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The Fully Connected Stochastic Substitution System

This is my paper that I have recently had published in the Journal of Cellular Automata, if you download Latex you can edit copy edit past this code into it and you can read my paper, it relates to maths physics and areas of computer science.

Date : 30/8/2016

Ben

Author Information

Uploaded by : Ben
Uploaded on : 30/8/2016
Subject : Maths

This article investigates the properties and emergent behaviour of a new kind of discrete substitution system. The micro-states of these systems are modelled as complete weighted graphs over $mathbb{N}$, the weights of which are stored in a state matrix $S_t={{s_{ij}}_t}$, and evolve via a set of constituent specific, independent probabilistic substitution rules. The state is then embedded in $mathbb{R}^n$ by treating flat space geometrical violations as internal curvature within the system that may be minimized via a process of extit{stress minimization}. This paper gives arguments for a definition of energy within the system, observes an emergent intrinsic inertia affecting vertex clusters, and shows the emergence of particle like massive equilibrium states. An retentive effect due to clustering is also observed within this system.

INTRODUCTION

This paper introduces a new, computational, dynamic, discrete system, the Fully-Connected Stochastic Substitution (FCSS) system. The main aims of this paper are to implement the system, retain measurements of meaningful emergent behaviour, and finally, by analogy to space, time and localized particles, make meaningful comparisons with physical systems.


The motivation of the approach of this paper is justified as follows. Firstly, that programs built from simple fundamental rules produce emergent complex behaviour cite{CAintofQM}cite{EmergentCompexity}cite{IlachinskiBook}cite{NKS1}, implying that the emergent complexity of our universe may arise from fundamentally simple rules. Secondly, one may attempt to reconcile general relativity with quantum mechanics at very small scales by rigorously defining a dynamic discrete structure of space-time, cite{Manfred.2006}cite{Fay.2008}cite{Joe.2006}. When you combine these two concepts, within a reductionist viewpoint of physics, it leads to the established idea that our universe may operate as a simple discrete program at the microscopic scale and that its observed complex behaviour results from the simple fundamental rules underlying it cite{CAintofQM}cite{whywelive}cite{Seth.1.2002}cite{IlachinskiBook}. If space is discrete then the number of states the universe may occupy becomes finite and therefore theoretically computable cite{Seth.2013}. Therefore the aim of this paper is to implement a candidate program, and make observation of physically comparable behaviour.


The FCSS system consists of a network of causally related micro-states corresponding to complete weighted graphs that consists of two fundamental constituents, the extit{Point Particle (PP)} (the vertices or nodes of the graph), and the extit{Space Element (SE)} (the sole constituents of the edges or links of the graph, the edge weights are given by the number of SEs in an edge). Unlike similar models cite{Ilachinski1987}cite{RR.2015}cite{NR.1997} these fundamental constituents contain no internal states, and are therefore named. The system evolves in clock time, during each time step every constituent may undergo two possible substitutions corresponding to their respective global probability. The graph may be relaxed into an equilibrium configuration in $mathbb{R}^n$ by stress minimization, which minimizes the internal curvature of the graph in $mathbb{R}^n$. The macro-state of the system corresponds to the mean behaviour of all possible paths of causally related micro-states, this is depicted by expectation values and probability distributions.


The specific motivation for the structure and rules of the FCSS system is as follows.

Firstly that the structure has a direct analogy to that of space-time and particles, allowing for a direct and meaningful comparisons to be made with physical systems. Secondly, the rules themselves are very simple and symmetric, and yet allow for relative particle dynamics, and the potential emergence of complex behaviour.


Thirdly the system structure and rules result as a consequence of a reductionist philosophy concerning what can fundamentally exist. In a concise form the philosophy argues that all dimensions exist without a cause as a timeless infinitesimal point within zero dimensions (nothingness). The dimensions of this primary point may then have a lower, orthogonal set of dimensions contained within them that result in a secondary point within the first. This secondary point, by following a simple set of indeterministic rules outlined in this paper, may split and move, and as a result describe the proportions of the dimensions of the primary point, and the dynamics of the secondary point, or points, within it. Effectively creating a complex dynamic system from something that arguably does not require a cause.


