount of
knowledge of the present to yield a complete knowledge of all history.

In _type-m_ non-determinism, the system takes multiple paths simultaneously -
every path it can take, it will take (the “m” in “type-m” stands for
“multiple”). Quantum mechanics is often interpreted as displaying type-m non-
determinism; for example, in a double slit experiment, the particle is viewed
as traveling through both slits.

There are also mixed models. In non-deterministic automata, the input shows
type-i non-determinism (in that a single character in the input string will
not determine what the rest of the input must be), while the automata has
type-m non-determinism (as individual characters are read in, the automata
“non-deterministically samples” all possible transitions). Some quantum
mechanical interpretations invert that view - the system being experimented on
is seen as having type-m non-determinism while the experimental set-up is seen
as showing type-i non-determinism (it’s often further assumed that if the
system were to be deterministic, then the experimental set-up would also be
deterministic, and more specifically, classical). In a similar manner,
decoherence often entails an unspoken assumption that a system displays type-m
non-determinism if the non-diagonal elements of the density matrix do not
vanish, and displays type-i non-determinism otherwise; in this sense, the non-
determinism is considered to be type-i in so far the probabilities are
additive, and type-m in so far as the probabilities are non-additive.

It’s important to stress that type-i, type-m, and mixed models are not the
only possible models for non-determinism; many-worlds interpretations, for
example, provide yet another type of model.

#### Two Models for Non-Additivity

Two models will now be given for systems that display both non-determinism &
non-additive probabilities.

Systems with type-i non-determinism can have non-additive probabilities if
interactions with the measuring devices can not be made arbitrarily small
(e.g., if the fields mediating the interactions are quantized). As an example,
imagine experimental outcome $X$ and outcomes $\\{X_{1},X_{2}\\}$ s.t
$X=X_{1}\bigcup X_{2}$ and $X_{1}\bigcap X_{2}=\emptyset$. The minimal
interactions required to determine $X$ will in general be different from the
minimal interactions to determine $X_{1}$ or $X_{2}$. These differing
interactions will cause different deflections to the system paths, which can
result in differences between the statistical likelihood of the outcome being
$X$ and the statistical likelihood of the outcome being $X_{1}$ or $X_{2}$.
Such a state of affairs can be referred to as the “intuitive model”, because
it allows our physical intuition to be applied.

For type-m non-determinism, one manner in which non-additive probabilities can
appear is if the various paths that the system takes interfere with one other.
In this case, for outcome $X$, all paths corresponding to $X$ may interfere,
whereas for $\\{X_{1},X_{2}\\}$ paths can only interfere if they correspond to
the same $X_{i}$; the differences in interference then lead to different
probabilities for the outcomes. This can be referred to as the “orthodox
model”, because it is shared by many of the most widely accepted
interpretations of quantum mechanics.

Once again, these are just two “sample” models for the non-additivity; there
are many others.

##  II Dynamics

To begin, we require a theory of dynamic systems. Existing theories tend to
make assumptions that are violated by quantum systems, so here a simple,
lightweight theory will be presented; a theory in which all excess assumptions
will be stripped away.

###  II.1 Parameters

Dynamic systems are systems that can change over time; they are parametrized
by time. In a sense, the one feature that all dynamic systems have in common
is that the are parametrized. We therefore begin by quickly reviewing the
concept of a parameter.

Parameters come in variety of forms. For example, some systems have discrete
parameters, while others have continuous parameters. None the less, all
parameters share several basic features. Specifically, a parameter must be
totally ordered, and must support addition. Thus, parameters are structures
with signature $(\Lambda,<,+,0)$, where $<$ is a total ordering, $+$ is the
usual addition function, and $0$ is the additive identity. It is shown in
Appendix D that this, together with the requirement that there be an element
greater than $0$, yields the general concept of a parameter. The set of Real
numbers are parameters in this sense, as are the Integers , the Rationals, and
all infinite ordinals.

This is, however, a little too general. First, in this article we will only be
interested in parameters whose values are finite. Second, for reasons of
analysis, we will only be interested in parameters that are Cauchy convergent.
These two requirements are equivalent to adding a completeness axiom; this
axiom states that all subsets of $\Lambda$ that are bounded from above have a
least upper bound. This limits the models to only four, classified by whether
the parameter are discrete or continuous, and whether or not they are bounded
from below by $0$. These four types of parameters are closely related to the
canonical number systems, and can be readily constructed from them by
introducing the parameter value $\mathbf{1}$, and multiplication by a number.

If the parameter is discrete, assign “$\mathbf{1}$” to be the successor to
$0$; if it’s continuous, choose “$\mathbf{1}$” to be any parameter value
greater than $0$. The choice of $\mathbf{1}$ sets the scale (e.g., 1 second);
it is because parameters have a scale that multiplication is not defined on
parameters.

If $\Lambda$ is a parameter and $\lambda\in\Lambda$, $\lambda$ added to itself
$n$ times will be denoted $n\lambda$ (for example,
$3\lambda\equiv\lambda+\lambda+\lambda$). $0\lambda\equiv 0$.

As shown in Appendix D, constructing the four types of parameters from numbers
is now straightforward. For any discrete, bounded from below parameter,
$(\Lambda,<,+,0)$, the following will hold:
$\Lambda=\\{n\mathbf{1}:n\in\mathbb{N}\\}$,
$n\mathbf{1}+m\mathbf{1}=(n+m)\mathbf{1}$, and $n\mathbf{1}>m\mathbf{1}$ iff
$n>m$. Discrete, unbounded parameters are similar, but with the Integers
replacing the natural numbers. For continuous, unbounded parameters, take
$\mathbf{1}$ to be any positive value, and replace the natural numbers with
the Reals (multiplication by a real number is defined in Appendix D).
Similarly, for continuous, bounded parameters, replace the natural numbers
with the non-negative Reals. This close correspondence between parameters and
numbers is why the two concepts are often treated interchangeably.

We conclude this section by reviewing some standard notation. First, the
notation for parameter intervals:

######  Definition 1.

If $\Lambda$ is a parameter and $\lambda_{1},\lambda_{2}\in\Lambda$

$[\lambda_{1},\lambda_{2}]=\\{\lambda\in\Lambda\,:\,\lambda_{1}\leq\lambda\leq\lambda_{2}\\}$

$(\lambda_{1},\lambda_{2})=\\{\lambda\in\Lambda\,:\,\lambda_{1}<\lambda<\lambda_{2}\\}$

and similarly for $[\lambda_{1},\lambda_{2})$ and $(\lambda,\lambda_{2}]$.

$[\lambda_{1},\infty]=\\{\lambda\in\Lambda\,:\,\lambda_{1}\leq\lambda\\}$

$[-\infty,\lambda_{1}]=\\{\lambda\in\Lambda\,:\,\lambda\leq\lambda_{1}\\}$

and similarly for $(\lambda,\infty)$, etc. (note that
$[\lambda_{1},\infty)=[\lambda_{1},\infty]$).

Next, functions for least upper bound and greatest lower bound:

######  Definition 2.

If $\Lambda$ is a parameter and $\chi\subset\Lambda$ then if $\chi$ is bounded
from above, $lub(\chi)$ is $\chi$’s least upper bound, and if $\chi$ is
bounded from below, $glb(\chi)$ is $\chi$’s greatest lower bound.

$Min(\chi)$ is equal to $glb(\chi)$ if $\chi$ is bounded from below, and
$-\infty$ otherwise. Similarly, $Max(\chi)$ is equal to $lub(\chi)$ if $\chi$
is bounded from above, and $\infty$ otherwise.

And lastly, the definition of subtraction:

######  Definition 3.

If $\Lambda$ is a parameter and $\lambda,\lambda^{\prime}\in\Lambda$:

$\lambda-\lambda^{\prime}\equiv\begin{cases}0&if\,Min(\Lambda)=0\,and\,\lambda<\lambda^{\prime}\\\
\Delta\lambda&where\,\Delta\lambda+\lambda=\lambda^{\prime},\>otherwise\end{cases}$

It follows from the numeric constructions outlined above that
$\lambda-\lambda^{\prime}$ always yields a unique element of $\Lambda$.

###  II.2 Parametrized Functions and Dynamic Sets

Parametrized functions and dynamic sets are the rudimentary concepts on which
all else will be built. This section introduces them, along with their
notational language. We start with parametrized functions.

######  Definition 4.

A _parametrized function_ is a function whose domain is a parameter.

If $f$ is a parametrized function, and $[x_{1},x_{2}]$ is an interval of
$Dom(f)$, then $f[x_{1},x_{2}]$ is $f$ restricted to domain $[x_{1},\,x_{2}]$
(values of $x_{1}=-\infty$ and/or $x_{2}=\infty$ are allowed.)

We define one operation on parametrized functions, concatenation.

######  Definition 5.

If $f$ and $g$ are parametrized functions, $Dom(f)=Dom(g)$, and
$f(\lambda)=g(\lambda)$, then $f[x_{1},\lambda]\circ g[\lambda,x_{2}]$ is the
function on domain $[x_{1},x_{2}]$ s.t.

$f[x_{1},\lambda]\circ
g[\lambda,x_{2}](\lambda^{\prime})=\begin{cases}f(\lambda^{\prime})&if\,\lambda^{\prime}\in[x_{1},\lambda]\\\
g(\lambda^{\prime})&if\,\lambda^{\prime}\in[\lambda,x_{2}]\end{cases}$

That is all that’s needed for parametrized functions. They will now be used to
define dynamic sets.

######  Definition 6.

A _dynamic set_ , $S$, is any non-empty set of parametrized functions s.t. all
elements share the same parameter

If $f\in S$ then $f$ can be referred to as a _path_ , and will be generally be
written $\bar{p}$

With $\bar{p}\in S$, $\Lambda_{S}\equiv Dom(\bar{p})$ (that is, $\Lambda_{S}$
is the parameter that all elements of $S$ share)

$\mathcal{P}_{S}\equiv\bigcup_{\bar{p}\in S}Ran(\bar{p})$; the elements of
$\mathcal{P}_{S}$ are _states_

$S[x_{1},x_{2}]\equiv\\{\bar{p}[x_{1},x_{2}]:\bar{p}\in S\\}$

For $\lambda\in\Lambda_{S}$,
$S(\lambda)\equiv\\{p\in\mathcal{P}_{S}:for\>some\>\bar{p}\in
S,\>\bar{p}(\lambda)=p\\}$ (That is, $S(\lambda)$ is the set of possible
states at time $\lambda$)

$Uni(S)\equiv\\{(\lambda,p)\in\Lambda_{S}\otimes\mathcal{P}_{S}:p\in
S(\lambda)\\}$ (“$Uni(S)$” is the “universe” of $S$ \- the set of all possible
time-state pairs)

In the above definition, “time” was occasionally mentioned. In what follows,
we will refer only to “the parameter”, and not “time”. This is because “time”
has grown into an over-loaded concept. For example, in relativity the time
measured on a clock is a function of the path the clock takes. Time measured
by a clock is generally referred to as the “proper time”. On the other hand,
in relativity theory a particle state is generally taken to be a four
dimensional vector, $\overrightarrow{x}$, and coordinates are often chosen so
that the “$0^{th}$” element parametrizes the dynamics. This coordinate is
generally referred to as the “time coordinate”, and this notion of time
referred to as “coordinate time”. For such single particle systems, the
elements of $Uni(S)$ will be of the form $(\lambda,\overrightarrow{x})$, and
for coordinates that have a “time coordinate”, the $0^{th}$ element of
$\overrightarrow{x}$ will always equal $\lambda$. Since coordinates may be
used to describe all paths, and “proper time” is path dependent, it follows
that in such this case $\lambda$ may not equal the proper time.

In this paper, all such complications will be shrugged off. First, we will
make no attempt to map states onto coordinates. Moreover, there will be a
preference to reserve the term “time” for the quantity that is measured by
clocks. Since this may or may not equal the quantity used to parametrize the
system dynamics (depending on the path the clock takes), we retire the term
“time” and speak only of “parameters”.

Back to dynamic sets.

The concatenation operation can be extended to sets of path-segments.

######  Definition 7.

If $S$ is a dynamic set and $A$ and $B$ are sets of partial paths of $S$ then
$A\circ
B\equiv\\{\bar{p}_{1}[x_{1},\lambda]\circ\bar{p}_{2}[\lambda,x_{2}]:\bar{p}_{1}[x_{1},\lambda]\in
A,\>\bar{p}_{2}[\lambda,x_{2}]\in
B\>and\>\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)\\}$

The following notation for various sets of path-segments will prove quite
useful.

######  Definition 8.

If $S$ is a dynamic set,
$(\lambda,p),(\lambda_{1},p_{1}),(\lambda_{2},p_{2})\in Uni(S)$, and
$\lambda_{1}\leq\lambda_{2}$

$S_{\rightarrow(\lambda,p)}\equiv\\{\bar{p}[-\infty,\lambda]:\bar{p}\in
S\>and\>\bar{p}(\lambda)=p\\}$

$S_{(\lambda,p)\rightarrow}\equiv\\{\bar{p}[\lambda,\infty]:\bar{p}\in
S\>and\>\bar{p}(\lambda)=p\\}$

$S_{(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2})}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}\in
S,\>\bar{p}(\lambda_{1})=p_{1},\>and\>\bar{p}(\lambda_{2})=p_{2}\\}$

If $S$ is a dynamic set and $p,p_{1},p_{2}\in\mathcal{P}_{S}$

$S_{\rightarrow p}\equiv\\{\bar{p}[-\infty,\lambda]:\bar{p}\in
S,\>\lambda\in\Lambda_{S},\>and\>\bar{p}(\lambda)=p\\}$

$S_{p\rightarrow}\equiv\\{\bar{p}[\lambda,\infty]:\bar{p}\in
S,\>\lambda\in\Lambda_{S},\>and\>\bar{p}(\lambda)=p\\}$

$S_{p_{1}\rightarrow
p_{2}}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}\in
S,\>\lambda_{1},\lambda_{2}\in\Lambda_{S},\>\bar{p}(\lambda_{1})=p_{1},\>and\>\bar{p}(\lambda_{2})=p_{2}\\}$

If $S$ is a dynamic set, $Y,Z\subset\mathcal{P}_{S}$

$S_{Y\rightarrow Z}\equiv\bigcup_{p\in Y,p^{\prime}\in Z}S_{p\rightarrow
p^{\prime}}$

…and similarly for $S_{\rightarrow Z}$, $S_{Y\rightarrow}$,
$S_{(\lambda,Y)\rightarrow(\lambda,Z)}$, etc.

This notation may be extended as needed. For example, $S_{\rightarrow
p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}\rightarrow}$ is the set
of all paths in $S$ that pass through $p_{1}$, then through
$(\lambda_{2},p_{2})$, then through $p_{3}$.

One of the most basic properties of a dynamic systems is whether or not its
dynamics can change over time. A dynamic set is homogeneous if its dynamics
are the same regardless of when a state occurs. More formally:

######  Definition 9.

A dynamic set, $S$, is _homogeneous_ if for all
$(\lambda_{1},p),(\lambda_{2},p)\in Uni(S)$, $\lambda_{1}\leq\lambda_{2}$,
$\bar{p}\in S_{\rightarrow(\lambda_{1},p)\rightarrow}$ iff there exists a
$\bar{p}^{\prime}\in S_{\rightarrow(\lambda_{2},p)\rightarrow}$ s.t. for all
$\lambda\in\Lambda_{S}$,
$\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda+\lambda_{2}-\lambda_{1})$.

Note that if $\Lambda_{D}$ is bounded from below, $S$ can still be
homogeneous; it would simply mean that the paths running through
$(\lambda_{1},p)$ and those running through $(\lambda_{2},p)$ only differ by a
shift & by the fact the initial part of the paths running though
$(\lambda_{1},p)$ are cut off.

A related property is whether or not a given state, or set of states, can
occur at any time.

######  Definition 10.

If $S$ is a dynamic set, $p\in\mathcal{P}_{S}$ is _homogeneously realized_ if
for every $\lambda\in\Lambda_{S}$, $p\in S(\lambda)$.

$A\subset\mathcal{P}_{S}$ is _homogeneously realized_ if every $p\in A$ is.

###  II.3 Dynamic Spaces

Dynamic sets embrace quite a broad concept of dynamics. This can make them
difficult to work with, and so it is often helpful to make further assumptions
about the system dynamics. A common assumption for closed systems is that the
system’s possible future paths are determined entirely by its current state.
This notion is captured by the following type of dynamic set:

######  Definition 11.

A _dynamic space_ , $D$, is a dynamic set s.t. if $\bar{p},\bar{p}^{\prime}\in
D$, $\lambda\in\Lambda_{D}$, and $\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda)$,
then $\bar{p}[-\infty,\lambda]\circ\bar{p}^{\prime}[\lambda,\infty]\in D$.

