By Paul Almond, 30 April 2006

Previous Articles in this Series

Reading of the following articles is suggested before reading this article:

Occam's Razor Part 1: What Is Occam's Razor? http://www.paul-almond.com/OccamsRazorPart01.htm. [1]

Occam's Razor Part 2: Principles of Language http://www.paul-almond.com/OccamsRazorPart02.htm. [2]

Occam's Razor Part 3: Assumptions About Reality http://www.paul-almond.com/OccamsRazorPart03.htm. [3]

Occam's Razor Part 4: An Overview of How Occam's Razor Works http://www.paul-almond.com/OccamsRazorPart04.htm. [4]

Occam's Razor Part 5: How Mapping Can Work http://www.paul-almond.com/OccamsRazorPart05.htm. [5]

Occam's Razor Part 6: Partial Models as Envelopes http://www.paul-almond.com/OccamsRazorPart06.htm. [6]

The following article, although not part of this series, is closely related to it and is a prequel to it:

What is a Low Level Language? http://www.paul-almond.com/WhatIsALowLevelLanguage.htm. [7]

Introduction

The previous article Occam's Razor Part 6: Partial Models as "Envelopes" [6] continued the process of deriving a formal version of Occam's razor. It was stated that the issue of representing models was important to this and would need to be dealt with adequately before further progress could be made.

It was proposed that partial models are best described by meaning extraction algorithms. A meaning extraction algorithm is an algorithm that operates on a matrix of observations and returns some result. This way of representing models did not include any concept of hierarchy.

This article will show how the previous concept of partial models as meaning extraction algorithms can be extended to allow construction of hierarchical models. There will also be an ontological discussion in which we relate this to the concept of "existence".

Previous Work

We have previously defined a partial model as follows:

Let B = an n-dimensional matrix of binary digits (bits) representing observations of reality that is generated by the actual model. B is the matrix of bits that we regard as being reality and which conforms to the assumption of algorithmic description.

Let i = an index, which is an n-dimensional vector giving a position in the n-dimensional matrix of bits B.

Let P = a sequence of bits provided as a parameter to any meaning extraction algorithm.

Let M(P,B,i) be a partial model - a meaning extraction algorithm - that accepts B as input data and returns some sequence of bits R corresponding to meaning that it has extracted.

That is to say:

R = M(P,B,i) for some partial model (meaning extraction algorithm) acting on an n-dimensional matrix of observations B.

The Need for Hierarchy

The model representation system proposed previously does not allow for hierarchies, yet hierarchy is an important characteristic of human models. Any modelling system that ignored hierarchy would not be able to express human worldviews.

There is a further reason for using hierarchy in a modelling system, and it is probably the reason for us using hierarchy. Computation is needed to generate partial models. Generating partial models described by long algorithms that relate to complex patterns in reality is likely to require a lot of computation and the task of generating such partial models is likely to be NP-hard, which for all practical purposes may as well mean "impossible" for reasonably long partial model algorithms.

The use of hierarchy gives us a way of "cheating" with the issue of NP-hardness. It is not perfect, but it allows us to extract some meaningful patterns that would require very long algorithms to express them. It does this by allowing us to extract simple patterns that are described by small algorithms and then to use these as "building blocks" to facilitate the task of extracting more complex patterns that would otherwise be described by longer algorithms.

Object-Like Partial Model Results

In the last article in this series [6] I described various kinds of partial model. All of these are basically the same kind of partial model, used in different ways and I described these various types to try to reduce the risk of the ontological system later being found to be too narrow in scope.

The concepts of physics-like and object-like partial models were discussed. It is mainly Boolean object-like partial models that will be relevant in this article and it should be assumed that this is what is meant when partial models are mentioned.

I will explain again what a Boolean object-like partial model is. This will involve some repetition of the previous article:

The term object-like will be used to describe a particular set of results returned by a particular partial model without the requirement of any interesting degree of physics-like behaviour. We will decide what set of results constitute the object-like behaviour for the algorithm and when the algorithm returns a result that is a member of that set then the result will be considered object-like. When the result is not within that set it will not be considered object-like - unless, of course, it happens to be part of a different set also declared to constitute object-like behaviour for the same partial model.

A particularly convenient application of this is likely to be with Boolean partial models that are not physics-like. One of the two possible returned values may be considered to constitute the entire set for object-like behaviour. This means that we may consider the returned result of "1" to mean that the result is object-like and a returned result of "0" to mean that the result is not object-like. I will refer to this as object detection.

As the index i for an algorithm is changed it may return results that are sometimes object-like and sometimes not object-like.

A simple way of thinking about a Boolean object-like partial model is as an algorithm that looks for a specific pattern in some array of "1"s and "0"s, returning a "1" when it finds the pattern and "0" or another value when it does not find it.