Many mainstream approaches to describing a discrete space-time are much more direct,

cite{Joe.2006}cite{AJL.2006}cite{KMS1.2006}cite{BMRS.2006}, reconciling the inconsistencies between continuous space time geometry and quantum mechanics within established mathematical frameworks. The approach taken in this paper is somewhat the opposite, here we construct generalised dynamic topologies that may show signs of emergent physically comparable behaviour at some scale. Once such behaviour is observed then the system should be investigated in more detail and variations of the system devised and implemented.


The FCSS system varies from other similar Graph based physically comparable Cellular Automata cite{KlausS}cite{Ilachinski1987}cite{Manfred.1998} and Graph Re-writing systems cite{KMS2}cite{Grzegorz.1997}, in the sense that it evolves probabilistically by a set of global, as opposed to local, transition rules, and acts as a generalisation of these systems, with the added constraint of completeness of the state graph. This means that the FCSS system is therefore not formally a Cellular Automata cite{NKS2}. The potential of computation within the FCSS system correlates directly to the that of the Kolmorov-Uspenski machine cite{Uspenski1992}, rather than a classical Turing machines. These machines have been shown to be the computational mechanism by which biological substrates of Physium compute cite{Andy}, indicating that there is a tendency within natural systems towards this form of computation.


The results shown in this paper, although somewhat tentative, do, by comparison with physical systems, provide the foundations for a formal definition of energy and mass within the FCSS system. There are many developments and variations on the general FCSS system. These are not outlined in this paper however they provide basis for future investigations and will likely lead to further physical comparisons. This paper is to act both as a descri ptive introduction to the FCSS system, and by showing some of its emergent properties, promote further investigation into this system and others like it in the future.




Descri ption


The micro-state of the FCSS system is described by a complete edge weighted graph, called the extit{state graph}, $G=(V,E)$ over $mathbb{N}$, where $v_iin V$ is the vertex set and $s_{ij,t}in E$ is the edge set. In this system edges are referred to as extit{interaction edges} (IE), and their weight, given by the elements of the state matrix, $S_t={s_{ij}}_t$ , $s_{ij,t}in mathbb{N}$, describe their associated state at a given time.


Each IE evolves independently as a one dimensional stochastic substitution system cite{NKS2}. Therefore for all possible transitions there exists an associated transition probability $P(s_{ij,t_1}, s_{ij,t_2}, gamma)$, where gamma represents the set of input variables of the system that dictate the probabilities of the associated rules.


The system evolves probabilistically through a series of extit{time moments (TM)} connected by extit{time steps (TS)}, where the TMs contain the micro-states, and the TSs contain the transitions.



The Space Element (SE)

SEs are the sole constituents of the IE system. They have proportionate size and line up consecutively to form a one dimensional array, the number of SEs in the IE gives the state of the IE $s_{ij,t}$ (this is determined by the model for IE evolution). Each SE may undergo one of a set of two actions through each TS, with an associated constant global probability.

1. extit{Duplication}


egin{figure}[h]

includegraphics[scale=0.4]{SpaceElementDuplication}

centering

end{figure}


A SE that undergoes a duplication through one TS is substituted by two SEs in the following TM at the location of the duplicating SE, $1se ightarrow 2se$. The associated universal probability of duplication is given by $p_d$, $0


2. extit{Reduction}


egin{figure}[h]

includegraphics[scale=0.4]{SpaceElementReduction}

centering

end{figure}


A SE that undergoes a reduction through one TS is substituted by zero SEs in the following TM at the location of the reducing SE, $1se ightarrow 0se$. The associated universal probability of reduction is given by $p_r$, $0


egin{equation}

label{}

p_r+p_dleq 1

end{equation}

egin{equation}

label{}

p_r+p_d+p_0=1

end{equation}

Where $p_0$ is the associated probability of a SE performing no action though a TS.


The Point Particle (PP)

PPs are the vertices of the state graph, $v_iin V$, where $1leq v_ileq m$, and where $m$ represents the number of PPs in the system. They have no internal parameters however occupy locations within the system. For two PPs to be distinct they must be separated by an IE of at least one SE. Like SEs PPs may undergo one of a set of two actions through each TS.