Thus, dynamic spaces are closed under concatenation. This will prove to be an
enormous simplification. Closed systems are generally assumed to have this
property, and so in what follows, closed systems will always be assumed to be
dynamic spaces. (Note that a system being experimented on interacts with
experimental equipment, and so is not closed. As a result, such a system might
not be a dynamic space.)

######  Definition 12.

If $D$ is a dynamic space, $D_{\square_{1}\rightarrow\square_{2}}$ may be
written $\square_{1}\rightarrow\square_{2}$. (For example,
$D_{p_{1}\rightarrow p_{2}}$ may by written $p_{1}\rightarrow p_{2}$).

Similarly, $D_{\square_{1}\rightarrow\square_{2}}\circ
D_{\square_{2}\rightarrow\square_{3}}$ may be written
$\square_{1}\rightarrow\square_{2}\rightarrow\square_{3}$,
$D_{\rightarrow\square_{1}}\circ D_{\square_{1}\rightarrow\square_{2}}\circ
D_{\square_{2}\rightarrow}$ may be written
$\rightarrow\square_{1}\rightarrow\square_{2}\rightarrow$, etc.

If $D$ is a dynamic space then, for example,
$D_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow
p_{3}}=D_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ
D_{(\lambda_{2},p_{2})\rightarrow p_{3}}$; if, on the other hand, $S$ is
simply a dynamic set, we could only assert that
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}}\subset
S_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ S_{(\lambda_{2},p_{2})\rightarrow
p_{3}}$. This is why, for dynamic spaces, we may relax the notation and simply
refer to $p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}$, while for
dynamic sets we need to be clear about whether we mean
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ
S_{(\lambda_{2},p_{2})\rightarrow p_{3}}$ or
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}}$.

For dynamic spaces the use of outer arrows, such as
$\rightarrow(\lambda_{1},p_{1})\rightarrow p_{2}\rightarrow$, may be extended
to arbitrary sets of partial paths:

######  Definition 13.

If $D$ is a dynamic space, and $A$ is a set of partial paths s.t. if
$\bar{p}[x_{1},x_{2}]\in A$ then $\bar{p}[x_{1},x_{2}]\in D[x_{1},x_{2}]$,

$\bar{p}\in\,\rightarrow A\rightarrow$ if $\bar{p}\in D$ and for some
$\bar{p}^{\prime}[x_{1},x_{2}]\in A$,
$\bar{p}[x_{1},x_{2}]=\bar{p}^{\prime}[x_{1},x_{2}]$.

$\bar{p}[x,\infty]\in\,A\rightarrow$ if $\bar{p}[x,\infty]\in D[x,\infty]$ and
for some $\bar{p}^{\prime}[x,x_{1}]\in A$,
$\bar{p}[x,x_{1}]=\bar{p}^{\prime}[x,x_{1}]$.

$\bar{p}[-\infty,x]\in\,\rightarrow A$ if $\bar{p}[-\infty,x]\in D[-\infty,x]$
and for some $\bar{p}^{\prime}[x_{0},x]\in A$,
$\bar{p}[x_{0},x]=\bar{p}^{\prime}[x_{0},x]$.

###  II.4 Special Sets

Although the notation introduced in Secs II.2 and II.3 will be sufficient for
nearly all circumstances, two straightforward additions will prove useful in
the discussion of experiments.

First, it will be useful to isolate the subset of $X\rightarrow Y$ consisting
of the path-segments that don’t re-enter $X$ after their start; it will
similarly be useful to isolate the path-segments that don’t enter $Y$ until
their end.

######  Definition 14.

If $S$ is a dynamic set and $X,Y,Z\subset\mathcal{P}_{S}$:

$S_{X\rightarrow Y\upharpoonright
Z}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\in S_{X\rightarrow
Y}:for\>\lambda\in[\lambda_{1},\lambda_{2}),\>\bar{p}(\lambda)\notin Z\\}$

$S_{Z\upharpoonleft X\rightarrow
Y}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\in S_{X\rightarrow
Y}:for\>\lambda\in(\lambda,\lambda_{2}],\>\bar{p}(\lambda)\notin Z\\}$

$S_{X\rightarrow Y\upharpoonright Y}$ is then the set of path segments from
$S$ that start at $X$ and end at $Y$, but do not enter $Y$ before the end of
the segment. Similarly for $S_{X\upharpoonleft X\rightarrow Y}$.

$S_{X/Y}$, to be introduces momentarily, bears a resemblance to
$S_{X\rightarrow Y\upharpoonright Y}$, but overcomes a difficulty that occurs
in the continuum:

######  Definition 15.

$S_{X/Y}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}[\lambda_{1},\infty]\in
S_{X\rightarrow},\>Y\bigcap
Ran(\bar{p}[\lambda_{1},\infty])\neq\emptyset,\>and\>\lambda_{2}=glb(\bar{p}^{-1}[\lambda_{1},\infty][Y])\\}$

In the above definition, $\lambda_{2}$ is either the first time $Y$ occurs in
$\bar{p}[\lambda_{1},\infty]$ or, if the “first time” can’t be obtained in the
continuum, the moment before $Y$ first appears.

######  Definition 16.

If $S$ is a dynamic set and $X,Y\subset\mathcal{P}_{S}$,
$(X/Y)\equiv\\{p\in\mathcal{P}_{S}\,:\,for\>some\>\bar{p}[\lambda_{1},\lambda_{2}]\in
S_{X/Y},\>p=\bar{p}(\lambda_{2})\\}$

######  Theorem 17.

If $D$ is a homogeneous dynamic space then
$D_{X/Y}=X\rightarrow(X/Y)\upharpoonright Y$

###### Proof.

If $\bar{p}[\lambda_{1},\lambda_{2}]\in D_{X/Y}$ then $\bar{p}(\lambda_{1})\in
X$, $\bar{p}(\lambda_{2})\in(X/Y)$, and for all
$\lambda\in[\lambda_{1},\lambda_{2})$, $\bar{p}(\lambda)\notin Y$, so
$D_{X/Y}\subset X\rightarrow(X/Y)\upharpoonright Y$. (Note that this doesn’t
require $D$ to be homogeneous, or a dynamic space.)

It remains to show that $X\rightarrow(X/Y)\upharpoonright Y\subset D_{X/Y}$.
Take $\bar{p}[\lambda_{1},\lambda_{2}]\in X\rightarrow(X/Y)\upharpoonright Y$,
$p\equiv\bar{p}(\lambda_{2})\in(X/Y)$. By the definition of $(X/Y)$, there’s a
$\bar{p}^{\prime}[\lambda_{3},\infty]\in p\rightarrow$ s.t.
$\lambda_{3}=glb(\bar{p}^{\prime}[\lambda_{3},\infty]^{-1}[Y])$. By
homogeneity, there’s a $\bar{p}^{\prime\prime}$ s.t. for all
$\lambda\in\Lambda_{D}$,
$\bar{p}^{\prime\prime}(\lambda)=\bar{p}^{\prime}(\lambda+(\lambda_{3}-\lambda_{2}))$
(assuming $\lambda_{3}\geq\lambda_{2}$ or $\Lambda_{D}$ is unbounded from
below; otherwise
$\bar{p}^{\prime}(\lambda)=\bar{p}^{\prime\prime}(\lambda+(\lambda_{2}-\lambda_{3}))$
may be asserted for all $\lambda\in\Lambda_{D}$). By considering
$\bar{p}[\lambda_{1},\lambda_{2}]\circ\bar{p}^{\prime\prime}[\lambda_{2},\infty]$,
it follows that $\bar{p}[\lambda_{1},\lambda_{2}]\in D_{X/Y}$. ∎

###  II.5 Limits and Closed Sets

n this final section, system dynamics will be used to define limits. Four ways
in which a point $p\in\mathcal{P}_{S}$ can be a limit point of set
$X\subset\mathcal{P}_{S}$ at $\lambda$ will be defined. They correspond to: if
the system is in state $p$ at $\lambda$ then $X$ must be about to occur, if
the system is in state $p$ at $\lambda$ then $X$ might be about to occur, if
the system is in state $p$ at $\lambda$ then $X$ might have just happened, and
if the system is in state $p$ at $\lambda$ then $X$ must have just happened.
For formally:

######  Definition 18.

If $S$ is a dynamic set, $p\in\mathcal{P}_{S}$ , $\lambda\in\Lambda_{S},$ and
$X$ is a non-empty subset of $\mathcal{P}_{S}$ then

$p\in lim_{\lambda}^{\vartriangleright}X$ if for every
$\bar{p}[\lambda,\infty]\in p\rightarrow$, every $\lambda^{\prime}>\lambda$,
there’s a $\lambda^{\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$

$p\in lim_{\lambda}^{>}X$ if for every $\lambda^{\prime}>\lambda$, there’s a
$\bar{p}[\lambda,\infty]\in p\rightarrow$ and a
$\lambda^{\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$

$p\in lim_{\lambda}^{<}X$ if $\lambda>Min(\Lambda_{D})$ and for every
$\lambda^{\prime}<\lambda$, there’s $\bar{p}[-\infty,\lambda]\in\,\rightarrow
p$ and a $\lambda^{\prime\prime}\in(\lambda^{\prime},\lambda)$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$

$p\in lim_{\lambda}^{\vartriangleleft}X$ if $\lambda>Min(\Lambda_{D})$ and for
every $\bar{p}[-\infty,\lambda]\in\,\rightarrow p$ s.t. , every
$\lambda^{\prime}<\lambda$, there’s a
$\lambda^{\prime\prime}\in(\lambda^{\prime},\lambda)$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$

If $S$ is homogeneous, then the $\lambda$ subscript may be dropped, and we may
simply write $p\in lim^{\vartriangleright}X$, etc.

######  Definition 19.

If $S$ is a dynamic set, $X\subset\mathcal{P}_{S}$, and
$\lambda\in\Lambda_{S}$ then $X_{\lambda}^{\vartriangleright}\equiv X\bigcup
lim^{\vartriangleright}X$, $X_{\lambda}^{>}\equiv X\bigcup lim^{>}X$,
$X_{\lambda}^{<}\equiv X\bigcup lim^{<}X$, and
$X_{\lambda}^{\vartriangleleft}\equiv X\bigcup lim^{\vartriangleleft}X$.

Once again, if $S$ is homogeneous, then the $\lambda$ subscript may be
dropped, and one may simply write $X^{\vartriangleright}$, etc.

A set is closed if it contains all its limit points. The following theorem
lays out conditions under which $X^{\vartriangleright}$, etc., are closed.

######  Theorem 20.

1) If $S$ is a homogeneous dynamic set, and $X\subset\mathcal{P}_{S}$, then
$X^{\vartriangleright\vartriangleright}=X^{\vartriangleright}$ and
$X^{\vartriangleleft\vartriangleleft}=X^{\vartriangleleft}$.

1) If $D$ is a homogeneous dynamic space, and $X\subset\mathcal{P}_{D}$, then
$X^{>>}=X^{>}$ and $X^{<<}=X^{<}$.

###### Proof.

If $\Lambda_{D}$ is discrete, this holds because if $\Lambda_{D}$ is discrete
then for all $X\subset D$, $lim^{\square}X=\emptyset$.

Take $\Lambda_{D}$ to be continuous.

1) Assume $p\in lim^{\vartriangleright}X^{\vartriangleright}$ and take
$\bar{p}[\lambda,\infty]\in S_{p\rightarrow}$ & $\lambda^{\prime}>\lambda$.
There’s a
$\lambda^{\prime\prime}\in(\lambda,\lambda+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime})\in X^{\vartriangleright}$. Because
$\bar{p}(\lambda^{\prime\prime})\in X^{\vartriangleright}$, either
$\bar{p}(\lambda^{\prime\prime})\in X$ or $\bar{p}(\lambda^{\prime\prime})\in
lim^{\vartriangleright}X$, so there’s a
$\lambda^{\prime\prime\prime}\in[\lambda^{\prime\prime},\lambda^{\prime\prime}+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime\prime})\in X$. Therefore there’s a
$\lambda^{\prime\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime\prime})\in X$, and so $p\in
lim^{\vartriangleright}X$, and so $p\in X^{\vartriangleright}$.

$\vartriangleleft$ is similar

2) Assume $p\in lim^{>}X^{>}$. For any $\lambda$, $\lambda^{\prime}>\lambda$
there’s a $\bar{p}[\lambda,\infty]\in p\rightarrow$ and a
$\lambda^{\prime\prime}\in(\lambda,\lambda+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime})\in X^{>}$. Because
$\bar{p}(\lambda^{\prime\prime})\in X^{>}$, either
$\bar{p}(\lambda^{\prime\prime})\in X$ or $\bar{p}(\lambda^{\prime\prime})\in
lim^{>}X$, so there’s a $\bar{p}^{\prime}[\lambda^{\prime\prime},\infty]\in
p\rightarrow$,
$\lambda^{\prime\prime\prime}\in[\lambda^{\prime\prime},\lambda^{\prime\prime}+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}^{\prime}(\lambda^{\prime\prime\prime})\in X$. Take
$\bar{p}^{\prime\prime}[\lambda,\infty]=\bar{p}[\lambda,\lambda^{\prime\prime}]\circ\bar{p}^{\prime}[\lambda^{\prime\prime},\infty]$.
$\bar{p}^{\prime\prime}\in p\rightarrow$,
$\lambda^{\prime\prime\prime}\in(\lambda,\lambda^{\prime})$, and
$\bar{p}^{\prime\prime}(\lambda^{\prime\prime\prime})\in X$, so $p\in
lim^{\vartriangleright}X$, and so $p\in X^{\vartriangleright}$.

$<$ is similar. ∎

Under the conditions of Theorem 20, the collection of all closed sets of type
“$>$” are closed under intersections and finite unions. They therefore form a
topology on $\mathcal{P}_{D}$; the lower-limit topology. The closed sets of
type “$<$” also form a topology; the upper-limit topology.

The closed sets of type “$\vartriangleright$”, and those of type
“$\vartriangleleft$”, are closed under intersection, but not union. They
therefore do not form topologies. That is because these limits are fairly
strict, and so can not always determine whether or not some point $p$ is local
to some set $X$.

######  Theorem 21.

If $S$ is a homogeneous dynamic set, and $X,Y\subset\mathcal{P}_{S}$, then
$(X/Y)\subset Y^{>}$

###### Proof.

By the definition of $(X/Y)$, $p\in(X/Y)$ iff for some $\bar{p}\in S$,
$\lambda_{1},\lambda_{2}\in\Lambda_{D}$ ($\lambda_{1}\leq\lambda_{2}$),
$\bar{p}(\lambda_{2})=p$, $\bar{p}(\lambda_{1})\in X$,
$[\lambda_{1},\lambda_{2})\bigcap\bar{p}^{-1}[Y]=\emptyset$, and for all
$\lambda^{\prime}>\lambda_{2}$
$[\lambda_{2},\lambda^{\prime})\bigcap\bar{p}^{-1}[Y]\neq\emptyset$. In this
case, $\bar{p}[\lambda_{2},\infty]\in p\rightarrow$ and for all
$\lambda^{\prime}>\lambda_{2}$ there’s a
$\lambda^{\prime\prime}\in[\lambda_{2},\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in Y$, so either $p\in Y$ or $p\in lim^{>}Y$.
∎

##  III Experiments

###  III.1 Shells

In this part, experimentation on dynamic systems will be formalized. As
mentioned in the Introduction, this will be analogous to creating a formal
theory of computation via automata. In computation theory, sequences of
characters are read into an automata, and the automata determines whether they
are sentences in a given language. One of the goals of the theory is to
determine what languages automata are capable of deciding. Experiments are
similar; system paths are “read into” an experimental set-up, which then
determines which outcomes these paths belong to. Our goal will be to determine
what sets of outcomes experiments are capable of deciding.

In this first section, we will not divide the experiment into the system being
experimented on, and the environment containing the experimental apparatus;
for now, we consider the closed system that encompasses both. The dynamic
space of this closed system will be used to formalize some of the external
properties of experiments; namely, that experiments are re-runnable, they have
a clearly defined start, once started they must complete, and once complete
they “remember” that the experiment took place.

######  Definition 22.

A _shell_ is a triple, $(D,I,F)$, where $D$ is a dynamic space and
$I,F\subset\mathcal{P}_{D}$ are the sets of initial and final states. These
must satisfy:

1) $D$ is homogeneous

2) $I$ is homogeneously realized

3) $I\rightarrow\,=(I\upharpoonleft I\rightarrow F)\rightarrow$ (That is,
$I\rightarrow\,=D_{I\upharpoonleft I\rightarrow F}\rightarrow$)

4) $\rightarrow F=\,\rightarrow(I\upharpoonleft I\rightarrow F)$

In the above definition, $I$ represents the set of initial states, or start
states; when the system enters into one of these states, the experiment
starts. $F$ represents the set of final states; when the system enters into
one of these states, the experiment has ended.

Axioms (1) and (2) ensure that the experiment is reproducible. Axiom (3)
ensures that once the experiment begins, it must end. Axiom (4) ensures that
the system can only enter a final state via the experiment.