The partial model can be applied at different places in the array by altering its index value. The index value can be thought of as where the partial model is "looking" in the array, although it is not quite as simple as this: the index value is merely a coordinate offset and a partial model, with a particular index, could actually examine values anywhere in the array.

This article will be discussing the use of Boolean object-like partial models in a hierarchical way. The arrangement of "1"s and "0"s (or other values that indicate failure) generated by a partial model as it succeeds or fails in finding its pattern when applied with different indices can be used to form another matrix and this matrix can be acted on by another partial model, as if it is the "reality" from which that partial model is extracting meaning and in which it is detecting or failing to detect a pattern (an object) for different indices.

The Idea Behind Hierarchy

A partial model is supposed to act on reality to extract meaning from it. Reality is viewed as being a matrix of observations. The idea is that one "layer" of partial models can form a kind of "created" reality on which another layer of partial models can be used. This means that the partial models on the next layer can be shorter because, rather than operating on "raw" reality, they are operating on a reality which has been "partly digested" by the previous layer. These partial models do not need to finish the job. They can simply do some more of the work of pattern extraction to create a slightly more manageable version of reality which is provided to the next layer, and so on.

How it Works

R=M(P,B,i) is a partial model that acts on a matrix of observations B. We can take B as being direct observations of reality or a matrix of observations extracted by another partial model.

Any partial model R=M(P,B,i) implies a matrix of results. This is because one of the arguments taken by the partial model is the index i, a vector describing a position in the n-dimensional matrix B from which the partial model will extract its meaning. If we consider the partial model R=M(P,B,i) for every possible index i then we obtain a matrix - each element in the matrix being the meaning R extracted for a different index i.

We can express this slightly differently to make it obvious that M(P,B,i) implies a matrix:

R(i) = M(P,B,i)

where R(i) means "the value in matrix R at coordinates i".

We will now be discussing more than one partial model, so we will need to use some notation to differentiate between them. We will start with a partial model M₁ and we will say that it accepts an index i and a parameter P₁ and extracts its meaning from matrix B₁ and, by considering the extracted meanings for all indices, implies a matrix R₁:

R₁=M₁(P₁,B₁,i)

Expressing this differently to make it obvious that M₁(P₁,B₁,i) implies a matrix we have:

R₁(i)=M₁(P₁,B₁,i)

We will also define a second meaning extraction algorithm M₂ which accepts an index i and a parameter P₂ and extracts its meaning from a different matrix of observations B₂, by considering the extracted meanings for all indices, implies a matrix R₂:

R₂=M₂(P₂,B₂,i)

R₂(i)=M₂(P₂,B₂,i)

Any partial model, including M₂, implies a matrix and any partial model, including M₁ extracts meaning from a matrix, so we can use the matrix implied by the partial model M₂ to provide the "reality" on which the partial model M₁ operates and from which it extracts meaning:

R₁=M₁(P₁,R₂,i)

and expressing this differently to make it obvious that M₁(P₁,B₁,i) implies a matrix we have:

R₁(i)=M₁(P₁,R₂,i)

That is to say, the partial model M₁ is a meaning extraction algorithm which, instead of being applied to reality directly, is applied to the output from the partial model M₂. I should point out, however, that this idea that M₁ is not applied to reality directly can be considered from a different semantic point of view, which I will discuss shortly.

This ontology is probably not complete enough to describe anything in reality. It is only a first attempt and can be modified later: for now, I just want to establish the basic idea.

What this Means

Some readers may think of this in terms of one partial model calling another partial model as a subroutine. From a mechanistic point of view that is essentially what is happening: the meaning extraction algorithm M₁ can be considered to be calling the meaning extraction algorithm (partial model) M₂ to run with a particular index on a matrix of observations. This view, however, does not really express the idea of what is going on very well. Although M₂ passes information to M₁ this does not necessarily imply that M₁ is needed for M₂ to run. The idea is that the partial model M₁ implies an extra layer of information, regardless of whether any higher level partial model instructs it to run.

We can validly say that the partial model M₂ operates on reality while the partial model M₁ operates on something else. We can take this view as an outsider, looking at the entire system, but I prefer the view that, from the point of view of the individual algorithms, both M₁ and M₂ operate directly on reality. The only difference is that the reality in which M₁ exists happens to contain various simple elements that correspond to patterns that could be extracted by M₂.

This may seem to be philosophical hair-splitting, but it is important because it implies the possibility of an entire computer design philosophy in which we regard an artificially machine as being built in layers such that the way in which the "bottom" level interacts with reality is no different in any respect from the way in which each "higher" level interacts with the one below it.