1. extit{Splitting}


egin{figure}[h]

includegraphics[scale=0.4]{PointParticleSplitting}

centering

end{figure}


A PP that undergoes a split through one TS is replaced by two PPs with one SE separating them, $1pp ightarrow 2pp + 1se$. The associated global probability of splitting is given by $p_s$. Splitting increases the size of the vertex set of the state graph.


$$vert V vert=m ightarrow vert V vert=m+1$$


When a split occurs all the IEs incident on the PP that has split are extit{cloned}. If $N_e(v_i,t)$ defines the set of neighbouring edges of the vertex $v_i$ at time $t$, and $v_i equiv_t v_j$ means that two PPs were split from one PP in the previous TS, then if $v_i equiv_t v_j$, $N_e(v_i,t)=N_e(v_j,t)$. This makes sense, both from the point of view of conserving symmetry and completeness of the state graph. Furthermore, if $e$ is the number of IEs in the state graph when a single split occurs the size of the edge set is increased.


$$vert E vert=e ightarrow vert E vert =e+(m-1) $$


Note that splitting and SE duplication and reduction occur simultaneously, therefore all actions that occur within any cloning IEs will effect both IEs in that TS. After the TS where the PP splits, however, the IEs evolve completely independently.


2. extit{Merging}


egin{figure}[h]

includegraphics[scale=0.4]{PointParticleMerging}

centering

end{figure}


Unlike splitting, merging is not a constituent specific action and the associated probability is not global. Two PPs will merge if all SEs in the connecting IE reduce in the same TS. For this we say the corresponding IE has extit{fully reduced}. Therefore the associated probability is related to the number of SEs in the connecting IE. If $n$ is the number of SEs within a given IE, $s_{ij,t} in E $, and $v_i,v_jin V$ with $i eq j$ at time $t$, then the probability that $v_i$ and $v_j$ merge, from $t ightarrow t`=t+1$, is given by,


egin{equation}

label{}

p_m(v_i,v_j,t)=p_{r}^{s_{ij,t}}

end{equation}


and ${v_i,v_j}(t) ightarrow v_k(t`=t+1)$.


When a merge occurs between two PPs, $v_i$ and $v_j$, this has the effect of extit{joining} the corresponding IEs in the neighbouring IE sets $N_e(v_i)$ and $N_e(v_j)$. The joined IEs are given by the set of sets of extit{joined} IEs.

egin{equation}

label{}

M_e(v_i,v_j):=N_e(v_i)Join N_e(v_j)

end{equation}


egin{equation}

label{}

M_e(v_i,v_j)=igcuplimits_{kin V,k eq i,k eq j} {{s_{ikt},s_{kjt>

end{equation}


The elements in this set are then operated on by a join operation $phi:E ightarrow E$ such that ${s_{ikt},s_{kjt}} mapsto s_{ik(t+1)}$.


The method used for IE joining in this paper is extit{IE Averaging} and is defined as followsfootnote{Using multi-IEs is likely to be a more natural implementation for this system, however it would be too computationally intensive for this paper, and has not been used.}.


Pair IE states and average the values over $mathbb{N}$. We say $phi$ joins neighbouring sets by IE averaging. In this case $phi$ is defined as follows,


egin{equation}

label{}

phi(M_e(v_i,v_j)):=igcuplimits_{kin V,k eq i,k eq j} {lceilfrac{s_{ikt}+s_{kjt}}{2} ceil}

end{equation}


egin{equation}

label{}

N_e(v_i(t+1))=phi (M_e(v_i,v_j))

end{equation}


In the case where the sum of the two edge weights is odd the resultant average will be a fraction of two, since $s_{ijt}in mathbb{N}$ this is required to be rounded. A ceiling is used to conserve edge number. The situation of a floor being used need not be ruled out, and may be implemented in future implementations.




The Interaction Edge (IE) System}


IE systems form the edges of the state graph of the full system micro-state. The mechanism of their evolution determines the transition probabilities of corresponding edge weights, otherwise referred to as IE states $s_{ij,t}in mathbb{N}^l$ where $l=vert s_{ij,t}vert$, $S_t={s_{ij}}_t$ $P(s_{ij,t},s_{ij,t`},Delta t=t`-t)$ . In the case where the IE averaging model is employed $l=1$, since there is only ever one IE between two PPs..