A note of explanation may be in order for this final axiom. $D$ is assumed to
be a closed system encompassing everything that has bearing on the experiment,
including the person performing the experiment. As a result, all recording
equipment is considered part of $D$, including the experimenter’s memory. So
if the final axiom were violated, when the final state is reached, you
wouldn’t be able to remember whether or not the experiment had taken place.

In (3) and (4), $I\upharpoonleft I\rightarrow F$ is used rather than
$I\rightarrow F$ in order to ensure that there’s a clearly defined space in
which the experiment takes place. For example, $\rightarrow F=\,\rightarrow
I\rightarrow F$ would allow for a path, $\bar{p}[-\infty,0]\in\,\rightarrow
F$, s.t. for all $n\in\mathbb{N}^{+}$, $\bar{p}(\frac{-1}{2n})\in I$ and
$\bar{p}(\frac{-1}{2n+1})\in F$; in this case, the $F$ state at $\lambda=0$
can not be paired with any particular experimental run. Similarly,
$I\rightarrow\,=I\rightarrow F\rightarrow$ would allow a path to cross $I$
several times before crossing $F$; in this case it would not be clear which
crossing of $I$ represented the start of the experiment.

An experiment is considered to be in progress while the space is in a path
segment that runs from $I$ to $F$. The set of states in those path segments
constitute the shell interior, or more formally:

######  Definition 23.

If $Z=(D,I,F)$ is a shell, $Int(Z)=\\{p\in\mathcal{P}_{D}:I\rightarrow
p\upharpoonright F\neq\emptyset\\}$ is the _shell interior_.

######  Theorem 24.

If $Z=(D,I,F)$ is a shell then
$Int(Z)=Ran(D_{I/F})=Ran(I\rightarrow(I/F)\upharpoonright F)$. (The
abbreviation “$Ran(A)$” stands for the set of states
$\bigcup_{\bar{p}[x_{1},x_{2}]\in A}Ran(\bar{p}[x_{1},x_{2}])$.)

###### Proof.

By Thm 17 and shell axiom 1, $D_{I/F}=I\rightarrow(I/F)\upharpoonright F$.

If $p\in Ran(I\rightarrow(I/F)\upharpoonright F)$ then for some
$\bar{p}[\lambda_{1},\lambda_{2}]\in I\rightarrow(I/F)\upharpoonright F$, some
$\lambda_{3}\in[\lambda_{1},\lambda_{2}]$, $\bar{p}(\lambda_{3})=p$, so
$\bar{p}[\lambda_{1},\lambda_{3}]\in I\rightarrow p\upharpoonright F$, and so
$p\in Int(Z)$.

If $p\in Int(Z)$ then there’s a $\bar{p}[\lambda_{1},\lambda_{2}]\in
I\rightarrow p\upharpoonright F$. Since $I\rightarrow\,=(I\upharpoonleft
I\rightarrow F)\rightarrow$ there must be a
$\bar{p}^{\prime}[\lambda_{1},\lambda_{3}]\in D_{I/F}$ s.t.
$\bar{p}^{\prime}[\lambda_{1},\lambda_{2}]=\bar{p}[\lambda_{1},\lambda_{2}]$,
so $p\in Ran(D_{I/F})$. ∎

The following items will prove quite useful:.

######  Definition 25.

If $Z=(D,I,F)$ is a shell, $Dom(Z)\equiv F\bigcup Int(Z)$ is the _shell
domain_

If $A\subset Dom(Z)$, $\omega_{A}\equiv\\{\bar{p}[0,\lambda]\in
I\upharpoonleft I\rightarrow A\\}=I\upharpoonleft(0,I)\rightarrow A$

If $A\subset F$, $\Theta_{A}\equiv\\{\bar{p}[0,\lambda]\in
D_{I/F}:for\>some\>\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{A},\>\bar{p}^{\prime}[0,\lambda]=\bar{p}[0,\lambda]\\}$

Given that our current state is in $A$, $\omega_{A}$ tells us what has
happened in the experiment; because shell dynamics are homogeneous, and $I$ is
homogeneously realized, it is sufficient only consider paths that start at
$\lambda_{0}=0$.

Given that the final state is in $A$, $\Theta_{A}$ captures what occurred
during the experiment.

The shell axioms validate these interpretations of $\omega_{A}$ and
$\Theta_{A}$.

###  III.2 E-Automata

Further structure will now be added to shells, starting with dividing the
shell domain into a system and its environment.

####  III.2.1 Environmental Shells

In the following definition: “$\bigotimes$” will refer to the Cartesian
product, and for n-ary relation, $R$, on $A_{1}\bigotimes...\bigotimes A_{n}$,
$\textrm{P}_{i}(R)$ refers the set of $a\in A_{i}$ s.t. for some $r\in R$,
$a=r_{i}$.

######  Definition 26.

An _environmental shell_ , $Z=(D,I,F)$, is a shell together with three sets,
$\mathcal{S}_{Z}$, $\mathcal{E}_{Int(Z)}$, and $\mathcal{E}_{F}$ satisfying

1)
$(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{Int(Z)})\bigcap\mathcal{P}_{D}=Int(Z)$

2) $(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{F})\bigcap\mathcal{P}_{D}=F$

3) $\mathcal{S}_{Z}=\textrm{P}_{1}(Dom(Z))$,
$\mathcal{E}_{Int(Z)}=\textrm{P}_{2}(Int(Z))$, and
$\mathcal{E}_{F}=\textrm{P}_{2}(F)$

$\mathcal{S}_{Z}$ is the _system_ and
$\mathcal{E}_{Z}\equiv\mathcal{E}_{Int(Z)}\bigcup\mathcal{E}_{F}$ is the
_environment._

Informally speaking, it’s assumed that the system ($\mathcal{S}_{Z}$) is being
observed, measured, recorded, etc., and that any observers, measuring devices,
recording equipment, etc., are in the environment. Therefore, it is components
in the environment which decide whether the experiment is complete; this is
why a subset of the environmental states, $\mathcal{E}_{F}$, determine whether
the shell is on $F$.

If $\bar{p}[\lambda_{1},\lambda_{2}]$ is a path segment from $I\upharpoonleft
I\rightarrow F$, $\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}$ is the
system component of the path segment, and
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{E}$ is the environmental
portion. More formally:

######  Definition 27.

For environmental shell $Z$:

If
$\bar{s}[\lambda_{1},\lambda_{2}]:[\lambda_{1},\lambda_{2}]\rightarrow\mathcal{S}_{Z}$,
$\bar{e}[\lambda_{1},\lambda_{2}]:[\lambda_{1},\lambda_{2}]\rightarrow\mathcal{E}_{Z}$,
and
$\bar{p}[\lambda_{1},\lambda_{2}]=(\bar{s}[\lambda_{1},\lambda_{2}],\bar{e}[\lambda_{1},\lambda_{2}])$
then
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}\equiv\bar{s}[\lambda_{1},\lambda_{2}]$
and
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{E}\equiv\bar{e}[\lambda_{1},\lambda_{2}]$.

If $A$ is any set of shell domain path segments,
$A\cdot\mathcal{S}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}:\bar{p}[\lambda_{1},\lambda_{2}]\in
A\\}$ and
$A\cdot\mathcal{E}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{\mathcal{E}}:\bar{p}[\lambda_{1},\lambda_{2}]\in
A\\}$.

This can be applied to $\omega_{A}$ and $\Theta_{A}$ to extract the system
information:

######  Definition 28.

If $Z$ is an environmental shell:

If $X\subset\mathcal{E}_{F}$,
$\mathcal{O}_{X}\equiv\Theta_{\mathcal{S}_{Z}\bigotimes X}\cdot\mathcal{S}$

If $A\subset Int(Z)$:

$\mathcal{\mbox{$\Sigma$}}_{A}\equiv\\{\bar{s}[0,\lambda_{1}]:for\>some\>\bar{p}[0,\lambda_{2}]\in\Theta_{F},\>\bar{p}(\lambda_{1})\in
A\>\&\>\bar{p}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}[0,\lambda_{1}]\\}$

$\Sigma_{A\rightarrow}\equiv\\{\bar{s}[\lambda_{1},\lambda_{2}]:for\>some\>\bar{p}[0,\lambda_{2}]\in\Theta_{F},\>\bar{p}(\lambda_{1})\in
A\>\&\>\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}=\bar{s}[\lambda_{1},\lambda_{2}]\\}$.

If an experiment completes with the environment in $X$, $\mathcal{O}_{X}$
tells us what happened in the system during the experimental run (the
“$\mathcal{O}$” stands for “outcome”). $\mathcal{O}_{\mathcal{E}_{F}}$ may be
abbreviated $\mathcal{O}_{F}$.

$\Sigma_{A}$ and $\Sigma_{A\rightarrow}$ give insight into what’s happening to
the system in the shell interior: $\Sigma_{A}$ gives the system paths from $I$
to $A$ and $\Sigma_{A\rightarrow}$ gives the system paths from $A$ to $(I/F)$.
For convenience, $\Sigma_{\sigma\bigotimes\mathcal{E}_{Int(Z)}}$ may be
written $\Sigma_{\sigma}$.

######  Theorem 29.

$\mathcal{O}_{F}=\Sigma_{(I/F)}$

###### Proof.

It’s clear that $\mathcal{O}_{F}\subset\Sigma_{(I/F)}$.

Assume $\bar{s}[0,\lambda]\in\Sigma_{(I/F)}$. For some
$\bar{p}[0,\lambda_{2}]\in\Theta_{F}$, $\bar{p}(\lambda_{1})\equiv p\in(I/F)$
and $\bar{p}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}[0,\lambda_{1}]$. Since
$p\in(I/F)$ and $D$ is homogeneous there must be a
$\bar{p}^{\prime}[\lambda_{1},\infty]\in p\rightarrow$ s.t. for all
$\lambda_{2}>\lambda_{1}$ there’s a
$\lambda^{\prime}\in[\lambda_{1},\lambda_{2})$ s.t.
$\bar{p}^{\prime}(\lambda^{\prime})\in F$. Therefore
$\bar{p}[0,\lambda_{1}]\circ\bar{p}^{\prime}[\lambda_{1},\infty]\in
I\rightarrow$ and $\bar{p}[0,\lambda_{1}]\in D_{I/F}$; this means that
$\bar{p}[0,\lambda_{1}]\in\Theta_{F}$, and so
$\bar{s}[0,\lambda_{1}]\in\mathcal{O}_{F}$. ∎

####  III.2.2 The Automata Condition

The automata condition demands that an experiment will terminate based solely
on what has transpired in the system. In particular, it specifies that if
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\omega_{F}$,
$\bar{p}_{1}[0,\lambda]\in\Theta_{F}$, and
$\bar{p}_{1}[0,\lambda]\cdot\mathcal{S}=\bar{p}_{2}[0,\lambda]\cdot\mathcal{S}$
then $\bar{p}_{2}[0,\lambda]\in\Theta_{F}$. Given shell axiom (3), this may be
rephrased as follows:

######  Definition 30.

An environmental shell satisfies _the automata condition_ if for every
$\bar{s}_{1}[0,\lambda_{1}],\,\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$
s.t. $\lambda_{2}>\lambda_{1}$,
$\bar{s}_{1}[0,\lambda_{1}]\neq\bar{s}_{2}[0,\lambda_{1}]$

The constraints this places on the closed system’s dynamics are summarized by
the following theorem.

######  Theorem 31.

The following assertions on environmental shell $Z$ are equivalent:

1) $Z$ satisfies the automata condition

2) For $p\in Int(Z)$, if $\Sigma_{p}\bigcap\mathcal{O}_{F}\neq\emptyset$ then
$p\in F^{\vartriangleright}$

3) $(I/F)\subset F^{\vartriangleright}$ and
$\Sigma_{Int(Z)-(I/F)}\bigcap\Sigma_{(I/F)}=\emptyset$

###### Proof.

$1\Rightarrow 2$: In order for the automata condition to hold, If $p\in
Int(Z)$ and $\Sigma_{p}\bigcap\mathcal{O}_{F}\neq\emptyset$ then either $p\in
F\bigcap(I/F)$, or all paths leaving $p$ must immediately enter $F$; either
way, $p\in F^{\vartriangleright}$.

$2\Rightarrow 3$: Since $\mathcal{O}_{F}=\Sigma_{(I/F)}$,
$\Sigma_{(I/F)}\bigcap\mathcal{O}_{F}\neq\emptyset$, and so $(I/F)\subset
F^{\vartriangleright}$.

If $p\in Int(Z)-(I/F)$ then the shell is not in $F$ and is not about to
transition into $F$, so $p\notin F^{\vartriangleright}$, and so
$\Sigma_{p}\bigcap\mathcal{O}_{F}=\emptyset$.

$3\Rightarrow 1$: Take
$\bar{p}_{1}[0,\lambda_{1}],\,\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ and
$\lambda_{2}\geq\lambda_{1}$. If
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{p}_{2}[0,\lambda_{1}]\cdot\mathcal{S}$
then
$\Sigma_{\bar{p}_{1}(\lambda_{1})}\bigcap\Sigma_{\bar{p}_{2}(\lambda_{1})}\neq\emptyset$
and so $\bar{p}_{2}(\lambda_{1})\in(I/F)$ (since
$\bar{p}_{1}(\lambda_{1})\in(I/F)$ and
$\Sigma_{Int(Z)-(I/F)}\bigcap\Sigma_{(I/F)}=\emptyset$). Since
$\bar{p}_{2}(\lambda_{1})\in(I/F)$ , $\bar{p}_{2}(\lambda_{1})\in
F^{\vartriangleright}$, and so $\bar{p}_{2}[0,\lambda_{1}]\in\Theta_{F}$, and
for all $\lambda_{3}>\lambda_{1}$
$\bar{p}_{2}[0,\lambda_{3}]\notin\Theta_{F}$. Therefore
$\lambda_{2}=\lambda_{1}$. So if
$\bar{p}_{1}[0,\lambda_{1}],\,\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ and
$\lambda_{2}>\lambda_{1}$ then
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}\neq\bar{p}_{2}[0,\lambda_{1}]\cdot\mathcal{S}$
∎

####  III.2.3 Unbiased Conditions

An environmental shell is unbiased if the environment does not influence the
outcome. In the strongest sense this demands that, while the environment may
record the system’s past, it has no effect on the system’s future. Naively,
this may be written: For every $(s,e_{1}),(s.e_{2})\in Int(Z)$,
$\Sigma_{(s,e_{1})\rightarrow}=\Sigma_{(s,e_{2})\rightarrow}$. However,
because in general $\Sigma_{(s,e_{1})}\neq\Sigma_{(s,e_{2})}$, the elements of
$\Sigma_{(s,e_{1})\rightarrow}$ and $\Sigma_{(s,e_{2})\rightarrow}$ may
terminate at different times; this leads to a definition whose wording is more
convoluted, but whose meaning is essentially the same.

######  Definition 32.

An environmental shell, $(D,I,F)$ is _strongly unbiased_ if for every
$\lambda\in Dom(\Theta_{F})$, every
$(s,e_{1}),(s.e_{2})\in\Theta_{F}(\lambda)$,
$\bar{s}_{1}[\lambda,\lambda_{1}]\in\Sigma_{(s,e_{1})\rightarrow}$ iff there
exists a $\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$
s.t., with $\lambda^{\prime}\equiv Min(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$.

This condition means that any effect the environment may have on the system
can be incorporated into the system dynamics, allowing the system to be
comprehensible without having to reference its environment. The following
theorem shows that if an environmental shell is strongly unbiased,
$\mathcal{O}_{F}$ behaves like a dynamic space.

######  Theorem 33.

If an environmental shell is strongly unbiased then for every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$,
s.t. for some $\lambda\in[0,Min(\lambda_{1},\lambda_{2})]$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)$, there exists an
$\bar{s}_{3}[0,\lambda_{3}]\in\mathcal{O}_{F}$ s.t., with
$\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$,
$\bar{s}_{3}[0,\lambda^{\prime}]=\bar{s}_{1}[0,\lambda]\circ\bar{s}_{2}[\lambda,\lambda^{\prime}]$

###### Proof.

First note that, regardless of whether the environmental shell is unbiased,
for any $p\in Int(Z)$, if $\bar{s}[0,\lambda]\in\Sigma_{p}$ and
$\bar{s}^{\prime}[\lambda,\lambda^{\prime}]\in\Sigma_{p\rightarrow}$ then
$\bar{s}[0,\lambda]\circ\bar{s}^{\prime}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}$.