Combining Multiple Partial Models

The hierarchy that has been described so far is limited in that a partial model can only act on information generated by a single "lower level" partial model. The ontology available to humans is more sophisticated than this: we can have a single partial model which extracts patterns from a number of different partial models. The hierarchy that has been proposed therefore needs to be extended to accommodate this.

R₁=M₁(P₁,B₁, B_2,…B_n,i)

Similarity with the Hawkins System

The kind of hierarchy being proposed here, and the problems that it would be expected to solve in artificial intelligence, shares its basic approach - extracting meaning in hierarchical stages - with the idea of pattern extraction proposed by Jeff Hawkins [8,9]. I will be developing some things differently however, with regard to the conceptual view of the system, the details of how models are made and how such a modelling system would contribute to an artificially intelligent system.

Hawkins has selected a neurological implementation of his system. I am not intending anything that specific yet, while keeping an open mind about how systems like this can be implemented - in particular with respect to the possibility of "mapping" the ontology onto some neurological system later.

Ontology

There are two ontological positions that we could take about hierarchy and I will refer to them as ontologically weak hierarchy and ontologically strong hierarchy.

Ontologically weak hierarchy is a view that hierarchical systems like the one being discussed here are useful tools for doing things such as describing reality, evaluating scientific theories and artificial intelligence, but that this should not be taken as far as actually making statements about the nature of reality. The idea of ontologically weak hierarchy is that "things" have some sort of "thingness" independent of any consideration of hierarchy and algorithms and that partial models can be used to reveal this "thingness".

Ontologically strong hierarchy also takes the view that hierarchical systems like this have practical uses, but it goes further. It suggests that things are not merely conveniently described by partial models: they only exist by virtue of corresponding meanings being extractable by partial models in a hierarchy.

I think that ontologically strong hierarchy is the correct position. Some readers may challenge me on this by asking me to prove that objects only have such a reality. I would reply by asking them for any evidence of anything beyond this or any coherent idea of what it means to "exist" beyond this. In the absence of any such evidence we are only justified in assuming what we must assume.

This distinction between strong and weak ontology may seem difficult for some readers, so I think it is best illustrated with an example as follows:

Atoms exist. Atoms come together in all kinds of patterns to make things. Let us suppose that we have one of these things - a car - in front of us. A car is just a pattern of atoms so, in principle, if we use the correct partial model on any observations of atoms it should be able to tell us if any cars are present. Let us suppose also that we want a partial model that corresponds to the existence of a car. We would have some matrix corresponding to observations of the atoms. The partial model would be applied to this matrix with some index value and would generate a result that either signified "detection" of a car or failure to detect one. Different indices could give different results.

An ontologically weak position would regard this as a useful technique for "finding cars". The matrix of observations of atoms from which the "car" partial model extracted its meaning would really be a matrix extracted by lower level partial models, and the hierarchy of partial models would also be useful for finding the atoms like this. With ontologically weak hierarchy, however, that is as far as it goes. The car and the atoms have a reality quite apart from these algorithms.

The ontologically weak position states that:

• Atoms exist.
• Cars exist.
• Partial models can be used to confirm the existence of atoms.
• Partial models can be used to confirm the existence of cars.

The ontologically strong position goes further. It does not concern itself with the first two of the above statements, simply including existence with the idea of partial models.

The ontologically strong position states that:

• Atoms exist by virtue of the fact that partial models corresponding to "atoms" can indicate such existence.
• Cars exist by virtue of the fact that partial models corresponding to "cars" can indicate such existence.

This may initially seem the same as the weak position. Is not the weak position also saying that something exists if it can be "detected" by a partial model? The difference is that the strong position defines existence in these terms.

One area in which this will make a decisive philosophical difference is in consideration of the existence of something which cannot be described by a partial model. In an ontologically weak position we view this as possible, even if the thing is forever beyond human experience (unless we use some more general modelling system). In an ontologically strong position, however, the idea of any sort of existence beyond the scope of partial models is not merely viewed as impossible, but actually as incoherent.

I should point out that the atoms and cars example that I just gave is simplistic in the sense that extraction of patterns in human brains or any artificially intelligent computers that we make is unlikely to proceed in such a simple, direct way. Humans do not use a single partial model to extract something as complicated as a car from atoms in one leap. In practice, many levels of hierarchy would be required to ascend from something like atoms to cars, each involving partial models described by short algorithms (or their neurological equivalents) that could be more easily found. Furthermore, although the hierarchies generated by human science contain the concept of "atoms" it is unlikely that the concept of "cars" is really extracted from the concept of atoms for most people: it is actually extracted from basic sensory data. None of this, however, really matters with regard to how the example was just used.