There are various evolutionary mechanisms that could dictate our IEs evolution, however in this paper we describe and implement the most general case of extit{Instantanious Action (IA)}. The instantaneous action system takes the IE state to be the total number of SEs within the system at any given time. That is to say that the effect of the action (duplication or reduction) is instantaneous and affects the state of the IE in the same TS that in which the action occurred.


egin{figure}[H]

includegraphics[scale=0.7]{CAIAdistribution100}

caption{ extit{Positional probability distribution for varying $p_d$ and $p_r$, where $p_r=p_d$ after 1 TSs for 100000 runs. For this simulation the values of $p_d$ and $p_r$ have been taken to be equal. Here it can clearly be seen how the rate of actions increases the standard deviation of the positional probability distribution.}}

centering

end{figure}



egin{figure}[H]

includegraphics[scale=0.7]{EvolutionofIAsytem}

centering

caption{ extit{The evolution of the positional probability distribution of the state of the IA IE system through 20 TS. As you can see the spread of the distribution disperses as the system evolves from a single microstate. As the number of possible states increases with the number of TSs the distribution of the system begins to decay causing an increase in the standard deviation of the distribution and the range of possible acquired states of the system.}}

end{figure}


egin{figure}[H]

includegraphics[scale=0.84]{CAIAavevolution1}

centering

caption{ extit{Evolution of expectation value of $s_{ijt}$ through 25 TSs averaged over 10000 runs. Here the simple behaviour can clearly be seen, not that $p_r+p_d=0.1$ for all evolutions on this graph.}}

end{figure}


There is a qualatitive comparison that may be made at this point between the rate of duplication and reduction and energy. Figure 1 and 2 illustrate that as $p_d$ and $p_r$ increase, it is easier for the system to occupy more states, by analogy to statistical mechanics an increase in accessible states relates directly to an increase in energy of the physical system cite{TschoeglB.2000}. Figure 3 shows how the potential relative velocities and accelerations of the enclosing PP is directly related to both the magnitute of the rates of duplication and reduction and to the ratio between the two. For example an increase in $p_d$ causes an increase in the recessional acceleration of the enclosing PPs, the greater the number of duplications or reductions the further the enclosing PPs may be displaced in space from each other cite{ShapiroB.2003}footnote{Further developments on this system, such as the extit{Action Propagation} mechanism, yeald futher comparisons between the rate of action occurence and system energy, and result in a limit on the rate of growth of the system for $p_d>p_r$.}, providing a somewhat more qualitative comparison between these action rates and system energy.


Fully Connected Instantanious Action System (FCIA)


The FCIA system corresponds to a network of causally related, weighted complete graphs, where the evolution of each edge weight is determined by the IA IE system. The causal relations result from the transitions TS. If so there exists a corresponding transition probability.


In the FCIA system there is no information stored within the IE system besides the number of SEs, and therefore the complete state of a given micro-state is purely defined as the state matrix $S_t={s_{ij}}_t$.


Splitting increases the size of the state matrix by one, since the vertex set is increased by one where the IEs generated by the splitting are clones of the nearest neighbour IE incident on the splitting PP. IE averaging is implemented in the investigations into the FCIA system within this paper, but the FCIA is not constrained to just this IE evolutionary mechanism.


System Curvature and Embedding of the State Matrix into $mathbb{R}^n$


In order to make comparisons between the FCSS system and physical processes in space-time, we assign every PP $v_iin V$ an external state corresponding to it`s location within $mathbb{R}^n$, $v_i={x_0,x_1,...,x_{n-1}}$, where the IE weights correspond to the relative distances between PPs. By treating violations of Euclidean constraints by the state matrix as internal curvature within the system`s representation in $mathbb{R}^n$ and relating that curvature to internal stress on the IEs we embed our state matrix within $mathbb{R}^n$. By doing this a direct comparison can then be drawn between the observed emergent behaviour, and physical particle dynamics in nature, which is the main goal of this paper.


The approach implemented in this paper is built upon techniques employed in cite{YKS.2012}. Each IE when embedded has some stress value associated,

egin{equation}

label{}

phi_{ij}=||v_i-v_j||-s_{ij}

end{equation}

A natural embedding into $mathbb{R}^n$ would be one that minimizes $omega$ the sum of the squares of all the IEs within the system.

egin{equation}

label{}

omega=sum_{|V|geq i>j}phi_{ij}^2

end{equation}


egin{equation}

label{}

V={v_iin mathbb{R}^n|0leq ileq m}

end{equation}

Where $m=|V|$.


egin{equation}

label{}

V`={v_j`in mathbb{R}^n|0leq ileq m, min{omega(v_j`)}}

end{equation}


Therefore $V`$ corresponds to the configoration of the state matrix in $ mathbb{R}^n$ that minimizes the stress, and therefore minimizes the internal curvature of the state matrix.