Taking $s=\bar{s}_{1}(\lambda)$, for some
$e_{1},e_{2}\in\mathcal{E}_{Int(Z)}$,
$\bar{s}_{1}[0,\lambda]\in\Sigma_{(s,e_{1})}$ and
$\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$. Since the
shell is unbiased, there is a
$\bar{s}_{3}[\lambda,\lambda_{3}]\in\Sigma_{(s,e_{1})\rightarrow}$ s.t.
$\bar{s}_{3}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$
($\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$). By the considerations
of the prior paragraph,
$\bar{s}_{1}[0,\lambda]\circ\bar{s}_{3}[\lambda,\lambda_{3}]\in\mathcal{O}_{F}$.
∎

There are experiments for which the strong unbiased condition can fail, but
the experiment still be accepted as valid. Taking an example from quantum
mechanics, consider the case in which a particle’s position is measured
$\lambda_{1}$, and if the particle is in region $A$, the particle’s spin will
be measured along the y-axis at $\lambda_{2}$, and if particle is not in
region $A$, the spin will be measured along the z-axis at $\lambda_{2}$. Now
consider two paths, $\bar{s}_{1}$ and $\bar{s}_{2}$, s.t. $\bar{s}_{1}$ is in
region $A$ at $\lambda_{1}$, $\bar{s}_{2}$ is not in region $A$ at
$\lambda_{1}$, and for some $\lambda\in(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)\equiv s$.
$\bar{s}_{1}[-\infty,\lambda]\circ\bar{s}_{2}[\lambda,\infty]$ is not a
possible path because a particle can’t be in region $A$ at $\lambda_{1}$ and
have its spin polarized along the z-axis at $\lambda_{2}$. Therefore the set
of particle paths is not a dynamic space (though it is still a dynamic set).
Since the strong unbiased condition insures that the system can be described
by a dynamic space, the unbiased condition must have failed.

To see this, assume that the e-automata is in state $(s,e_{1})$ at $\lambda$
in the case where the system takes path $\bar{s}_{1}$, and state $(s,e_{2})$
in the case where the system takes path $\bar{s}_{2}$. $e_{1}$ and $e_{2}$
determine different futures for the particle paths, $e_{1}$ insures that the
spin will be measured along the y-axis at $\lambda_{2}$ while $e_{2}$ insures
that the spin will be measured along the z-axis at $\lambda_{2}$. This
violates the strong unbiased condition. However, it doesn’t necessarily create
a problem for the experiment because $e_{1}$ and $e_{2}$ know enough about the
system history to know $s_{1}[-\infty,\lambda]$ and $s_{2}[-\infty,\lambda]$
must reside in separate outcomes (one belongs to the set of “$A$” outcomes,
the other to the set of “not $A$” outcomes); once paths have been
differentiated into separate outcomes, they may be treated differently by the
environment. This motivates a weak version of the unbiased condition, one that
holds only when $(s,e_{1})$ and $(s,e_{2})$ have not yet been differentiated
into separate outcomes.

For an environmental shell, $Z=(D,I,F)$, $p,p^{\prime}\in Int(Z)$ have not
been differentiated if for some $e\in\mathcal{E}_{F}$, $\lambda\in\Lambda$,
$\Sigma_{p}\bigcap\mathcal{O}_{e}[0,\lambda]\neq\emptyset$ and
$\Sigma_{p^{\prime}}\bigcap\mathcal{O}_{e}[0,\lambda]\neq\emptyset$. If
$(s,e)$ and $(s,e^{\prime})$ have not been differentiated then the unbiased
condition should hold for them. Similarly, if $(s,e_{1})$ and $(s,e_{2})$ have
not been differentiated, and neither have $(s,e_{2})$ and $(s,e_{3})$, then
the unbiased condition should hold for $(s,e_{1})$ and $(s,e_{3})$. This
motivates the following definition

######  Definition 34.

For e-automata $Z$, $e\in\mathcal{E}_{F}$, $\lambda\geq 0$

$|\mathcal{O}_{e}[0,\lambda]|_{0}\equiv\mathcal{O}_{e}[0,\lambda]$

$|\mathcal{O}_{e}[0,\lambda]|_{n+1}\equiv\\{\bar{s}[0,\lambda]:for\>some\>e^{\prime}\in\mathcal{E}_{F},\>\mathcal{O}_{e^{\prime}}[0,\lambda]\bigcap|\mathcal{O}_{e}[0,\lambda]|_{n}\neq\emptyset,\>\&\>\bar{s}[0,\lambda]\in\mathcal{O}_{e^{\prime}}[0,\lambda]\\}$

$|\mathcal{O}_{e}[0,\lambda]|\equiv\bigcup_{n\in\mathbb{N}}|\mathcal{O}_{e}[0,\lambda]|_{n}$

Note that if $\bar{s}[0,\lambda_{0}]\in\mathcal{O}_{e}$ then
$\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$ iff
$\lambda_{0}\geq\lambda$.

######  Definition 35.

For environmental shell, $Z=(D,I,F)$, $p,p^{\prime}\in Int(Z)$ are
_undifferentiated_ if for some $e\in\mathcal{E}_{F}$, $\lambda\in\Lambda$,
$\Sigma_{p}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$ and
$\Sigma_{p^{\prime}}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$

Now define the weak unbiased condition in the same way as the strong
condition, except that it only applies to undifferentiated states.

######  Definition 36.

An environmental shell, $(D,I,F)$ is _weakly unbiased_ if for every
$\lambda\in Dom(\Theta_{F})$, every
$(s,e_{1}),(s.e_{2})\in\Theta_{F}(\lambda)$ s.t. $(s,e_{1})$ and $(s.e_{2})$
are undifferentiated,
$\bar{s}_{1}[\lambda,\lambda_{1}]\in\Sigma_{(s,e_{1})\rightarrow}$ iff there
exists a $\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$
s.t. with $\lambda^{\prime}\equiv Min(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$.

######  Theorem 37.

If an environmental shell is weakly unbiased then for every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$,
s.t. for some $\lambda\in[0,Min(\lambda_{1},\lambda_{2})]$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)=s$ and for some
$e\in\mathcal{E}_{F}$,
$\bar{s}_{1}[0,\lambda],\bar{s}_{2}[0,\lambda]\in|\mathcal{O}_{e}[0,\lambda]|$,
there exists an $\bar{s}_{3}[0,\lambda_{3}]\in\mathcal{O}_{F}$ s.t., with
$\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$,
$\bar{s}_{3}[0,\lambda^{\prime}]=\bar{s}_{1}[0,\lambda]\circ\bar{s}_{2}[\lambda,\lambda^{\prime}]$

###### Proof.

Choose $e_{1},e_{2}\in\mathcal{E}_{Int(Z)}$ s.t.
$\bar{s}_{1}[0,\lambda]\in\Sigma_{(s,e_{1})}$,
$\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$,
$\Sigma_{(s,e_{1})}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$, and
$\Sigma_{(s,e_{2})}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$. Such
$e_{1}$and $e_{2}$ must exist because
$\bar{s}_{1}[0,\lambda],\bar{s}_{2}[0,\lambda]\in|\mathcal{O}_{e}[0,\lambda]|$.
$(s,e_{1})$ and $(s,e_{2})$ are undifferentiated, and so the proof now
proceeds similarly to that for Thm 33. ∎

####  III.2.4 E-Automata

######  Definition 38.

An environmental shell is an _e-automata_ if it is weakly unbiased and
satisfies the automata condition.

######  Theorem 39.

If $Z$ is an e-automata, $p_{1},p_{2}\in Int(Z)$, and
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}\neq\emptyset$ then
$\Sigma_{p_{1}\rightarrow}=\Sigma_{p_{2}\rightarrow}$

###### Proof.

Since $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}\neq\emptyset$, $p$ and $p^{\prime}$
are undifferentiated, so the result follows from $Z$ being weakly unbiased &
satisfying the automata condition ∎

As mentioned earlier, an important goal will be to determine what sets of
outcomes can be decided by various types of e-automata. Towards that end, the
meaning of an e-automata “deciding a set of outcomes” will now be defined. For
the moment we’ll concentrate measurements on dynamic spaces; the more general
case will be taken up in a later section.

######  Definition 40.

If $X$ is a set, $C$ is a _covering_ of $X$ if it is a set of subsets of $X$
and $\bigcup C=X$. (Note: it is more common to demand that $X\subset\bigcup
C$; the restriction to $\bigcup C=X$ will allow for some mild simplification.)

$C$ is a _partition_ of $X$ if it is a pairwise disjoint covering

…

######  Definition 41.

If $Z=(D,I,F)$ is an e-automata, $\lambda\in\Lambda_{D}$, $A\subset F$, and
$X\subset\mathcal{E}_{F}$:

$\Theta_{A}^{\lambda}\equiv\\{\bar{p}[\lambda,\lambda^{\prime}]\in
D_{I/F}:for\>some\>\bar{p}^{\prime}[\lambda,\lambda^{\prime\prime}]\in
I\upharpoonleft I\rightarrow
A,\>\bar{p}^{\prime}[\lambda,\lambda^{\prime}]=\bar{p}[\lambda,\lambda^{\prime}]\\}$

$\mathcal{O}_{X}^{\lambda}\equiv\Theta_{\mathcal{S}_{Z}\bigotimes
X}^{\lambda}\cdot\mathcal{S}$

Because $D$ is homogeneous and $I$ is homogeneously realized,
$\Theta_{A}^{\lambda}$ and $\mathcal{O}_{X}^{\lambda}$ are simply $\Theta_{A}$
and $\mathcal{O}_{X}$ shifted by $\lambda$, and theorems for $\Theta_{A}$ and
$\mathcal{O}_{X}$ can be readily translated to theorems for
$\Theta_{A}^{\lambda}$ and $\mathcal{O}_{X}^{\lambda}$ .

When an e-automata decides a set of outcomes on a dynamic space, it is fairly
straightforward to define the relationship between the dynamic space and the
e-automata’s outcomes.

######  Definition 42.

A covering, $K$, of dynamic space $D$ is _decided_ by e-automata
$Z=(D_{Z},I,F)$ if $\Lambda_{D}=\Lambda_{D_{Z}}$ and, for some
$\lambda\in\Lambda_{D}$,
$K\equiv\\{\rightarrow\mathcal{O}_{e}^{\lambda}\rightarrow:e\in\mathcal{E}_{F}\\}$.

Note that $Z$ can not decide $K$ if starts after some $\lambda$. This may seem
to violate experimental reproducibility. In a later section it will be seen
that any such a measurement can be consistent with experimental
reproducibility. It’s not difficult to see how. If we were interested, for
example, in the likelihood that a system is in state $y$ at time $1\>sec$
given that is was in state $x$ at time $0$, we do not mean that we are only
interested in those probabilities at a particular moment in the history of the
universe; it is assumed that the system can be assembled at any time, and the
time at which it is assembled is simply assigned the value of $0$.

If $D$ is a dynamic space, and $A\subset D$, an e-automata can determine
whether or not $A$ occurs if it decides a covering on $D$, $K$, s.t. and for
all $\alpha\in K$, either $\alpha\subset A$ or $\alpha\bigcap A\,=\emptyset$.
While this concept of measurement is adequate for many purposes, it’s a little
too broad for quantum physics; in the next section a subclass of e-automata
will be introduced that will eliminate the problematic measurements.

###  III.3 Ideal E-Automata

As they stand, e-automata draw no clear distinction between two different
types of uncertainty: extrinsic uncertainty - uncertainty about the state of
the environment, particularly with regard to knowing precisely which
$e\in\mathcal{E}_{F}$ occurred, and intrinsic uncertainty - uncertainty about
the system given complete knowledge of the environment.

It is often necessary to treat with these two type of uncertainty separately.
Such is the case in quantum physics: if $\bar{s}_{1}$ and $\bar{s}_{2}$ are
system paths, there’s a difference between “either $\bar{s}_{1}$ or
$\bar{s}_{2}$ was measured, but we don’t know which” (extrinsic uncertainty)
and “$\\{\bar{s}_{1},\bar{s}_{2}\\}$ was measured” (intrinsic uncertainty),
because in general $P(A\bigcup B)\neq P(A)+P(B)$.

In this section two properties will be introduced that together will remove
the ambiguity between these types of uncertainty. These are the only
refinement to e-automata that will be required.

####  III.3.1 Boolean E-Automata

######  Definition 43.

An e-automata is _boolean_ if for every $e,e^{\prime}\in\mathcal{E}_{F}$
either $\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}$ or
$\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}=\emptyset$.

######  Theorem 44.

If $K$ is decided by a boolean e-automata then it is a partition

###### Proof.

Follows from the boolean & automata conditions. ∎

If an e-automata is not boolean, it’s possible to have
$e,e^{\prime}\in\mathcal{E}_{F}$ s.t.
$\mathcal{O}_{e}\neq\mathcal{O}_{e^{\prime}}$ and
$\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}\neq\emptyset$. If it’s known
that $e$ occurred, and so $e^{\prime}$ did not occur, it’s not clear whether
the elements of $\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}$ could have
occurred. Boolean e-automata eliminate this sort of ambiguity; if $e$ occurred
and $\mathcal{O}_{e}\neq\mathcal{O}_{e^{\prime}}$ then none of the paths in
$\mathcal{O}_{e^{\prime}}$ could have been taken. This means that for boolean
e-automata, one can always separate uncertainty about which path(s)
corresponding to $\mathcal{O}_{e}$ occurred from uncertainty about which
$e\in\mathcal{E}_{F}$ occurred.

######  Definition 45.

For a boolean e-automata, $e\in\mathcal{E}_{F}$,
$[e]\equiv\\{e^{\prime}\in\mathcal{E}_{F}:\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}\\}$

####  III.3.2 All-Reet E-Automata

It is possible for the environment to record a piece of information about the
system at one point, and then later forget it, so that it is not ultimately
reflected in the experimental outcome. This leads ambiguity of the type
discussed above; should the outcome be understood to be as $A\bigcup B$ or as
$A\>or\>B$ (or as something distinct from either)?

An all-reet e-automata is all retaining, once it records a bit of information
about the system, it never forgets it. In order to define this type of
e-automate, it will be helpful to first define the subsets of
$\omega_{Dom(Z)}$, $\Sigma_{Int(Z)}$, $\Theta_{F}$, and $\mathcal{O}_{F}$
containing just those path-segments that are consistent with a given
environmental path-segment.

######  Definition 46.

For $\bar{e}:\Lambda\rightarrow\mathcal{E}_{Z}$,

$\omega_{\bar{e}[0,\lambda]}\equiv\\{\bar{p}[0,\lambda]\in\omega_{Dom(Z)}\,:\,\bar{p}[0,\lambda]\cdot\mathcal{E}=\bar{e}[0,\lambda]\\}$

$\Sigma_{\bar{e}[0,\lambda]}\equiv\\{\bar{s}[0,\lambda]:(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\in\omega_{Int(Z)}\\}$

If $\bar{e}(\lambda)\in\mathcal{E}_{F}$:

$\Theta_{\bar{e}[0,\lambda]}\equiv\\{\bar{p}[0,\lambda^{\prime}]\in\Theta_{F}:for\>some\>\bar{p}^{\prime}[0,\lambda]\in\omega_{\bar{e}[0,\lambda]},\>\bar{p}^{\prime}[0,\lambda^{\prime}]=\bar{p}[0,\lambda^{\prime}]\\}$

$\mathcal{O}_{\bar{e}[0,\lambda]}\equiv\Theta_{\bar{e}[0,\lambda]}\cdot\mathcal{S}$

######  Theorem 47.

If $Z$ is an e-automata and $e\in\mathcal{E}_{F}$ then
$\mathcal{O}_{e}=\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$

###### Proof.

Since $F=(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{F})\bigcap\mathcal{P}_{D}$,
for any
$(\bar{s}_{1}[0,\lambda],\bar{e}[0,\lambda]),(\bar{s}_{2}[0,\lambda],\bar{e}[0,\lambda])\in\omega_{F}$,
any $\lambda^{\prime}\leq\lambda$,
$\bar{s}_{1}[0,\lambda^{\prime}]\in\mathcal{O}_{\bar{e}(\lambda)}$ iff
$\bar{s}_{2}[0,\lambda^{\prime}]\in\mathcal{O}_{\bar{e}(\lambda)}$; therefore
$\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}\subset\mathcal{O}_{e}$.

If $\bar{s}[0,\lambda^{\prime}]\in\mathcal{O}_{e}$ then there must exist a
$\bar{p}[0,\lambda]\in\omega_{e}$ s.t.consistent
$\bar{s}[0,\lambda^{\prime}]=\bar{p}[0,\lambda^{\prime}]\cdot\mathcal{S}$, in
which case
$\bar{s}[0,\lambda^{\prime}]=\mathcal{O}_{\bar{p}[0,\lambda^{\prime}]\cdot\mathcal{E}}$.
Therefore
$\mathcal{O}_{e}\subset\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$.
∎

Let’s assume that as an experiment unfolds, the environment sequentially
writes everything it discovers about the system to incorruptible memory. The
state of the environment would include the state of this memory, so for every
$\bar{e}_{1}[0,\lambda_{1}],\bar{e}_{2}[0,\lambda_{2}]\in\omega_{F}\cdot\mathcal{E}$,
if
$\mathcal{O}_{\bar{e}_{1}[0,\lambda_{1}]}\neq\mathcal{O}_{\bar{e}_{2}[0,\lambda_{2}]}$
then $\bar{e}_{1}(\lambda_{1})\neq\bar{e}_{2}(\lambda_{2})$. Since
$\mathcal{O}_{e}=\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$,
we are lead to the following definition.