A Justification of Ontologically Strong Hierarchy

I think that a good justification for strong ontologically is provided by the following reasoning:

1. If we do employ such a hierarchy of pattern extraction to get our model of the world, then every object that we recognize is only recognized by virtue of a corresponding meaning being extracted by some equivalent of a partial model in our brains at some level in a hierarchy.
2. If we are going to talk about what it means to "exist" we should look at what all the things that we think exist have in common. All that we can know them to have in common is that they are all known to us only by virtue of a corresponding meaning being extracted by some equivalent of partial models in our brains at some level in hierarchy.
3. We do not know about all objects that exist. Some objects clearly exist that are not represented in the models in our brains. It would therefore be a kind of solipsistic fallacy to suggest that objects only exist by virtue of their meanings actually being extracted by partial models in our brains.
4. For an object to be said to exist, however, only makes sense if we can at least imagine knowing about it, which means representing it in our modelling system.
5. For a corresponding meaning to be extracted by a partial model and represented in a human brain it follows that there is the logical possibility of extraction of such meaning from lower levels of the hierarchy; that is to say, that an algorithm which extracts such meaning can be consistently described. This deals with the issues of things which we know to exist because they are represented in our brains and things that exist without our knowledge: both types of things can be said to exist by virtue of the logical possibility of constructing appropriate meaning extraction algorithms (partial models).
6. All objects that we know, or can know, to exist therefore share this characteristic of the logical possibility of the construction of an algorithm to extract corresponding meaning from a lower level of the hierarchy that already exists (being implied by the logical possibility of construction of algorithms to extract corresponding meanings for that level from a still lower level of the hierarchy). It is therefore reasonable to define this characteristic as "existence".

Do things exist because we think they do?

The short answer is: "No", but I will go into more detail.

This is a possible misunderstanding that could easily follow from what I have said. I have suggested that things exist only by virtue of their patterns being "findable" by the meaning extraction algorithms corresponding to partial models in a hierarchy. Some readers may think that this means that the process of observing something and thinking that it exists must involve the application of partial models which cause things to exist. This would be similar to the popular idea, which is flawed in my opinion, that observation by a conscious entity has some central role to play in quantum mechanics.

This is not what I have said at all. Partial models used in a brain or a computer may enable that brain or computer to recognize something that exists but I do not make any suggestion that existence is contingent on the actual application of a partial model's algorithm, or something equivalent to it, by a physical system such as a brain.

The idea of the ontologically strong position is that a thing exists because the capability for a partial model to extract a corresponding meaning exists in a hierarchy of other partial models that have the capability to extract their corresponding meanings. Whether or not any physical system actually implements these partial models is immaterial to the existence of things. The mere logical possibility of stating a partial model which detects the existence of a thing at some level in a hierarchy implies the existence of a thing and in the ontologically strong position is the existence of that thing.

As a mathematical analogy, we think that 2+2=4. A human brain can assert this, or maybe we could make a machine assert this in some way. Whatever arrangement of neurons or brain cells does this asserting is implementing this equality, but do we seriously think that 2+2=4 because we think about it? Of course not! We think that 2+2=4 not because of implementations of this equality but because of some logically implied equality - which means that there is the logical possibility of making implementations of it in brains or computers. Similarly, there are two sorts of partial models and parts of the hierarchy: those that are logically implied and those that are implemented in some system such as a brain or a computer. Things exist because their partial models in the hierarchy are logically implied, which means that they could be physically implemented in things like brains or computers, irrespective of whether or not this implementation takes place.

Does this mean that things do not really exist?

It may appear that I am saying that we delude ourselves that things exist, when in fact they are just implied by partial models in hierarchies. I am not saying this, however. Saying things exist because they are implied by partial models does not mean that their existence is fake: it is simply saying what existence means.

Going to an Ensemble Position

In a hierarchy, partial models provide an extra layer of reality from which other partial models can extract meaning - corresponding to the existence of things. This means that one or more partial models imply further partial models "higher up" in the hierarchy - all those partial models that can be applied to the meaning that they extract to extract meaning, and all the partial models which can be used to extract meaning from these partial models, and so on.

There are actually an infinite number of partial models which can be applied to the meanings extracted by any partial model or group of partial models to extract meaning and this means that each partial model implies an infinite number of partial models that can directly extract meaning from its own meaning. Each of these partial models in turn implies an infinite number of partial models that directly extract meaning from its own meaning, and so on.

This has become an ensemble view of reality, in which anything that can be logically described by a partial model will exist. We are supposed to be considering Occam's razor, although we have had to get involved in the issue of model representation. If such an ensemble view contains everything how can it make sense to talk about different worldviews and theories and which ones are most likely to be "true"?