To fix the orientation of plain $mathbb{R}^n$ we define the extit{coordinate orientation vertex set}, $Q={v_0,...,v_{n-1}}$ where $|Q|=n$ where $n$ is the dimension of the plane we are embedding into. In the general case $|Q|=n$ and $Q={v_0,v_1,...,v_{n-1}}$ and $v_i={x_{1i},x_{2i},...,x_{ii},0,...,0}$, where $x_{ij}$, $j


One ways to minimize $omega$, is by minimizing directly by use of specifically devised computational functions such as extit{NMinimize[]} in Mathematica. The drawback of this is that is sometimes leads to the system relaxing into a local minima which creates a very discontinued effect with the systems evolution.


The mechanism used in this paper is a specifically devised algorithm named the extit{Localized Edge Stress Minimization} algorithmfootnote{The LESM algorithm works iteratively. In each iteration each PP is shifted by a vector that is constructed by multiplying unit vectors, corresponding to each incedent IE, by the difference between the weight of the IE and the cartesian distance of that PP in the configoration at that iteration. The result is that the system slows to an effective halt at some localized minima of the system stress.}. This algorithm takes as its inputs the state matrix $S_t$ and the specifically chosen starting configuration $V$. After multiple iterations of the algorithm the system tends towards a configuration $V`$ that is a local minima of $omega$ resulting from the input of $V$. When implemented in system evolutions this allows for a feedback loop, where the $V`$ of $t_0=t$ is the $V$ of $t_1=t+1$. This resolves the problem of discontinuity of evolutions due to the embedding procedure by favouring local minima of the configuration before the last time step.


RESULTS


In this section we outline the core finding of this paper that areas of increased density of PPs, or PP clusters, have intrinsic inertia, that these clusters also exhibit a retentive effect on neighbouring PPs, and that certain configurations of the global parameters produce clusters of constant size and PP density. Both findings draw upon the comparison introduced in section 2.3 between rate of actions and internal system energy.


Intrinsic Inertia}


Figures 5,6,7 and 8, show a how a specifically constructed system responds to an instantaneous insertion of duplications affecting the state of one of the IEs incident on a PP within a PP cluster of increasing size. Here $Delta s=5$, and $s_{02}=s_{01}=s_{12}=10$. Here by PP cluster size we mean extit{PP number} $(n)$ of a PP cluster, and it is the number of PPs within the cluster. For all relations in this section all clusters are treated as extit{tightly bound}, all the PPs within the cluster are a proportionately small distance from all other PPs within the cluster. This can be approximated by a unit cluster (a cluster where all inter-PP distances are taken to be 1), as done in the following implementation.


egin{figure}[H]

includegraphics[scale=0.5]{IISEinsert1PP}

(a) hspace{2.7cm} (b)

caption{ extit{Effect of SE insertion on position of PP cluster. $s_{02} ightarrow s_{02}+5$, the resultant displacement within $mathbb{R}^n$ is $Delta d= 6.376 SE$. Here the PP cluster contains only extbf{1 PP}. The slight increase in distance is due to system stress cause by flat space violations of the system.}}

end{figure}


vspace{2cm}


egin{figure}[H]

includegraphics[scale=0.5]{IISEinsert2PP}

(a) hspace{2.7cm} (b)

caption{ extit{Effect of SE insertion on position of PP cluster. $s_{02} ightarrow s_{02}+5$, the resultant displacement within $mathbb{R}^n$ is $Delta d= 3.020 SE$. Here the PP cluster contains extbf{2 PPs}.}}

end{figure}


vspace{2cm}


egin{figure}[H]

includegraphics[scale=0.5]{IISEinsert4PP}

(a) hspace{2.7cm} (b)

caption{ extit{Effect of SE insertion on position of PP cluster. $s_{02} ightarrow s_{02}+5$, the resultant displacement within $mathbb{R}^n$ is $Delta d= 1.479 SE$. Here the PP cluster contains extbf{4 PPs}.}}

end{figure}


vspace{2cm}


egin{figure}[H]

includegraphics[scale=0.5]{IISEinsert20PP}

(a) hspace{2.7cm} (b)

caption{ extit{Effect of SE insertion on position of PP cluster. $s_{02} ightarrow s_{02}+5$, the resultant displacement within $mathbb{R}^n$ is $Delta d= 0.291 SE$. Here the PP cluster contains extbf{20 PPs}.}}