######  Definition 48.

For e-automata $Z=(D,I,F)$, $e\in\mathcal{E}_{F}$ is _reet_ if for all
$\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$,
$\mathcal{O}_{\bar{e}[0,\lambda]}=\mathcal{O}_{e}$

$Z$ is _all-reet_ if every $e\in\mathcal{E}_{F}$ is reet.

######  Theorem 49.

If $Z$ is an all-reet e-automata, $e\in\mathcal{E}_{F}$,
$\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$, and
$\bar{e}[0,\lambda](\lambda^{\prime})=e^{\prime}\in\mathcal{E}_{F}$ then
$\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}$

###### Proof.

Take $\lambda_{1}=glb(\bar{e}^{-1}[0,\lambda][\mathcal{E}_{F}])$; since
$\bar{e}[0,\lambda](\lambda^{\prime})\in\mathcal{E}_{F}$,
$\lambda_{1}\leq\lambda^{\prime}$.
$\Theta_{\bar{e}[0,\lambda]}=\Theta_{\bar{e}[0,\lambda^{\prime}]}=\omega_{\bar{e}[0,\lambda_{1}]}$,
and so
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}=\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\mathcal{O}_{e^{\prime}}$.∎

######  Theorem 50.

If $Z$ is an all-reet e-automata then for every $e\in\mathcal{E}_{F}$, every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{e}$,
$\lambda_{1}=\lambda_{2}$

###### Proof.

Take any $\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$;
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}$. Assume that
$\lambda_{1}<\lambda_{2}$; since $(I/F)\subset F^{\vartriangleright}$ there’s
a $\lambda^{\prime}\in[\lambda_{1},\lambda_{2})$ s.t.
$\bar{e}(\lambda^{\prime})\in\mathcal{E}_{F}$. However, in that case
$\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{e}$ and
$\bar{s}_{2}[0,\lambda_{2}]\notin\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\mathcal{O}_{\bar{e}(\lambda^{\prime})}$,
contrary to Thm 49.∎

######  Definition 51.

If $Z$ is an all-reet e-automata, $e\in\mathcal{E}_{F}$ and
$\bar{s}[0,\lambda]\in\mathcal{O}_{e}$ then
$\Lambda(\mathcal{O}_{e})\equiv\lambda$.

Thm 50 implies a similar statement for all of $Int(Z)$.

######  Theorem 52.

If $Z=(D,I,F)$ is an all-reet e-automata then for every $p\in Int(Z)$, every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\Sigma_{p}$,
$\lambda_{1}=\lambda_{2}$

###### Proof.

Assume $\lambda_{1}\leq\lambda_{2}$. Take any
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\omega_{p}$ and any
$\bar{p}[\lambda_{1},\lambda_{3}]\in I\upharpoonleft p\rightarrow F$. Because
$D$ is homogeneous there must be a
$\bar{p}^{\prime}[\lambda_{2},\lambda_{3}+\lambda_{2}-\lambda_{1}]\in
I\upharpoonleft p\rightarrow F$ s.t. for all
$\lambda\in[\lambda_{1},\lambda_{3}]$,
$\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda+\lambda_{2}-\lambda_{1})$. Define
$\bar{p}_{3}[0,\lambda_{3}]\equiv\bar{p}_{1}[0,\lambda_{1}]\circ\bar{p}[\lambda_{1},\lambda_{3}]$
and
$\bar{p}_{4}[0,\lambda_{3}+\lambda_{2}-\lambda_{1}]\equiv\bar{p}_{2}[0,\lambda_{2}]\circ\bar{p}^{\prime}[\lambda_{2},\lambda_{3}+\lambda_{2}-\lambda_{1}]$.
If $\bar{p}_{3}[0,\lambda_{x}]\in\Theta_{F}$ then
$\bar{p}_{4}[0,\lambda_{x}+\lambda_{2}-\lambda_{1}]\in\Theta_{F}$, so by Thm
50, $\lambda_{1}=\lambda_{2}$.consistent ∎

The next two theorems will prove to be quite useful.

######  Theorem 53.

If $Z$ is an all-reet e-automata then for any $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$\mathcal{O}_{e}=\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$.

###### Proof.

Clearly
$\mathcal{O}_{e}\subset\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$.

Take any
$\bar{s}[0,\Lambda(\mathcal{O}_{e})],\bar{s}^{\prime}[0,\Lambda(\mathcal{O}_{e})]\in\mathcal{O}_{e}$
s.t. $\bar{s}(\lambda)=\bar{s}^{\prime}(\lambda)$. For
$\bar{e}[0,\lambda^{\prime}]\in\omega_{e}\cdot\mathcal{E}$,
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}\cdot\mathcal{S}$.
$(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\circ(\bar{s}_{1}[\lambda,\Lambda(\mathcal{O}_{e})],\bar{e}[\lambda,\Lambda(\mathcal{O}_{e})])\in\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}$,
so since
$\mathcal{O}_{e}=\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}\cdot\mathcal{S}$,
$\bar{s}[0,\lambda]\circ\bar{s}[\lambda,\Lambda(\mathcal{O}_{e})]\in\mathcal{O}_{e}$.∎

######  Theorem 54.

If $Z$ is an all-reet e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{e}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$

###### Proof.

First note that there exists a
$\bar{e}[0,\lambda^{\prime}]\in\omega_{e}\cdot\mathcal{E}$ s.t.
$\bar{e}(\lambda)=e_{\lambda}$.

A:
$\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}\subset\mathcal{O}_{e}[0,\lambda]$

\- Take any $s_{\lambda}\in\mathcal{O}_{e}(\lambda)$. Since $Z$ is all-reet,
$s_{\lambda}\in\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}(\lambda)$, so there
must be a
$\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{\bar{e}[0,\lambda^{\prime}]}$
s.t. $\bar{p}^{\prime}(\lambda)=(s_{\lambda},e_{\lambda})$. Since
$\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{\bar{e}[0,\lambda^{\prime}]}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in
I\upharpoonleft(s_{\lambda},e_{\lambda})\rightarrow e$. Now take any
$\bar{s}[0,\lambda]\in\Sigma_{(s_{\lambda},e_{\lambda})}$; there must exist a
$(\bar{s}[0,\lambda],\bar{e}^{\prime}[0,\lambda])\in\omega_{(s_{\lambda},e_{\lambda})}$.
$(\bar{s}[0,\lambda],\bar{e}^{\prime}[0,\lambda])\circ\bar{p}[\lambda,\Lambda(\mathcal{O}_{e})]\in\Theta_{e}$
so $\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$. -

B:
$\mathcal{O}_{e}[0,\lambda]\subset\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$

\- If $\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$ then
$\bar{s}[0,\lambda]\in\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}[0,\lambda]$,
so with $s=\bar{s}(\lambda)\in\mathcal{O}_{e}(\lambda)$,
$\bar{s}[0,\lambda]\in\Sigma_{(s,e_{\lambda})}$. \- ∎

The following are some of the consequences of Thm 54.

######  Theorem 55.

If $Z$ is an all-reet e-automata, $p,p_{1},p_{2}\in Int(Z)$,
$e\in\mathcal{E}_{F}$, and $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$

1) If $(s_{1},e_{1}),(s_{2},e_{2})\in\Theta_{e}(\lambda)$ then
$(s_{1},e_{2})\in\Theta_{e}(\lambda)$

2) If $(s,e_{1}),(s,e_{2})\in\Theta_{e}(\lambda)$ then
$\Sigma_{(s,e_{1})}=\Sigma_{(s,e_{2})}$

3) If $Z$ is boolean and all-reet and
$(s,e_{1}),(s,e_{2})\in\Theta_{[e]}(\lambda)$ then
$\Sigma_{(s,e_{1})}=\Sigma_{(s,e_{2})}$

###### Proof.

1 and 2 are immediate from Thm 54. 3 follows from Thm 54 and the definition of
$[e]$. ∎

####  III.3.3 Ideal E-Automata

######  Definition 56.

An e-automata is _ideal_ if it is boolean and all-reet.

Ideal e-automata lack the ambiguities mentioned at the beginning of the
section. The remainder of this section will seek to establish a central
property ideal e-automata, to be given in Thm 59.

######  Theorem 57.

For ideal e-automata $Z$, for any $p_{1},p_{2}\in Int(Z)$

1) If $p_{1}\cdot\mathcal{S}=p_{2}\cdot\mathcal{S}$, and for some
$e\in\mathcal{E}_{F}$, $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$p_{1},p_{2}\in\Theta_{[e]}(\lambda)$ then $\Sigma_{p_{1}}=\Sigma_{p_{2}}$, in
all other cases $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$.

2) If $\Sigma_{p_{1}}=\Sigma_{p_{2}}$ then for all $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, $p_{1}\in\Theta_{[e]}(\lambda)$ iff
$p_{2}\in\Theta_{[e]}(\lambda)$

###### Proof.

1) If the conditions hold then $\Sigma_{p_{1}}=\Sigma_{p_{2}}$by Thm 55.3.

If $p_{1}\cdot\mathcal{S}\neq p_{2}\cdot\mathcal{S}$ then clearly
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$.

If there’s doesn’t exist a $e\in\mathcal{E}_{F}$ s.t. $p_{1},p_{2}\in
Ran(\Theta_{[e]})$ then, since $Z$ is boolean,
$\Sigma_{p_{1}}\circ\Sigma_{p_{1}\rightarrow}\bigcap\Sigma_{p_{2}}\circ\Sigma_{p_{2}\rightarrow}=\emptyset$.
By Thm 39, $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$

If there exits an $e\in\mathcal{E}_{F}$ s.t. $p_{1},p_{2}\in
Ran(\Theta_{[e]})$, but no $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$ s.t.
$p_{1},p_{2}\in\Theta_{[e]}(\lambda)$ then
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$ by Thm 52.

2) By Thm 39
$\Sigma_{p_{1}}\circ\Sigma_{p_{1}\rightarrow}=\Sigma_{p_{2}}\circ\Sigma_{p_{2}\rightarrow}$;
since $Z$ is boolean $p_{1}\in Ran(\Theta_{[e]})$ iff $p_{2}\in
Ran(\Theta_{[e]})$. From Thm 52 it then follows that
$p_{1}\in\Theta_{[e]}(\lambda)$ iff $p_{2}\in\Theta_{[e]}(\lambda)$. ∎

For ideal e-automata, the sets $|\mathcal{O}_{e}[0,\lambda]|$ (see Section
III.2.3) are particularly useful; they tell you what has been measured as of
$\lambda$.

######  Theorem 58.

If $Z$ is an ideal e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{[e]}\cdot\mathcal{E})(\lambda)$,
$|\mathcal{O}_{e}[0,\lambda]|=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|(\lambda)}\Sigma_{(s,e_{\lambda})}$

###### Proof.

A: If $Z$ is an ideal e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{[e]}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$

\- Immediate from Thm 54 and Thm 55.3 -

It is sufficient to show that for every $n\in\mathbb{N},$
$|\mathcal{O}_{e}[0,\lambda]|_{n}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{n}(\lambda)}\Sigma_{(s,e_{\lambda})}$.
By (A) this holds for $n=0$. Assume it holds for $n=i$ and consider $n=i+1$.

For any $\mathcal{O}_{e^{\prime}}$ s.t.
$\mathcal{O}_{e^{\prime}}[0,\lambda]\bigcap|\mathcal{O}_{e}[0,\lambda]|_{i}\neq\emptyset$
for every $e_{2}\in(\Theta_{[e^{\prime}]}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e^{\prime}}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e^{\prime}}(\lambda)}\Sigma_{(s,e_{2})}$.
By assumption
$|\mathcal{O}_{e}[0,\lambda]|_{i}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{i}(\lambda)}\Sigma_{(s,e_{\lambda})}$,
so by Thm 57.1 there’s an
$s\in|\mathcal{O}_{e}[0,\lambda]|_{i}(\lambda)\bigcap\mathcal{O}_{e^{\prime}}(\lambda)$
s.t. $\Sigma_{(s,e_{2})}=\Sigma_{(s,e_{\lambda})}$; by Thm 57.2
$(s,e_{\lambda})\in\Theta_{[e^{\prime}]}(\lambda)$, so by (A)
$\mathcal{O}_{e^{\prime}}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e^{\prime}}(\lambda)}\Sigma_{(s,e_{\lambda})}$
. Since $|\mathcal{O}_{e}[0,\lambda]|_{i+1}$is the union over all such
$\mathcal{O}_{e^{\prime}}[0,\lambda]$ it follows immediately that
$|\mathcal{O}_{e}[0,\lambda]|_{i+1}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{i+1}(\lambda)}\Sigma_{(s,e_{\lambda})}$.∎

######  Theorem 59.

If $Z$ is an ideal e-automata then for any $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$\mathcal{O}_{e}=|\mathcal{O}_{e}[0,\lambda]|\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$

###### Proof.

With $e_{\lambda}\in(\Theta_{e}\cdot\mathcal{E})(\lambda)$:

$\mathcal{O}_{e}=\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 53)

$=(\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})})\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 54)

$=(\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|(\lambda)}\Sigma_{(s,e_{\lambda})})\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$

$=|\mathcal{O}_{e}[0,\lambda]|\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 58) ∎

###  III.4 Ideal Partitions

This section will be concerned with the necessary and sufficient conditions
for a partition of a dynamic set to be the set of outcomes of an ideal
e-automata. As a warm up, we’ll start by picking up where we left off in
Section III.2.4 and consider the case where the system is a dynamic space; the
more general case, where the system is a dynamic set, will be handled
immediately thereafter. In order to attack these problems, several basic
definitions first have to be given.

######  Definition 60.

If $S$ is a dynamic set, $\alpha\subset S$ is _bounded from above (by_
$\lambda$) if $\alpha=\alpha[-\infty,\lambda]\circ S[\lambda,\infty]$,
$\alpha$ is _bounded from below (by_ $\lambda$) if
$\alpha=S[-\infty,\lambda]\circ\alpha[\lambda,\infty]$, and $\alpha$ is
_bounded_ if it is both bounded from above and bounded from below.

If $A$ is a set of subsets of $S$, $A$ is _bounded from above/below (by_
$\lambda$) every element of $A$ is bounded from above/below by $\lambda$; it’s
_bounded_ if it’s bounded from both above and below.

“Bounded from below by $\lambda$” may be abbreviated “$bb\lambda$” and
“Bounded from above by $\lambda$” may be abbreviated “$ba\lambda$”.

Next to transfer the concept of $|\mathcal{O}_{e}[0,\lambda]|$ to partitions.

######  Definition 61.

If $K$ is a covering of dynamic set $S$, $\alpha\in\gamma$,
$\lambda\in\Lambda_{D}$, and $x\leq\lambda$ is either $-\infty$ or an element
of $\Lambda_{S}$

$|\alpha[x,\lambda]|_{0}\equiv\alpha[x,\lambda]$

$|\alpha[x,\lambda]|_{n+1}\equiv\\{\bar{s}[x,\lambda]:for\>some\>\beta\in
K\>s.t.\>\beta[x,\lambda]\bigcap|\alpha[x,\lambda]|_{n}\neq\emptyset,\>\bar{s}[x,\lambda]\in\beta[x,\lambda]\\}$

$|\alpha[x,\lambda]|\equiv\bigcup_{n\in\mathbb{N}}|\alpha[x,\lambda]|_{n}$

By far the most important case is when $x=-\infty$; this is a because of its
use in defining ideal partitions. Ideal partitions will initially be defined
just for dynamic spaces.

######  Definition 62.

If $\gamma$ a partition of dynamic space $D$, it is an _ideal partition_
(_ip_) if it is bounded from below, all $\alpha\in\gamma$ are bounded from
above, and for all $\alpha\in\gamma$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.

The result to be attained in this section is that a covering is decidable by
an ideal e-automata if and only if the covering is an ideal partition. The
first step towards that result is the following theorem.

######  Theorem 63.

If a covering, $K$, of dynamic space $D$ is decided by an ideal e-automata,
$Z=(D_{Z},I,F)$, then $K$ is an ideal partition

###### Proof.

Since $K$ is decided by $Z$, for some $\lambda_{0}\in\Lambda_{D}$,
$K\equiv\\{\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow:e\in\mathcal{E}_{F}\\}$.
For each $\alpha=\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow\,\in K$
define $\lambda_{\alpha}\equiv\lambda_{0}+\Lambda(\mathcal{O}_{e})$.