One way of dealing with this is to take a "commonsense" approach. Although an infinite number of partial models are available at each level in the hierarchy, this is only because conceptually, there is no limit on the lengths of the algorithms that describe partial models: only a small subset of these will be short algorithms. If we take the view that a partial model can only be part of a "real" description of reality if its associated partial models do not exceed a certain length then almost all partial models in the hierarchy disappear. If we start from simple matrix from which will be extracting our hierarchy of meaning then there will now be a finite number of partial models that can exist for each layer in the hierarchy. The hierarchy is infinite if we go up an infinite number of layers, but only a finite "slice" through this hierarchy is now permitted and it could be argued that this disposes of "silly" partial models and allows something like a sensible worldview.

The problem with such a commonsense approach is that there is no justification for excluding algorithms beyond a certain length apart from convenience. The decision about what length of algorithm to permit is arbitrary and based on nothing more than what would seem to give an acceptable worldview. This, however, should not mean that we write it off totally: it can give an idea about what is involved in a "reasonable" model and, even if the philosophical grounds for an arbitrary maximum algorithm length are poor, such an approach may be of use in practical implementations of Occam's razor. In fact, limits on computing power would force the use of such an approach, or something like it, in a practical situation.

If we need something better than a commonsense approach what is there? The obvious answer is measure. The idea of measure is already well established in the area of ensemble computational models and also with the many-worlds interpretation of quantum mechanics. Using measure would involve looking at the frequency with which things occur in the hierarchy and linking this to our idea of probability.

"Frequency of what?" is an obvious question. This does not mean the frequency of partial models. A partial model is merely an algorithm that extracts some meaning. The measure of a partial model would be the frequency with which a partial model of that type, applied to different parts of the hierarchy and with different meanings, extracted a given result signifying the existence of something.

We can, and should, take this further. Any idea of probability should take account of an observer's particular situation. As an example, if someone shows me a box and asks me to guess what is in it how would I determine the probability that a football is in the box? If I just used the measure with which partial models that "find" footballs actually find them anywhere in the hierarchy, and look at this measure relative to the measure with which other things are found, taking this as an indication of probability I would be mistaken. Quite apart from the fact that we will find an infinite number of footballs in the hierarchy, none of this would deal with my own context - the fact that I want to know the chances of finding a football in the box that is in front of me right now.

The solution would be to look at all the partial models describing situations in which a person could be. All of the partial models that do not describe your situation could then be eliminated, leaving only those that are consistent with your experiences. Your probability of finding a football in the box should be taken as the probability that the partial model that correctly describes your situation happens to be one of the possible partial models that describes the box as containing a football.

These partial models would be based on meaning extracted from lower level partial models and your probability of finding a football in the box would be based on the measure of partial models which have their description of what is in the box based on extraction of information from a "football" partial model, relative to the measure of partial models in which the description of what is in the box based on other things.

Ensemble models tend to go even further than this, regarding the observer him/herself as being embedded in the ensemble. In the context of what is being discussed here this means that partial models would emerge from the hierarchy that describe the observer's own existence, as well as his/her situation and measure, from the point of view of an observer, would involve a counting of these partial models and comparing the numbers of them which have various features.

Non-Existent Things

If a Boolean object-like partial model "finds" an object for a particular index value it may respond by returning "1", but what about the times it does not return "1"? Some readers may ask why we should make the arbitrary decision that "1" means a successful object detection.

We do not have to use "1" for successful object detection. These are just arbitrary symbols. If we wanted we could do it the other way round and use "0" for successful detection - or we could use some completely different symbols.

So why should we be able to select any particular symbol as corresponding to successful object detection? The answer is that we are defining a conceptual structure to find objects and we can arbitrarily choose any conventions that we want. This, however, does not solve another problem: for any algorithm which outputs a particular symbol when a particular pattern is found, we should be able to design an algorithm that outputs that same symbol when the pattern is not found.

As an example:

Let us suppose that we have a "car detection" partial model. In any real system like a brain or a computer such a partial model would never work on just one level of the hierarchy: the partial model would only be able to detect something as complicated as a car by using lots of meaning fed up the hierarchy by lower level partial models. We do not need to worry about this, though. Let us suppose that the partial model returns a "1" when it finds a pattern that we would call a "car" and some other value otherwise. We should easily be able to make another algorithm that does not return "1" when it finds a "car" pattern, and which does return "1" whenever it does not find a "car" pattern: such an algorithm would merely be our car detection partial model with its output switched round.

We cannot use any arguments about convention or the right to choose whatever symbols we choose now: this null object partial model is obeying the rules. We are faced with a partial model that seems to be detecting the absence of an object in the same way as other algorithms detect objects. Why should we be able to say that this is not really an object?

We have to accept this situation at face value. There is no reason why we can eliminate the partial model which returns an object detection in the absence of a car: we have to accept that, by the standards we are using, it is detecting an object that is just as real as the car.