end{figure}


egin{figure}[H]

includegraphics[scale=0.9]{Displacementofclustervsenergyinput1}

caption{ extit{Displacement of PP cluster vs SE insertion for varying cluster PP numbers. This figure shows the direct inertial effect of increasing the PP number $n$ in the cluster by embedding the system into one dimension and plotting the effective displacement $Delta d$ of the PP cluster versus the SE insertion $Delta s$ between the origin of the system. }}

end{figure}


vspace{2cm}


egin{figure}[H]

includegraphics[scale=1]{ClusterPPnumbervsimpedimentratio}

caption{ extit{Cluster PP number vs Impedement Ratio. The relation between SE insertion into one IE and effecting PP cluster displacement is linear. The gradient of this linear relation, the extit{impediment ratio} $Delta s/Delta d$, is decreased by increasing the cluster PP number.}}

end{figure}


vspace{2cm}


egin{figure}[H]

includegraphics[scale=0.72]{InterPPPPclusterdistance1}

caption{ extit{This graph shows the relative distance $Delta x$ between the effected PP and the PP cluster excluding the effected PP. The location of the cluster is taken as the mean distance of all PPs within the cluster. Here it can be seen that increasing the cluster PP number reduces the relative displacement of the effected PP.}}

end{figure}




egin{figure}[H]

includegraphics[scale=0.92]{standdevvsclusterPPnumber}

centering

caption{ extit{Cluster PP number ($n$) vs Standard deviation of PP clusters positional probability density distribution (a). This figure shows the effect of increasing the cluster PP number $n$ on the standard deviation of the positional probability distribution of the average position of the PP cluster, $a$, in one dimension for varying levels of SE action activity under the IA assumption.}}

end{figure}


In this section we examined how the FCSS system operates when PP clusters form, under the assumption that the clusters are tightly packed. Three fundamental effects of increasing the cluster PP number are observed. The first two are linked and depicted by figures 9 and 10. SEs were inserted into one of the IEs incident on the PP cluster from outside the system. The PP that is connected to the chosen IE is then displaced, the amount that the PP is displaced is decreased as the cluster PP number is increased. In other words the PPs in the cluster have the effect of pulling the affected PP back, giving a retentive effect. This pulling along with the increase in the number of PPs determining position then has the effect of decreasing the average displacement of the PP cluster. Finally, figure 11 shows that an increase in the PP number of a cluster decreases the standard deviation, or spread, of the resulting positional probability distribution.


Equilibrium Cluster States


In all of the observations made in the previous section the assumption is made that the cluster is extit{tightly packed}. In this section we implement IA evolutions of the state matrix to show that it naturally relaxes into configurations that correspond to clusters of constant PP density and size.


We treat the complete system as a extit{Isolated PP Cluster}, this means that we examine the behaviour of a system that is so far away from any other PPs in the system that we can treat their effects as negligible. This means that we can run the complete system and treat it`s behaviour as an isolated cluster.


To keep track of the evolution of these systems two quantities of the system are measured. The extit{cluster PP number} $n$, the number of PPs contained within the cluster, and the extit{cluster scale factor} $q$, the average value of the IEs contained within the cluster. To observe massive equilibrium states what we are looking for are isolated PP systems where $n$ and $q$ tend to a constant value, where the size and mass of the PP cluster reaches an equilibrium.



egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd10pr12ni10}

caption{ extit{Evolution of an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.12$, the initial configurations are $n=10$ and $q=5$. As can be seen the system quickly tends towards an equilibrium state where $n$ and $q$ are constant.}}

end{figure}




egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum100ps10pd10pr12ni10ts250}

caption{ extit{Evolution of an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.12$, the initial configurations are $n=100$ and $q=5$. Here it can be seen that the equilibrium state of the system for this case is completely unaffected by the initial state of the PP cluster as the system with the same parameters tends to the same equilibrium configuration.}}

end{figure}


vspace{2.5cm}


egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd10pr11ni10ts500}

caption{ extit{Evolution of an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.11$, the initial configurations are $n=10$ and $q=5$. Similar behaviour can be seen, now the probability of reduction is reduced by 0.01.}}

end{figure}


vspace{2.5cm}


egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd10pr106ni10ts500}

caption{ extit{Evolution of the an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.106$. $n$ has not reached a stable equilibrium and the system is still growing, however $q$ appears to tend to a constant value.}}