A: $K$ is $bb\lambda_{0}$ :

\- Immediate from the definition of
$\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow$ -

B: Every $\alpha\in K$ is bounded from above

\- Immediate from Thm 50 and
$K=\\{\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow:e\in\mathcal{E}_{F}\\}$
-

C: $K$ is a partition

\- Immediate from Thm 44 -

D: For all $\lambda>\lambda_{0}$, $\alpha\in K$,
$|\alpha[-\infty,\lambda]|=\,\rightarrow|\alpha[\lambda_{0},\lambda]|$

\- It is sufficient to show that for all $n\in\mathbb{N}$,
$|\alpha[-\infty,\lambda]|_{n}=\rightarrow|\alpha[\lambda_{0},\lambda]|_{n}$.
By (A), this holds for $n=0$. Assume it holds for $n=i$. By (A) and assumption
on $i$, for all $\beta\in K$,
$\beta[-\infty,\lambda]\bigcap|\alpha[-\infty,\lambda]|_{i}\neq\emptyset$ iff
$\beta[\lambda_{0},\lambda]\bigcap|\alpha[\lambda_{0},\lambda]|_{i}\neq\emptyset$.
Since $|\alpha[x,\lambda]|_{i+1}$ is equal to the union over the $\beta$’s
that intersect $|\alpha[x,\lambda]|_{i}$, $|\alpha[-\infty,\lambda]|_{i}$ &
$|\alpha[\lambda_{0},\lambda]|_{i}$ are intersected by the same set of
$\beta\in K$, and because all such $\beta$ are $bb\lambda$,
$|\alpha[-\infty,\lambda]|_{i+1}=\rightarrow|\alpha[\lambda_{0},\lambda]|_{i+1}$
-

It remains to show that for all $\alpha\in K$, $\lambda\in\Lambda_{D}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$. Clearly
$\alpha\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$, so it
only needs to be shown that
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]\subset\alpha$.

For $\lambda\leq\lambda_{0}$: Immediate from (A).

For $\lambda\in(\lambda_{0},\lambda_{\alpha})$: By Thm 59,
$\alpha[\lambda_{0},\lambda_{\alpha}]=|\alpha[\lambda_{0},\lambda]|\circ\alpha[\lambda,\lambda_{\alpha}]$.
By (B) and (D)
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=(\rightarrow|\alpha[\lambda_{0},\lambda]|)\circ(\alpha[\lambda,\lambda_{\alpha}]\rightarrow)=\,\rightarrow(|\alpha[\lambda_{0},\lambda]|\circ\alpha[\lambda,\lambda_{\alpha}])\rightarrow\,=\,\rightarrow\alpha[\lambda_{0},\lambda_{\alpha}]\rightarrow\,=\alpha$.

For $\lambda\geq\lambda_{\alpha}$: By (B) and (C),
$|\alpha[-\infty,\lambda]|=\alpha[-\infty,\lambda]$, so
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset\alpha[-\infty,\lambda]\rightarrow\,=\alpha$.
∎

Because ideal e-automata need only be weakly unbiased, it is insufficient to
only consider measurements on dynamic spaces, so let’s now to consider the
more general case of a system that’s a dynamic set.

The first thing to do is to extend the “outer $\rightarrow$” notation to
dynamic sets.

######  Definition 64.

If $S$ is a dynamic set and $A$ is a set of partial paths then $\rightarrow
A\rightarrow$, $\rightarrow A$, and $A\rightarrow$ are defined as before:

$\bar{p}\in\rightarrow A\rightarrow$ if there exists a
$\bar{p}^{\prime}[x_{1},x_{2}]\in A$ s.t. $\bar{p}\in
S[-\infty,x_{1}]\circ\bar{p}^{\prime}[x_{1},x_{2}]\circ S[x_{2},\infty]$, etc.

$\rightharpoondown A\rightharpoondown$ is $\rightarrow A\rightarrow$
relativized to $S$: $\rightharpoondown A\rightharpoondown\equiv(\rightarrow
A\rightarrow)\bigcap S$;

similarly for $\rightharpoondown A$ and $A\rightharpoondown$.

As a convenient shorthand, $+A\equiv\rightharpoondown A\rightharpoondown$ .

If $S$ is a dynamic space then $\rightharpoondown A=\,\rightarrow A$ and
$A\rightharpoondown\,=A\rightarrow$ .

It will also be useful to also extend the notion of boundedness to cover
various situations that can arise with dynamic sets. (Note that the definition
for bounded from above/below was given for dynamic sets, and so still holds.)

######  Definition 65.

If $S$ is a dynamic set and $\alpha\subset S$:

If $\alpha=\alpha[-\infty,\lambda]\rightharpoondown$ then $\alpha$ is _weakly
bounded from above (by $\lambda$)_; if
$\alpha=\,\rightharpoondown\alpha[\lambda,\infty]$ then $\alpha$ is _weakly
bounded from below (by $\lambda$)_.

If for all $\lambda^{\prime}\leq\lambda$,
$\alpha=\,\rightarrow\alpha[-\infty,\lambda^{\prime}],$ $\alpha$ is _strongly
bounded from below (by $\lambda$);_ similarly for _strongly bounded from above
(by $\lambda$)._

If $A$ is a set of subsets of $S$, $A$ is _strongly/weakly bounded from
above/below (by_ $\lambda$) every element of $A$ is strongly/weakly bounded
from above/below (by $\lambda$).

“Weakly bounded from above by $\lambda$” may be abbreviated “$wba\lambda$”,
“strongly bounded from above by $\lambda$” may be abbreviated “$sba\lambda$”,
etc.

If $S$ is a dynamic space there is no difference between being bounded from
above/below, weakly bounded from above/below, and strongly bounded from
above/below.

######  Theorem 66.

If $\alpha$ is $wba\lambda$ and $\lambda^{\prime}>\lambda$ then $\alpha$ is
$wba\lambda^{\prime}$

###### Proof.

Assume
$\bar{p}_{1}[-\infty,\lambda^{\prime}]\in\alpha[-\infty,\lambda^{\prime}]$,
$\bar{p}_{2}[\lambda^{\prime},\infty]\in S[\lambda^{\prime},\infty]$, and
$\bar{p}=\bar{p}_{1}[-\infty,\lambda^{\prime}]\circ\bar{p}_{2}[\lambda^{\prime},\infty]\in
S$. Since $\alpha$ is $wba\lambda$ and
$\bar{p}[-\infty,\lambda]=\bar{p}_{1}[-\infty,\lambda]\in\alpha[-\infty,\lambda]$,
$\bar{p}\in\alpha$. ∎

The most inclusive notion of decidability on a dynamic set would be:
e-automata $(D_{Z},I,F)$ decides covering $K$ on dynamic set $S$ if for some
$\lambda\in\Lambda_{S}$

1) For all $\bar{p}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}^{\lambda}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in S[\lambda,\lambda^{\prime}]$

2) $K=\\{+\mathcal{O}_{e}^{\lambda}:e\in\mathcal{E}_{F}\\}$.

However, this would mean that the interactions between the system and the
environment are only regulated while the experiment is taking place; before
and after the experiment, any kind of interactions would be allowed, even
those that would undermine the autonomy of the system. It is more sensible to
assume that these interactions are always unbiased. If the system and
environment don’t interact outside of the experiment, which would be a natural
assumption, then it’s certainly the case that the interactions are always
unbiased. Because nothing is measured prior to $\lambda$, prior to $\lambda$
there is no distinction between being weakly & strongly unbiased. From
$\lambda$ onward, Thm 37 ought to hold. This leads to the following with
regard to the make up of a system:

######  Definition 67.

If $S$ is a dynamic set, $K$ is a covering of $S$, and
$\lambda\in\Lambda_{S}$, _$S$ is unbiased with respect to $(\lambda,K)$_ if

1) $S$ is $sbb\lambda$

2) For all $a\in K$, all $\lambda^{\prime}>\lambda$, all
$\bar{p}_{1},\bar{p}_{2}\in+|a[-\infty,\lambda^{\prime}]|$, if
$\bar{p}_{1}(\lambda^{\prime})=\bar{p}_{2}(\lambda^{\prime})$ then
$\bar{p}_{1}[-\infty,\lambda^{\prime}]\circ\bar{p}_{2}[\lambda^{\prime},\infty]\in
S$.

It may be assumed that $K$ is $sbb\lambda$, but it is not demanded.

We’re now in a position to define decidability in general.

######  Definition 68.

A covering, $K$, of dynamic set $S$ is _decided_ by e-automata $Z=(D_{Z},I,F)$
if $\Lambda_{S}=\Lambda_{D_{Z}}$ and for some $\lambda\in\Lambda_{D}$:

1) For all $\bar{p}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}^{\lambda}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in S[\lambda,\lambda^{\prime}]$.

2) $S$ is unbiased with respect to $(\lambda,K)$.

3) $K\equiv\\{+\mathcal{O}_{e}^{\lambda}:e\in\mathcal{E}_{F}\\}$.

Ideal partitions now also need to be generalized for dynamic sets.

######  Definition 69.

If $\gamma$ a partition of dynamic set $S$, it is an _ideal partition_ (_ip_)
if it is strongly bounded from below, all $\alpha\in\gamma$ are weakly bounded
from above, and for all $\alpha\in\gamma$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.

In the case where $S$ is a dynamic set, this reduces to the prior definition.
Now to generalize Thm 63.

######  Theorem 70.

If a covering, $K$, of dynamic set $S$ is decided by an ideal e-automata,
$Z=(D_{Z},I,F)$, then $K$ is an ideal partition

###### Proof.

Since $K$ is decided by $Z$, for some$\lambda_{0}\in\Lambda_{S}$,
$K\equiv\\{+\mathcal{O}_{e}^{\lambda_{0}}:e\in\mathcal{E}_{F}\\}$. For each
$\alpha=+\mathcal{O}_{e}^{\lambda_{0}}\in K$ define
$\lambda_{\alpha}\equiv\lambda_{0}+\Lambda(\mathcal{O}_{e})$.

A: $K$ is $sbb\lambda_{0}$

\- Take any $\alpha\in K$, $\lambda\leq\lambda_{0}$; it is necessary to show
that $\alpha=\,\rightarrow\alpha[\lambda,\infty]$. For any
$\bar{p}_{1}[-\infty,\lambda]\in S[-\infty,\lambda]$,
$\bar{p}_{2}[\lambda,\infty]\in\alpha[\lambda,\infty]$ s.t.
$\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)$,
$\bar{p}=\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in S$
because $S$ is unbiased with respect to $(\lambda,K)$. Since
$\lambda\leq\lambda_{0}$,
$\bar{p}_{2}[\lambda_{0},\infty]\in\alpha[\lambda_{0},\infty]$, so since
$\alpha=+\alpha[\lambda_{0},\lambda_{\alpha}]$, $\bar{p}\in\alpha$.

B: Every $\alpha\in K$ is weakly bounded from above

-Immediate from Thm 50 and $K=\\{\rightharpoondown\mathcal{O}_{e}^{\lambda_{0}}\rightharpoondown:e\in\mathcal{E}_{F}\\}$ -

C: $K$ is a partition

\- Follows from the boolean & automata conditions -

D: For all $\lambda\in\Lambda_{S}$, $\alpha\in K$,
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset S$

\- Follows from $S$ being unbiased with respect to $(\lambda,K)$. -

E: For all $\lambda>\lambda_{0}$, $\alpha\in K$,
$|\alpha[-\infty,\lambda]|=\rightarrow|\alpha[\lambda_{0},\lambda]|$

\- Identical to (D) in Thm 63 -

It remains to show that for all $\alpha\in K$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$. As before,
$\alpha\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$, so it
only needs to be shown that
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]\subset\alpha$.

For $\lambda\leq\lambda_{0}$: Immediate from (A).

For $\lambda\in(\lambda_{0},\lambda_{\alpha})$: Assume
$\bar{p}_{1}[-\infty,\lambda]\in|\alpha[-\infty,\lambda]|$ and
$\bar{p}_{2}[\lambda,\infty]\in\alpha[\lambda,\infty]$. It follows from (E)
that $\bar{p}_{1}[\lambda_{0},\lambda]\in|\alpha[\lambda_{0},\lambda]|$, and
so from Thm 59 that
$\bar{p}_{1}[\lambda_{0},\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha[\lambda_{0},\lambda_{\alpha}]$.
It then follows from (A) that
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha[-\infty,\lambda_{\alpha}]$.
Since $S$ is unbiased with respect to $(\lambda,K)$,
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in S$.
Therefore, since $\alpha=\alpha[-\infty,\lambda_{\alpha}]\rightharpoondown$,
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha$.

For $\lambda\geq\lambda_{\alpha}$: By (B) and (C),
$|\alpha[-\infty,\lambda]|=\alpha[-\infty,\lambda]$, and by (D)
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset\alpha[-\infty,\lambda]\rightharpoondown$.
By (B) $\alpha=\mathbf{\alpha[-\infty,\lambda]\rightharpoondown}$. ∎

And now establish the inverse, that all ip’s are ideally decidable.

######  Theorem 71.

All ideal partitions are ideally decidable

###### Proof.

Assume $\gamma$ is an ideal partition of dynamic set $S$. An ideal e-automata,
$Z=(D,I,F)$, that decides $\gamma$ will be constructed.

1) Start by constructing a new dynamic space $D_{0}=S\otimes E_{0}$ as
follows: For every $\lambda\in\Lambda_{D}$ create a set $E(\lambda),$ and a
bijection $b_{\lambda}:|\gamma[-\infty,\lambda]|\rightarrow E(\lambda)$ s.t.
for all $\lambda_{1}\neq\lambda_{2}$, $E(\lambda_{1})\bigcap
E(\lambda_{2})=\emptyset$; $(\bar{s},\bar{e})\in D_{0}$ iff $\bar{s}\in S$ and
for all $\lambda\in\Lambda$, if $\bar{s}\in\alpha\in\gamma$ then
$\bar{e}(\lambda)=b_{\lambda}(|\alpha[-\infty,\lambda]|)$

2) From $D_{0}$ construct $Z$’s dynamic space, $D$:

If $\Lambda_{S}$ is unbounded from below, for every $\lambda\in\Lambda_{D}$
define $D_{\lambda}\equiv\\{\bar{p}:for\>some\>\bar{p}^{\prime}\in
D_{0},\>for\>all\>\lambda^{\prime}\in\Lambda_{S},\>\bar{p}^{\prime}(\lambda^{\prime}+\lambda)=\bar{p}(\lambda^{\prime})\\}$.
$D_{Z}\equiv\bigcup_{\lambda\in\Lambda}D_{\lambda}$.

If $\Lambda_{D}$ is bounded from below, take $D_{0}^{*}$ to be any dynamic
space s.t. $\Lambda_{D_{0}^{*}}$, is unbounded from below,
$D_{0}^{*}[0,\infty]=D_{0}$, and for all $\lambda<\lambda^{\prime}<0$,
$D_{0}^{*}(\lambda)\bigcap D_{0}^{*}(\lambda^{\prime})=\emptyset$ (this
already holds for all $\lambda>\lambda^{\prime}\geq 0$). Construct $D^{*}$ as
above and take $D=D^{*}[0,\infty]$.

3) Construct $I$: Select any $\lambda_{0}\in\Lambda_{D}$ s.t. $\gamma$ is
$bb\lambda_{0}$, $I\equiv D_{0}(\lambda_{0})$

4) Construct $F$: For every $\alpha\in\gamma$ select a
$\lambda_{\alpha}\in\Lambda_{D}$ s.t.
$\alpha=+\alpha[-\infty,\lambda_{\alpha}]$;
$F\equiv\bigcup_{\alpha\in\gamma}\alpha(\lambda_{\alpha})\otimes\\{b_{\lambda_{a}}(|\alpha[-\infty,\lambda_{\alpha}]|)\\}$.

Now to show that $(D,I,Z)$ is an ideal e-automata that decides $\gamma$.

A: $D_{0}$ is a dynamic space

\- Take any $(\bar{s}_{1},\bar{e}_{1}),(\bar{s}_{2},\bar{e}_{2})\in D_{0}$
s.t. for some $\lambda\in\Lambda_{S}$,
$(\bar{s}_{1}(\lambda),\bar{e}_{1}(\lambda))=(\bar{s}_{2}(\lambda),\bar{e}_{2}(\lambda))$.
Take $\bar{s}_{1}\in\alpha_{1}\in\gamma$ and
$\bar{s}_{2}\in\alpha_{2}\in\gamma$; note that since
$\bar{e}_{1}(\lambda)=\bar{e}_{2}(\lambda)$,
$|\alpha_{1}[-\infty,\lambda]|=|\alpha_{2}[-\infty,\lambda]|$.