We may as well give this object a name. The object exists wherever a car does not exist, so I will call this object "null-car". We can make partial models that detect cars and other partial models that detect null-cars - whatever notation conventions we use - and there is no way of saying that a null-car is not a valid object.

Why do we not recognize a null-car as a valid object? The answer is that it is of no practical use. The information that null-cars are everywhere does not help us fill in patterns, make predictions about the world or plan our actions. Null-cars may be everywhere, but the fact that they are everywhere, and that the description of a null-car includes almost anything, makes them trivial and worthless.

This can be said about any other object that we recognize. For any object that we recognize it should be easy to define another object in terms of absence of that object. There is no good reason why such objects are less real than others, but such objects are trivial. Our brains restrict those objects that they consider "real" to those which have some predictive value.

It could be suggested that we define "real" in terms of lack of such triviality and eliminate trivial objects from the class of "real" objects because of such triviality. This could be difficult and it may be hard to decide where to draw the line.

Another way of considering this "null object" issue is to regard the partial model for an object and the partial model for the absence of the object as the same algorithm stated differently.

Removal of the Need for the Sequence of Observations

In previous articles I have assumed that observations of reality over time can be represented as a sequence of bits. If we were implementing a hierarchical modelling system like this in some real system, such as an artificially intelligent computer, then this concept could be useful to us. The first layer of partial models would need to extract meaning from something and it would have to be from the sensory inputs that the system received. This would basically be the "sequence of observations". Although it could, in principle, be represented as a sequence it may be that some more sophisticated representation is more useful - for example, observations made using a camera may be best represented as a two-dimensional matrix of bits for each instant of time, giving a three-dimensional array to describe the input from a camera over a period of time. Such issues need not concern us now, however.

There is another way in which we can consider this hierarchy, however, and that is when we are looking at deeper ontological issues and considering the entire hierarchy that gives rise to the sequence of observations that an observer experiences and the observer him/herself. If the hierarchy always has to be based on some sequence of observations from which partial models extract meaning then this would suggest that there is some "real" sequence of observations right at the bottom of the hierarchy - that there is something that corresponds to "raw" or "basic" reality before any partial models have been applied to it and any meaning extracted. To some readers this "raw" reality would be real and what was extracted by partial models would be "less real". There would be an obvious implication here: things get less real as they are generated by partial models further up the hierarchy.

Now that we have moved on to an ensemble approach we should ask, however, if we really need to assume this "basic" level of reality. There are two arguments against retaining it:

If we were to assume that some bottom level was needed we could ask what it would need to contain. Even if there is a huge amount of information at higher levels in the hierarchy, it can be generated at higher levels in the hierarchy. Suppose that the bottom level did not contain anything at all - that it effectively did not exist - so that it were impossible for a partial model to read any data from it for any index value? This would not be an obstacle to partial models "extracting" meaning for the next level in the hierarchy: a partial model is not obliged to make any particular use of the information in the reality below it and it can generate a result without reading any of the data in the level below.

An infinite number of partial models are actually possible, each building a matrix of meanings for the level above, without any reference to the level below. Higher levels in the hierarchy would then be based on meaning extracted from this new level. This may seem a strange situation, because it means that the meaning that we extract need not have anything to do with the "bottom" level and the bottom level is not even required to contain any information. If we think of the partial models as being things that extract meaning from "the universe" then, in a way, the partial models in this example are capable of extracting meaning, and building a full hierarchy, from a universe that consists of just a single bit or does not even need to exist! It is quite literally possible to algorithmically analyse a single bit, or even nothing, and obtain as much meaning as you want from it, the meanings getting more and more complex at higher levels so that it ceases to be obvious that the basis is nothing. If there is a bottom level containing a sequence of binary digits then it is now of very much reduced importance anyway, because the meanings that are extracted many levels further up in the hierarchy may not have much to do with what is in the bottom level, even allowing for partial models reading this bottom layer. The statistics of the partial models at each level in the hierarchy are likely to have more influence.

This kind of situation should persuade us that assumption of some bottom level of the hierarchy which contains a sequence of basic "observations" is not necessary, although, as I said, the idea could still be useful to us in practical situations.

The second argument against the need to presume some bottom level of reality is that, just as any level of the hierarchy implies everything in the level above it, then any level of the hierarchy can in turn be the result of extraction of meaning from a lower level. This could be taken as meaning that any level of the hierarchy also implies the existence of a level underneath it from which its own meaning was extracted. If we take such a position then we would not actually need to worry about what happens on the bottom level as the hierarchy is infinitely deep.