end{figure}




egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd10pr102ni10ts250}

caption{ extit{Evolution of the an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.102$. As $p_r$ gets closer to the value for $p_d$ the rate of growth of $n$ increases while $q$ is now appears to still be sublinear.}}

end{figure}


vspace{2.5cm}


egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd10pr10ni10}

caption{ extit{Evolution of the an isolated PP cluster for $p_s=0.1$, $p_d=0.1$ and $p_r=0.1$. The growth of $n$ appears to be exponential and $q$ looks as if now it is increasing linearly.}}

end{figure}


vspace{2.5cm}


egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10pd20pr20ni10ts250}

caption{ extit{Evolution of the an isolated PP cluster for $p_s=0.1$, $p_d=0.2$ and $p_r=0.2$. The evolution of $n$ could be the start of an linear or exponential curve, while $q$ definitely appears to be increasing linearly.}}

end{figure}


egin{figure}[H]

includegraphics[scale=0.8]{finalevdiappnum10ps03pd10pr10ni10ts1000}

caption{ extit{Evolution of the an isolated PP cluster for $p_s=0.03$, $p_d=0.1$ and $p_r=0.1$. This graph shows what happens when we decrease $p_s$ but keep $p_d=p_r$, again we have $q$ increasing linearly although in this case the evolution of $n$ is indeterminable for the number of TSs in the evolution of this implementation.}}

end{figure}


From figures 12, 13 and 14 we are given the indication that for an isolated PP system stable equilibrium states are only reached when $p_r>p_d$, although it is possible that equilibrium states or temprorary equilibrium states may still exist for other global parameter values. It is also indicated from figures 15, 16 and 17 that as $p_r$ is decreased to $p_r=p_d$, $q$ tends towards increasing at a linear rate and $n$ tends towards increasing at an exponential rate.


PHYSICAL COMPARISONS AND CONCLUSION


In this paper we have implemented and investigated the time evolution of complete weighted graphs. The edge weights of these graphs evolve as stochastic substitution systems, and the vertex set varies via probabilistic splitting and merging rules. These systems were then embedded into $mathbb{R}^n$ via the process of extit{stress minimization}. By treating the rate of substitutions of the IEs as a definition of internal system energy, the embedding process resulted in an inertial effect due to clustering which is directly related to the number of PPs within the structure. Due to the process of PP splitting and merging the system produces stable clusters for certain values of the global system parameters $pr$, $pd$ and $ps$. These clusters stabilize to contain a constant average number of PPs and a constant average diameter. The embedding also had the effect of causing a retentive effect that was directly related to the number of PPs in the cluster.


The basis for physical comparison is that all motion, and energy transfer within the FCSS system is a consequence of changes in the proportions of the IEs corresponding to the reduction and duplication of SEs that alter the geometry of space and not some intrinsic quantity attributed to the PPs cite{RR.2015}cite{AJL.2006}.


The macro-state of the system evolves as probability amplitudes, yet can not be said to behave intrinsically quantum mechanically as no direct observation of wave behaviour has been observedfootnote{This behaviour, however may yet be observed in development specific developments on the IA IE evolution mechanism}. However in both classical physics and quantum physics alike the kinetic energy term, $K=p^2/2m$, of the Hamiltonian $H$ is an inverse function of a particles inertial mass cite{RicardB.2000}cite{ShapiroB.2003}. By building on the arguments proposed at the end of section 2.3, and combing that with the observed behaviour in 3.1 figure 11, we can see that localised particle clusters within this system have an intrinsic inertia and the size and density of these PP clusters may be treated as a measure of mass within the system.


Section 3.2 dealt with the emergence of clusters within the dynamically evolving macro-state of the system. Due to the global nature of the action probabilities and the complete graph structure of the system, behaviour in this system evolves somewhat homogeneously, therefore we were only able to obtain equilibrium states within the isolated PP cluster model, and only for $p_d


The results of this paper are very much preliminary and the results are still somewhat tentative, however the hope is to show that by using probabilistic substitution rules under global system parameters there is the basis for a toy universe model which may produce interesting dynamics, behaviour, and may even show physically comparable behaviour. Many further qualitative physical comparisons have been left out of this paper. Further natural developments of this system may still resolve obvious inconsistencies between this model and physical systems that may arise cite{S.Carlip.2012}, and lead to more direct physical comparisons.

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