Take
$(\bar{s},\bar{e})\equiv(\bar{s}_{1}[-\infty,\lambda]\circ\bar{s}_{2}[\lambda,\infty],\bar{e}_{1}[-\infty,\lambda]\circ,\bar{e}_{2}[\lambda,\infty])$.
$\bar{s}\in|\alpha_{1}[-\infty,\lambda]|\circ\alpha_{2}[\lambda,\infty]=|\alpha_{2}[-\infty,\lambda]|\circ\alpha_{2}[\lambda,\infty]=\alpha_{2}$,
so $(\bar{s},\bar{e})\in D_{0}$ iff for all $\lambda^{\prime}\in\Lambda_{D}$,
$\bar{e}(\lambda^{\prime})=b_{\lambda^{\prime}}(|\alpha_{2}[-\infty,\lambda^{\prime}]|)$,
which follows from the fact that $\bar{s}\in\alpha_{2}$. -

B: $D$ is a dynamic space

\- For any $p\in\mathcal{P}_{D_{0}}=\mathcal{P}_{D}$, $p$ is only realized at
a single $\lambda\in\Lambda_{S}$ in $D_{0}$. Because $D$ is a union over
“shifted” copies of $D_{0}$, paths in $D$ can only intersect if the belong to
the same copy; it follows that $D$ is a dynamic space if $D_{0}$ is. -

C: $D$ is homogeneous; $I$ is homogeneously realized

\- Because for any $p\in\mathcal{P}_{D_{0}}$, $p$ is only realized at a single
$\lambda\in\Lambda_{S}$ in $D_{0}$, $D_{0}$ must be homogeneous. From the
nature of the construction of $D$ from $D_{0}$ it’s clear that $D$ is also
homogeneous, and that all of $\mathcal{P}_{D_{0}}$ is homogeneously realized
in $D$. -

D: $I\rightarrow\>=I\upharpoonleft(I\rightarrow F)\rightarrow$

$\rightarrow F=\>\rightarrow I\upharpoonleft(I\rightarrow F)$

\- $D_{0}=\>\rightarrow I\upharpoonleft I\rightarrow F\rightarrow$ and so for
$D_{0}$, $I\rightarrow\>=I\upharpoonleft(I\rightarrow F)\rightarrow$ and
$\rightarrow F=\>\rightarrow I\upharpoonleft(I\rightarrow F)$. If this holds
for $D_{0}$, it must also hold for $D$ -

E: $Z$ is an environmental shell

\- All elements of $Dom(Z)$ can be decomposed into system & environmental
states, and $F$ has its own set of environmental states -

F: $Z$ is weakly unbiased

\- Recall that $I=D_{0}(\lambda_{0})$. Choose any $e_{1}\in\mathcal{E}_{F}$,
and take $\alpha\in\gamma$, $\lambda_{1}+\lambda_{0}\in\Lambda_{S}$ s.t.
$b_{\lambda_{0}+\lambda_{1}}(|\alpha[-\infty,\lambda_{0}+\lambda_{1}]|)=e_{1}$
(since $e_{1}\in\mathcal{E}_{F}$ there can only be one such $\alpha$). Take
any $e\in\mathcal{E}_{Int(Z)}$, $\lambda\in\Lambda_{D}$ s.t.
$\Sigma_{e}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$. It follows
that $e=b_{\lambda_{0}+\lambda}(|\alpha[-\infty,\lambda_{0}+\lambda]|)$.
Therefore, if $e,e^{\prime}\in\mathcal{E}_{Int(Z)}$ and for some
$\lambda\in\Lambda$,
$\Sigma_{e}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$ and
$\Sigma_{e^{\prime}}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$ then
$e=e^{\prime}$. $Z$ must then be weakly unbiased -

G: $Z$ is an e-automata

\- Take any
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$ s.t.
$\lambda_{2}\geq\lambda_{1}$ and
$\bar{s}_{1}[0,\lambda_{1}]=\bar{s}_{2}[0,\lambda_{1}]$. For
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ s.t.
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}_{1}[0,\lambda_{1}]$ and
$\bar{p}_{2}[0,\lambda_{2}]\cdot\mathcal{S}=\bar{s}_{2}[0,\lambda_{2}]$,
$\bar{p}_{1}(\lambda_{1})=\bar{p}_{2}(\lambda_{1})\in F$, so
$\lambda_{2}=\lambda_{1}$. -

H: $Z$ is all-reet

\- For any $e\in\mathcal{E}_{F}$ there’s only a single
$\bar{e}[0,\lambda]\in\omega_{F}\cdot\mathcal{E}$, so naturally
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}$ -

I: $Z$ is boolean

\- For any $\bar{s}[0,\lambda]\in\mathcal{O}_{F}$ there’s only one
$(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\in\Theta_{F}$. Since
$\bar{e}(\lambda)\in\mathcal{E}_{F}$ there’s only one $e\in\mathcal{E}_{F}$
s.t. $\bar{s}[0,\lambda]\in\mathcal{O}_{e}$ -

J: $Z$ decides $\gamma$

\- Given any $\alpha\in\gamma$, take
$e=b_{\lambda_{\alpha}}(|\alpha[-\infty,\lambda_{\alpha}]|)$;
$e\in\mathcal{E}_{F}$ and $\alpha=+\mathcal{O}_{e}^{\lambda_{0}}$ \- ∎

Note that $S$ need not be homogeneous, nor does any part of $\mathcal{P}_{S}$
need to be homogeneously realized.333Had any part of $\mathcal{P}_{S}$ needed
to be homogeneously realized it could have presented a problem with viewing
$S$ as playing out on the stage of space-time, because every point in space-
time can only be realized at a single $\lambda$.

######  Theorem 72.

If $\gamma$ is an ip and $\bigcup\gamma$ is a dynamic space, then $\gamma$ is
decided by a strongly unbiased ideal e-automate.

###### Proof.

Use the same construction as the prior theorem. For every $(s,e)\in D_{0}$,
$e=b_{\lambda}(|\alpha[-\infty,\lambda]|)$,
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}=(+|\alpha[-\infty,\lambda]|)_{(\lambda,s)\rightarrow}$.
If $\bigcup\gamma$ is a dynamic space, this means that
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}=(\bigcup\gamma)_{(\lambda,s)\rightarrow}$.
Since $\Sigma_{(s,e)\rightarrow}$ is simply
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}$ shifted by $\lambda_{0}$ and
truncated at the $\lambda_{\alpha}$’s (for definitions of $\lambda_{0}$ and
$\lambda_{\alpha}$, see (3) and (4) in the proof of the prior theorem), it
follows immediately that the e-automata is strongly unbiased. ∎

###  III.5 Companionable Sets & Compatible Sets

In the final section of this part, the make-up of ip’s will be investigated.

######  Definition 73.

If $S$ is a dynamic set:

$A\subset S$ is a _subspace_ (of $S$) if it is non-empty and for every
$\lambda\in\Lambda_{S}$, $A=A[-\infty,\lambda]\circ A[\lambda,\infty]$.

$A\subset S$ is _companionable_ if it is a subspace that’s weakly bounded from
above and strongly bounded from below.

######  Theorem 74.

If $\gamma$ is an ip and $\alpha\in\gamma$ then $\alpha$ is companionable

###### Proof.

For any $\lambda\in\Lambda$,
$\alpha\subset\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.
Since $\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$,
$\alpha=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]$.

By the definition of an ip, $\alpha$ must be strongly bounded from below and
weakly bounded from above.∎

######  Definition 75.

If $S$ is a dynamic set, $\Gamma$ is a non-empty set of subsets of $S$, and
$\alpha\in\Gamma$:

$|\alpha[-\infty,\lambda]|_{\Gamma}$ is defined identically to how it’s
defined for covering in definition 61

$(\alpha)_{\lambda}^{\Gamma}\equiv\\{\beta\in\Gamma:\beta[-\infty,\lambda]\subset|\alpha[-\infty,\lambda]|_{\Gamma}\\}$.

$\Gamma$ is _compatible_ if it is a pairwise disjoint set of companionable
sets, strongly bounded from below, and for all $\alpha,\beta\in\Gamma$, all
$\lambda\in\Lambda_{S}$, if $\beta\in(\alpha)_{\lambda}^{\Gamma}$ and
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$ then
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.

If the set $\Gamma$ is understood, $|\alpha[-\infty,\lambda]|_{\Gamma}$ may be
written $|\alpha[-\infty,\lambda]|$ and $(\alpha)_{\lambda}^{\Gamma}$ may be
written $(\alpha)_{\lambda}$.

######  Theorem 76.

If $S$ is a dynamic set, $\Gamma$ is a non-empty set of subsets of $S$, and
$\alpha,\beta\in\Gamma$, then

1) $\beta\in(\alpha)_{\lambda}^{\Gamma}$ iff there’s a finite sequence of
elements of $\Gamma$, $(a_{i})_{i\leq n}$, s.t. $a_{1}=\alpha$, $a_{n}=\beta$,
and for all $1\leq i<n$, $a_{i}[-\infty,\lambda]\bigcap
a_{i+1}[-\infty,\lambda]\neq\emptyset$.

2) $\beta\in(\alpha)_{\lambda}^{\Gamma}$ iff
$|\beta[-\infty,\lambda]|_{\Gamma}\bigcap|\alpha[-\infty,\lambda]|_{\Gamma}\neq\emptyset$
iff $|\beta[-\infty,\lambda]|_{\Gamma}=|\alpha[-\infty,\lambda]|_{\Gamma}$.

###### Proof.

Immediate from the definition of $|\alpha[-\infty,\lambda]|_{\Gamma}$.∎

######  Theorem 77.

$\Gamma$ is compatible iff it’s pairwise disjoint, strongly bounded from
below, all $\alpha\in\Gamma$ are weakly bounded from above, and for all
$\lambda\in\Lambda$,
$\alpha=|\alpha[-\infty,\lambda]|_{\Gamma}\circ\alpha[\lambda,\infty]$.

###### Proof.

$\Rightarrow$
$|\alpha[-\infty,\lambda]|_{\Gamma}=\bigcup_{\beta\in(\alpha)_{\lambda}^{\Gamma}}\beta[-\infty,\lambda]$.
Since $\Gamma$ is compatible, for all $\beta\in(\alpha)_{\lambda}^{\Gamma}$,
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$,
$\beta_{\rightarrow(\lambda,p)}=\alpha_{\rightarrow(\lambda,p)}$. Therefore
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]=|\alpha[-\infty,\lambda]|_{\Gamma}\circ\alpha[\lambda,\infty]$.
Since $\alpha$ is companionable,
$\alpha=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]$.

$\Leftarrow$ For $\alpha,\beta\in\Gamma$, if
$\beta\in(\alpha)_{\lambda}^{\Gamma}$ then
$|\alpha[0,\lambda]|=|\beta[0,\lambda]|$, so
$\beta=|\alpha[-\infty,\lambda]|\circ\beta[\lambda,\infty]$, and so for any
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$,
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.

$\alpha\subset\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=\alpha$,
so $\alpha$ is companionable.∎

######  Theorem 78.

A covering of a dynamic set is an ip iff it’s compatible

###### Proof.

Immediate from Thm 77. ∎

This establishes that if $\alpha$ is the element of some ip then it’s
companionable, and if $t$ is the subset of some ip then it’s compatible. For
dynamic spaces, the converse also holds. For any compatible set, $t$, an “all-
reet not $t$” can be constructed by taking the paths not in $t$ and grouping
them by when they broke off from $t$, and which $|\alpha[-\infty,\lambda]|$
they broke off from. $t$ together with the “all-reet not $t$” form an ip. When
the parameter is not discrete one mild complication is that, in dealing with
paths that broke off from $t$ at $\lambda$, we’ll need to distinguish between
paths that were in $t$ at $\lambda$, but aren’t in $t$ at any
$\lambda^{\prime}>\lambda$ from those that are not in $t$ at $\lambda$, but
were in $t$ at all $\lambda^{\prime}<\lambda$.

######  Definition 79.

If $t$ is compatible, $S$ is a dynamic set s.t. $\bigcup t\subset S$, and
$\alpha\in t$ then

$\sim t_{\lambda}\equiv\\{\bar{p}\in S:\bar{p}[-\infty,\lambda]\notin\bigcup
t[-\infty,\lambda]\\}$

$\sim|\alpha|_{\lambda}^{+}\equiv\\{\bar{p}\in
S:\bar{p}[-\infty,\lambda]\in|\alpha[-\infty,\lambda]|\>and\>for\>all\>\lambda^{\prime}>\lambda,\>\bar{p}[-\infty,\lambda^{\prime}]\notin\bigcup
t[-\infty,\lambda]|\\}$

$\sim|\alpha|_{\lambda}^{-}\equiv\\{\bar{p}\in
S:\bar{p}[-\infty,\lambda]\notin\bigcup
t[-\infty,\lambda]|\>and\>for\>all\>\lambda^{\prime}<\lambda,\>\bar{p}[-\infty,\lambda^{\prime}]\in|\alpha[-\infty,\lambda^{\prime}]|\\}$

These sets can be used to construct an ip containing $t$.

######  Theorem 80.

If $D$ is a dynamic space, $\alpha$ is a subset of $D$, and $t$ is a set of
subsets of $D$ then

1) If $\alpha$ is companionable then there exists an ip, $\gamma$, s.t.
$\alpha\in\gamma$

2) If $t$ is compatible then there exists an ip, $\gamma$, s.t.
$t\subset\gamma$

###### Proof.

(1) Follows immediately from (2). For (2), assume $t$ is $bb\lambda$ and take
$\gamma$ to be the set s.t. $t\subset\gamma$, if $\sim
t_{\lambda}\neq\emptyset$ then $\sim t_{\lambda}\in\gamma$, and for all
$\alpha\in t$, $\lambda^{\prime}>\lambda$, if
$\sim|\alpha|_{\lambda^{\prime}}^{+/-}\neq\emptyset$ then
$\sim|\alpha|_{\lambda^{\prime}}^{+/-}\in\gamma$. $\gamma$ is then partition
of $D$. (From here on, the qualifiers “if $\sim t_{\lambda}\neq\emptyset$” and
“if $\sim|\alpha|_{\lambda^{\prime}}^{+/-}\neq\emptyset$” will be understood.)

The following is key to establishing that $\gamma$ is an ip:

A: For all $\lambda_{1}<\lambda_{2}$,
$p\in\,\sim|\alpha|_{\lambda_{2}}^{-}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$

\- By the definition of $\sim|\alpha|_{\lambda}^{-}$ it’s clearly the case
that
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}\subset|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$.
Take any $\bar{p}_{1}\in+|\alpha[-\infty,\lambda_{2}]|$ and any
$\bar{p}_{2}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$ s.t.
$\bar{p}_{1}(\lambda_{1})=\bar{p}_{2}(\lambda_{1})=p$. Take
$\bar{p}\equiv\bar{p}_{1}[-\infty,\lambda_{1}]\circ\bar{p}_{2}[\lambda_{1},\infty]\in
D$ and note that for all $\lambda^{\prime}<\lambda_{2}$
$\bar{p}[-\infty,\lambda^{\prime}]\in|\alpha[-\infty,\lambda^{\prime}]|$.
Assume that for some $\beta\in t$,
$\bar{p}[-\infty,\lambda_{2}]\in\beta[-\infty,\lambda_{2}]$. That would mean
$|\beta[-\infty,\lambda_{1}]|=|\alpha[-\infty,\lambda_{1}]|$ and so
$\bar{p}_{2}[-\infty,\lambda_{1}]\circ\bar{p}[\lambda_{1},\lambda_{2}]=\bar{p}_{2}[-\infty,\lambda_{2}]$$\in\beta[-\infty,\lambda_{2}]$.
This contradicts $\bar{p}_{2}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$, so there
can be no such $\beta\in t$. Therefore
$\bar{p}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$, and so for all
$\lambda_{1}<\lambda_{2}$,
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$.
-

Similarly, for all $\lambda_{1}\leq\lambda_{2}$,
$p\in\,\sim|\alpha|_{\lambda_{2}}^{+}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda_{2}}^{+})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$

For any $\lambda^{\prime}>\lambda$ the following properties therefore hold for
$\sim|\alpha|_{\lambda^{\prime}}^{-}$ and
$\sim|\alpha|_{\lambda^{\prime}}^{+}$:

$\sim|\alpha|_{\lambda^{\prime}}^{-}$ is $ba\lambda^{\prime}$ and for all all
$\lambda_{1}<\lambda^{\prime}$,
$p\in\,\sim|\alpha|_{\lambda^{\prime}}^{-}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda^{\prime}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda^{\prime}]|_{\rightarrow(\lambda_{1},p)}$.

Similarly, for all $\lambda^{\prime\prime}>\lambda^{\prime}$,
$\sim|\alpha|_{\lambda^{\prime}}^{+}$ is $ba\lambda^{\prime\prime}$ and for
all $\lambda_{1}\leq\lambda^{\prime}$,
$p\in\,\sim|\alpha|_{\lambda^{\prime}}^{+}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda^{\prime}}^{+})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda_{1},p)}$
.

As an aside, it follows that the $\sim|\alpha|_{\lambda^{\prime}}^{+/-}$ are
$bb\lambda$.

Finally, since $t$ is $bb\lambda$, $\sim
t_{\lambda}=\,\rightarrow(\lambda,D(\lambda)-\bigcup t(\lambda))\rightarrow$.

These properties are sufficient to establish that $\gamma$ is an ip. ∎

This result may be generalized for systems that are not dynamic spaces.