There are then two ways of disposing of the need to assume anything outside the hierarchy: a hierarchy which is generated from nothing because there is a layer of partial models that do not reference any later below and an infinite hierarchy. The first of these will actually occur as a special case in an infinite hierarchy: in an infinitely deep hierarchy some contributions will be made by partial models that make no reference to any lower levels.

It should also be noted that an infinite hierarchy will involve infinite repetition. Partial models can imply identical layers of extracted meaning at many different places in the hierarchy and, for any two such identical layers, all of the other layers implied by these would also be identical. This would mean that the entire hierarchy is generated many times within itself. This has similarities with an infinite ensemble proposed by Schmidhuber [10].

It should be noted that, although I have talked about how much information exists at higher levels in the hierarchy, this would only make sense in a localized way. The total amount of information in a hierarchy like the one I described would be zero. An ensemble should be reasonably expected to contain zero information, a point made by Russell Standish [11].

Removing the Need for the Assumption of Algorithmic Description

At the start of this series of articles I described our view of reality as being a sequence of observations - a sequence of bits - produced by an algorithm - the actual algorithm. I then discussed computational modelling of this algorithm.

This involves a philosophical assumption, the assumption of algorithmic description, which I stated previously and will now restate:

The assumption of algorithmic description

If we consider all the possible algorithmically expressed descriptions models of reality that could be made by following the principles in Occam's Razor Part 2: Principles of Language [2], one of these is correct. We may not know which algorithm is correct, but there is a correct algorithm out there somewhere that describes reality.

This may seem a trivial assumption to some readers: after all we can clearly make algorithms that describe reality, but there is rather more to the assumption. It presumes that we have to explicitly assume that reality will behave as if it is being generated by a Turing machine. I have not said much about this assumption previously, but now that it needs dealing with we will need to consider it further. What is being assumed here is causality. I could have called this assumption the assumption of causality.

To see why this is an assumption we should imagine that the universe has time but lacks any causality. Predicting the future in such a universe would be impossible, even statistically, because past observations need have nothing to do with what is going to happen in the future.

Occam's razor would not help us in such a situation because it is based simply on statistics. A big part of this was previously stated as being the principle of statistical impartiality with respect to algorithms:

The principle of statistical impartiality with respect to algorithms

Any particular consistent model is as likely as any other to be the true representation of reality in the absence of any knowledge about reality that contradicts this.

It is this principle that makes any algorithm as likely as other, yet means that certain features of algorithms will be more likely than others as they will have greater measure across the set of all algorithms.

If, however, we abandoned the assumption of algorithmic description, the principle of statistical impartiality with respect to algorithms would be different. It would now make no sense to treat modelling algorithms impartially. It would only make sense to treat sequences of observations impartially. This would mean that any sequence of observations is as likely as another - equivalent to the assumption that in the absence of causality any universe is as likely as any other. There would now be no reason for the universe to obey past patterns and not go "off the rails". The objection "There are more ways for the universe to be sensible than weird" could carry little weight if causality is not assumed and if any universe is as likely as any other. There may be fewer algorithmic descriptions of universes that do not behave sensibly, but if causality is abandoned this is not how we are making our statistical sample.

Whether or not we make the assumption of algorithmic description, therefore, is not a trivial assumption. It has implications for how we expect reality to behave.

The assumption of algorithmic description is essentially a statistical assumption because it dictates how our set of possible realities is put together. It does this by working in conjunction with the principle of statistical impartiality with respect to algorithms. The principle of statistical impartiality says that any member of a given set is as good as any other as a possible candidate for reality and the assumption of algorithmic description defines how this set is put together and what constitutes a single member of this set - an algorithm which generates observations of reality.

Why did we just go through all this? It was necessary to establish that the assumption of algorithmic description really did involve assuming something, rather than just being trivial or self-evidently true.

All this is about to change now. The assumption of algorithmic description viewed a description of reality as an algorithm that generates a sequence of observations. The idea of some basic sequence of observations - some true bottom level of reality - has just been discarded and this means that the assumption of algorithmic description is not merely no longer assumed: as a basic assumption it is now incoherent!

The infinite hierarchy that was discussed previously does not feature the assumption of algorithmic description as a basic assumption. The assumption of algorithmic description is also equivalent to the assumption of causality and it is from causality that any idea of time must come. This all means that:

• There is no assumption of any bottom level of "reality" or sequence of observations on which all modelling is based.
• There is now no basic assumption of the assumption of algorithmic description.
• There is now no basic assumption of the causality.
• There is now no basic assumption of time.

This will be apparent from a short consideration of the hierarchical system of partial models that was earlier described: it is atemporal, meaning that its definition does not relate to time.

Surely, however, we do experience time and we do observe causality! Such experience may seem to contradict what I have just said. It may seem that I have said that time and causality do not exist, but I have merely said that the ontological view here does not assume them as basic things.