######  Theorem 81.

If $S$ is a dynamic set and $\gamma$ is an ip of $S$ then

1) If $\alpha\subset\beta\in\gamma$ and $\alpha$ is companionable then there
exists an ip of $S$, $\gamma^{\prime}$, s.t. $\alpha\in\gamma^{\prime}$

2) If $t$ is a compatible set of subsets of $S$ s.t. for each $\alpha\in t$
there’s a $\alpha\subset\beta\in\gamma$ then there exists an ip of $S$,
$\gamma^{\prime}$, s.t. $t\subset\gamma^{\prime}$

###### Proof.

Once again, (1) follows from (2).

Assume $t$ is $bb\lambda$ and this time take $\gamma^{\prime}$ to be the set
s.t. $t\subset\gamma^{\prime}$ and for all $\beta\in\gamma$ if
$\beta\bigcap(\sim t_{\lambda})\neq\emptyset$ then $\beta\bigcap(\sim
t_{\lambda})\in\gamma^{\prime}$, and for all $\alpha\in t$,
$\lambda^{\prime}>\lambda$, if
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+/-})\neq\emptyset$ then
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+/-})\in\gamma^{\prime}$.

The nature of these sets are similar to those of the prior proof, except that
they’re relativized to each $\beta\in\gamma$. The following properties are
sufficient to establish that $\gamma^{\prime}$ is an ip:

For all $\lambda^{\prime\prime}<\lambda^{\prime}$,
$p\in(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))(\lambda^{\prime\prime})$,
$(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))_{\rightarrow(\lambda^{\prime\prime},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$
(Assume $\alpha\subset\eta\in\gamma$; since
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-})\neq\emptyset$, and
$\lambda^{\prime\prime}<\lambda^{\prime}$,
$\beta[-\infty,\lambda^{\prime\prime}]\subset|\eta[-\infty,\lambda^{\prime\prime}]|$,
so
$\beta_{\rightarrow(\lambda^{\prime\prime},p)}=|\eta[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$,
and so
$|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}\subset\beta_{\rightarrow(\lambda^{\prime\prime},p)}$.)

$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-})=(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))[-\infty,\lambda^{\prime}]\circ\beta[\lambda^{\prime},\infty]$.

For all $\lambda^{\prime\prime}\leq\lambda^{\prime}$,
$p\in(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})(\lambda^{\prime\prime})$,
$(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})_{\rightarrow(\lambda^{\prime\prime},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$

For all $\lambda^{\prime\prime}>\lambda^{\prime}$,
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})=(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+}))[-\infty,\lambda^{\prime\prime}]\circ\beta[\lambda^{\prime\prime},\infty]$.

With $X\equiv[\beta\bigcap(\sim t_{\lambda})](\lambda)$, $\beta\bigcap(\sim
t_{\lambda})=\beta_{\rightarrow(\lambda,X)}\circ\beta_{(\lambda,X)\rightarrow}$.
∎

The first theorem follows from the second, since if $D$ is a dynamic space
then $\\{D\\}$ is an ip.

##  IV Probabilities

###  IV.1 Dynamic Probability Spaces

For a single ip, probabilities are no different than in classic probability
and statistics444In the Introduction two models were for non-determinism were
given. One of them, type-m non-determinism, encounters a well known difficulty
at this point: in order for the statistical view of probabilities to be
applicable, every individual run of an experiment must result in an individual
outcome being obtained. However, if an e-automata displays pure type-m non-
determinism, it will simultaneously take all paths for all possible outcomes,
not just paths which cross some particular $[e]$, and this will result in
multiple outcomes. In this controversy, experimental results have decided in
favor of individual outcomes; experimental apparatus always end up in a single
state, and as outcomes have been defined, each individual final environmental
state corresponds to an individual outcome.:

######  Definition 82.

An _ip probability space_ is a triple, $(\gamma,\Sigma,P)$, where $\gamma$ is
an ip, $\Sigma$ is a set of subsets of $\gamma$, and
$P:\Sigma\rightarrow[0,1]$ s.t.:

1) $\gamma\in\Sigma$

2) If $\sigma\in\Sigma$ then $\gamma-\sigma\in\Sigma$

3) If $\psi\subset\Sigma$ is finite then $\bigcup\psi\in\Sigma$

4) $P(\gamma)=1$

5) If $\psi\subset\Sigma$ is finite and pairwise disjoint then
$P(\bigcup\psi)=\Sigma_{\sigma\in\psi}P(\sigma)$

Very commonly, “finite” in (3) and (5) is replaced with “countable”. Countable
additivity is invaluable when dealing with questions of convergence, however
convergence will become a more multifaceted issue in the structures to be
introduced, so the countable condition has been relaxed & questions of
convergence delayed.

######  Theorem 83.

In the definition of an ip probability space, (2) and (3) can be replaced
with: If $\sigma_{1},\sigma_{2}\in\Sigma$ then
$\sigma_{1}-\sigma_{2}\in\Sigma$

###### Proof.

If (2) holds then (3) is equivalent to “If $\sigma_{1},\sigma_{2}\in\Sigma$
then $\sigma_{1}\bigcap\sigma_{2}\in\Sigma$”. The equivalence now follows from
$\sigma_{1}-(\sigma_{1}-\sigma_{2})=\sigma_{1}\bigcap\sigma_{2}$ and
$\sigma_{1}-\sigma_{2}=\sigma_{1}\bigcap(\gamma-\sigma_{2})$.∎

######  Definition 84.

If $S_{1}$ and $S_{2}$ are dynamic spaces and $\gamma_{1}$ and $\gamma_{2}$
are ip’s of $S_{1}$ and $S_{2}$ respectively, then ip probability spaces
$(\gamma_{1},\Sigma_{1},P_{1})$ and $(\gamma_{2},\Sigma_{2},P_{2})$ are
_consistent_ if

1) $\gamma_{1}\bigcap\gamma_{2}\in\Sigma_{1}$ and
$\gamma_{1}\bigcap\gamma_{2}\in\Sigma_{2}$

2) If $\sigma\subset\gamma_{1}\bigcap\gamma_{2}$ then $\sigma\in\Sigma_{1}$
iff $\sigma\in\Sigma_{2}$

3) For any $t\in\Sigma_{1}\bigcap\Sigma_{2}$, $P_{1}(t)=P_{2}(t)$

If $Y$ is a set of ip probability spaces, $Y$ is _consistent_ if, for any
$x,y\in Y$, $x$ and $y$ are consistent.

A dynamic probability space is, essentially, a consistent collection of ip
probability spaces.

######  Definition 85.

A _dynamic probability space_ (dps) is a triple $(X,T,P)$ where $X$ is a set
of dynamic sets, $T$ is a set of compatible sets, and $P:T\rightarrow[0,1]$
s.t.:

1) For every $S\in X$ there’s a $\gamma\in T$ s.t. $\gamma$ is an ip of $S$

2) For every $t\in T$ there’s a $\gamma\in T$ s.t. $\gamma$ is an ip of some
$S\in X$ and $t\subset\gamma$

3) If $t_{1},t_{2}\in T$ then $t_{1}-t_{2}\in T$

4) If $\gamma\in T$ is an ip of some $S\in X$ then $P(\gamma)=1$

5) If $t_{1},t_{2}\in T$ are disjoint, and $t_{1}\bigcup t_{2}\in T$ then
$P(t_{1}\bigcup t_{2})=P(t_{1})+P(t_{2})$

For dynamic probability space $(X,T,P)$

$G_{T}\equiv\\{\gamma\in T:\gamma\>is\>a\>partition\>of\>some\>S\in X\\}$

If $S$ is a dynamic space and $\alpha,\beta\subset S$, it’s important to
stress that axiom 5 means that
$P(\\{\alpha,\beta\\})=P(\\{\alpha\\})+P(\\{\beta\\})$ (assuming
$\\{\alpha,\beta\\},\\{\alpha\\},\\{\beta\\}\in T$); it does not mean that
$P(\\{\alpha,\beta\\})=P(\\{\alpha\bigcup\beta\\})$ (even if
$\\{\alpha,\beta\\},\\{\alpha\bigcup\beta\\}\in T$), and so in general
$P(\\{\alpha\bigcup\beta\\})\neq P(\\{\alpha\\})+P(\\{\beta\\})$.

Note that Thm 83 does not hold for dps’s. This is essentially because there is
no $Z\in T$ s.t. for all $t\in T$, $t\subset Z$; as a result, “not $t$” is not
uniquely defined, and arbitrary finite unions can not be expected to be
elements of $T$ (though arbitrary finite intersections are elements of $T$).

To formally describe the connection between dynamic probability spaces and
classic probability theory, the following definitions will be useful.

######  Definition 86.

If $(X,T,P)$ is a dps and $\gamma\in G_{T}$ then $T_{\gamma}\equiv\\{t\in
T\,:\,t\subset\gamma\\}$ and $P_{\gamma}\equiv P\mid_{T_{\gamma}}$ (that is,
$Dom(P_{\gamma})=T_{\gamma}$ and for all $t\in T_{\gamma}$,
$P_{\gamma}(t)=P(t)$)

If $\pi=(X,T,P)$ is a dps and $A\subset G_{T}$,
$A_{\pi}\equiv\\{(\gamma,T_{\gamma},P_{\gamma}):\gamma\in A\\}$

If $Y$ is a consistent set of ip probability spaces,
$X_{Y}\equiv\\{\bigcup\gamma:(\gamma,\Sigma,P)\in Y\\}$,
$T_{Y}\equiv\bigcup_{(\gamma,\Sigma,P)\in Y}\Sigma$ and
$P_{Y}:T_{Y}\rightarrow[0,1]$ s.t. if $(\gamma,\Sigma,P)\in Y$ and
$t\in\Sigma$ then $P_{Y}(t)=P(t)$.

The following theorem states that a dynamic probability space is simply a
consistent set of ip probability spaces.

######  Theorem 87.

1) If $\pi=(X,T,P)$ is a dps and $A\subset G_{T}$ then $A_{\pi}$ is a
consistent set of ip probability spaces

2) If $Y$ is a consistent set of ip probability spaces then
$(X_{Y},T_{Y},P_{Y})$ is dps

3) If $(X,T,P)$ is a dps and
$Y\equiv\\{(\gamma,T_{\gamma},P_{\gamma})\,:\,\gamma\in G_{T}\\}$ then
$X=X_{Y}$, $T=T_{Y}$ and $P=P_{Y}$

4) If $Y$ is a consistent set of ip probability spaces then
$Y=\\{(\gamma,(T_{Y})_{\gamma},(P_{Y})_{\gamma}):\gamma\in G_{(T_{Y})}\\}$

###### Proof.

(1) says that a dps can be rewritten as a consistent set of ip probability
spaces, (2) says that a consistent set of ip probability spaces can be
rewritten as a dps, and (3) & (4) say that in moving between dps’s &
consistent sets of ip probability spaces, no information is lost.

A: For dps $(X,T,P)$, if $t_{1},t_{2}\in T$ and $\gamma$ is any element of
$G_{T}$ s.t. $t_{1}\subset\gamma$ then $t_{1}-t_{2}\in T_{\gamma}$

\- Follows from $t_{1}-t_{2}=t_{1}-t_{1}\bigcap t_{2}=t_{1}-\gamma\bigcap
t_{2}$ (note that $\gamma\bigcap t_{2}\in T_{\gamma}$) -

(1) That the elements of $A_{\pi}$ are ip probability spaces follows from Thm
83. That they’re consistent follows from
$\gamma_{1}\bigcap\gamma_{1}=\gamma_{1}-(\gamma_{1}-\gamma_{2})$

(2) Axioms 1, 2, and 4 clearly hold. Axiom 3 follows from Thm 83 and (A).
Axiom 5 follows from axiom 2.

(3) & (4) That no information is lost in going from a dps to a set of ip
probability spaces follows from (A). It’s clear that no information is lost
when going from a set of ip probability spaces to a dps. ∎

###  IV.2 T-Algebras and GPS’s

Very little in the definition of a dps depends on $X$ being a set of dynamic
sets or $G_{T}$ being a set of ip’s. As things often get simpler as they get
more abstract, it will be useful to take a step back & generalize the
probability theory.

######  Definition 88.

A _t-algebra_ is a double, $(X,T)$, where $X$ is a set of sets and

1) For every $x\in X$ there’s a $\gamma\in T$ s.t. $\gamma$ is a partition of
$x$

2) For every $t\in T$ there’s a $\gamma\in T$ s.t. $\gamma$ is a partition of
some $x\in X$ and $t\subset\gamma$

3) If $t_{1},t_{2}\in T$ then $t_{1}-t_{2}\in T$

A t-algebra, $(X,T)$, may be referred to simply by $T$. As before,
$G_{T}\equiv\\{\gamma\in T:\gamma\>is\>a\>partition\>of\>some\>x\in X\\}$.

######  Definition 89.

A _generalized probability space_ (gps) is a triple, $(X,T,P)$, where $(X,T)$
is a t-algebra and $P:T\rightarrow[0,1]$ s.t.

1) If $t\in T$ is a partition of some $x\in X$ then $P(t)=1$

2) If $t_{1},t_{2}\in T$ are disjoint and $t_{1}\bigcup t_{2}\in T$ then
$P(t_{1}\bigcup t_{2})=P(t_{1})+P(t_{2})$

####  IV.2.1 $\neg t$ and $[t]$

######  Definition 90.

For t-algebra $(X,T)$, $t\in T$, $\neg t\equiv\\{t^{\prime}\in
T:t^{\prime}\bigcap t=\emptyset\>and\>t^{\prime}\bigcup t\in G_{T}\\}$.

For $A\subset T$, $\neg A\equiv\bigcup_{t\in A}\neg t$.

$\neg^{(1)}t\equiv\neg t$, $\neg^{(2)}t\equiv\neg\neg t$,
$\neg^{(3)}t\equiv\neg\neg\neg t$, etc.

######  Theorem 91.

For gps $(X,T,P)$, $t\in T$, $t^{\prime}\in\neg^{(n)}t$

1) If $n$ is odd, $P(t)+P(t^{\prime})=1$

2) If $n$ is even, $P(t)=P(t^{\prime})$

###### Proof.

Simple induction over $\mathbb{N}^{+}$.∎

######  Theorem 92.

For gps $(X,T,P)$, $t,t^{\prime}\in T$, $n,m>0$

1) $t^{\prime}\in\neg^{(n+m)}t$ iff for some $t^{\prime\prime}\in T$,
$t^{\prime\prime}\in\neg^{(n)}t$ and $t^{\prime}\in\neg^{(m)}t^{\prime\prime}$

2) $t^{\prime}\in\neg^{(n)}t$ iff $t\in\neg^{(n)}t^{\prime}$

3) $t\in\neg^{(2n)}t$

4) If $m<n$ then $\neg^{(2m)}t\subset\neg^{(2n)}t$ and
$\neg^{(2m+1)}t\subset\neg^{(2n+1)}t$

###### Proof.

(1) follows from induction over $m$. (2) follows from induction over $n$. (3)
follows from induction over $n$ and (1). (4) follows from (1) and (3).∎

######  Definition 93.

If $(X,T)$ is a t-algebra and $t\in T$,
$[t]\equiv\bigcup_{n\in\mathbb{N}^{+}}\neg^{(2n)}t$.

######  Theorem 94.

$[t]$ is an equivalence class

###### Proof.

By Thm 92.3, $t\in[t]$

By Thm 92.2, if $t_{1}\in[t_{2}]$ then $t_{2}\in[t_{1}]$

By Thm 92.1, if $t_{2}\in[t_{1}]$ and $t_{3}\in[t_{2}]$ then $t_{3}\in[t_{1}]$
∎

######  Theorem 95.

$\neg[t]=\bigcup_{n\in\mathbb{N}^{+}}\neg^{(2n+1)}t$

###### Proof.

$\neg[t]\equiv\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$.
$t^{\prime\prime}\in\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$ iff for some
$t^{\prime}\in[t]$, $t^{\prime\prime}\in\neg t^{\prime}$. $t^{\prime}\in[t]$
iff for some $n\in\mathbb{N}^{+}$, $t^{\prime}\in\neg^{(2n)}t$. So
$t^{\prime\prime}\in\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$ iff for some
$n\in\mathbb{N}^{+}$, $t^{\prime\prime}\in\neg^{(2n+1)}t$.∎

######  Theorem 96.

If $t^{\prime}\in\neg[t]$ then $t\in\neg[t^{\prime}]$

###### Proof.

Follows from Thm 95 and Thm 92.2∎

######  Definition 97.

A t-algebra is _simple_ if for every $t\in T$, $\neg t=\neg\neg\neg t$.

######  Theorem 98.

If $T$ is a simple t-algebra and $t\in T$ then $[t]=\neg\neg t$ and
$\neg[t]=\neg t$

###### Proof.

Clear ∎

####  IV.2.2 $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$

In general, dps’s will not have simple t-algebras. This can be seen from the