We accept that tables, chairs and people exist as well, yet neither are these things assumed to have any basic existence in the hierarchy of meaning. They exist because they are implied by the logical possibility for partial models to extract corresponding meanings over a number of levels in the hierarchy. Tables, chairs and people are emergent properties of the hierarchy.

We take the same view of causality and time. Rather than viewing them as part of the "framework" of reality we can view them as emergent properties of the hierarchy. The logical capability of making hierarchies of partial models that describe things implies them and makes them real and this is how time exists. It is not necessary for a description of a sequence of events to actually rely on some fundamental idea of time beyond the description in which the events occur.

This may seem strange, but it should really be quite natural to us. Examples of descriptions of events in time that do not rely on fundamental time are common in human society. The description of a moving picture film is expressed as a sequence of pictures, yet that description contains a description of events occurring in time within itself. A novel can describe a flow of events, yet those events are not set out in some time outside the story. The description of the story contains a description of some flow of time in which events occur. Most people would regard a moving picture film or a book as existing within the "real" time of the "real" world but in an infinite of hierarchy of meanings extracted by partial models even this "real" time would itself merely exist as part of the meaning that is implied by the logical possibility of it being extracted by some algorithm.

It may seem strange for us to treat time like this, but it is actually a natural thing to do because it is consistent with the way that everything else is viewed when we use a hierarchy of meaning. It will have been clear for some time that space is not being viewed as anything fundamental in the sorts of ontological systems being considered here and we are merely dealing with time in the same way.

This also means, incidentally, that time is probably not a really important feature of reality. Most patterns that can be extracted probably lack anything that can be described as temporal. Thinking beings like us will never see those because we are temporal and can only inhabit any tiny temporal "islands" in the hierarchy of meanings.

The idea that time is not fundamental is not new: a timeless view of reality is proposed by Julian Barbour [12].

So, time is not real?

I am not saying that time is not real. Rather, it depends on the semantics of the word "real". I am saying that time is not a fundamental idea. This puts it in the same category of tables, chairs and people - all things that exist by virtue of their patterns being logically extractable from fundamental physics (if you want to take a conventional view of things) or from other meanings in the hierarchy (if you want to take the sort of view that is being proposed here).

The best way of summarising this is:

• Time is not part of any "framework" in which things exist. Time is a thing.
• All things exist by virtue of it being logically extractable to extract meanings corresponding to those things from other things.
• Is time real? It depends on whether or not "things" are real.

I prefer to take the view that things are "real". Any word should allow us to differentiate between entities to which it applies and entities to which it does not apply. If we define the word "real" in such a way that nothing can be real, then the word becomes useless and we lose a perfectly good word from our language. Every real thing, however, is contingent on logical implication from other real things.

Is the hierarchy more "real" than the things in it?

The hierarchy of partial models does not really "exist" in the sense that things extractable by partial models in it exist. This may sound strange because it is suggesting that my car exists by virtue of its meaning being extractable in the hierarchy, but the hierarchy does not exist. Am I really saying that the hierarchy is not real, but things inside are real? I am not really proposing such an idea. It comes down to the meaning of the word "exist". "Existence" has been defined now in terms of the logical capability to extract meaning from other meanings. This relies on the hierarchy to do it and, by definition, the hierarchy itself cannot be produced in this way.

Some readers may argue that, because the hierarchy is infinite, then the entire hierarchy can be recreated in many places within itself from a single partial model and that, therefore, the hierarchy does exist as one of the things within it. The problem with this objection is that the definitions in use here only regard things as real when they are implied by partial models. Even if the entire hierarchy reoccurs within itself it will make all the things implied in the hierarchy real. It will not make the partial models themselves real. A partial model simply does not have any reality, although, of course, some real system can be made which models a partial model.

None of what I have said is a claim that this hierarchy is some "fundamental" reality or the true "reality". There is no "fundamental" reality. As for it being the "true" reality, the true reality is simply all the things implied by the partial models within the hierarchy. It might be said that, while the hierarchy of partial models is not real, the hierarchy of meanings or "things" extracted by them is real, but really it is just the individual things extracted in the hierarchy that are real.

The hierarchy of partial models itself is simply a conceptual structure used to show how things are implied. A partial model does not have any reality.

What is doing the implying?

I have said that things exist because they are implied by partial models. I have also said that partial models are not real. Some readers may wonder what is doing the implying.

Readers asking this question probably image the hierarchy as some big calculation machine, superior to everything else in reality, that is supposed to bring things into existence by computing corresponding meanings and then computing meanings from these meanings and so on. What I have just said though, about the non-reality of partial models, should make it clear that I am not proposing that this at all.

Nothing is doing the implying. Nothing is "running" the partial models. Things exist because they are implied by the logical possibility of partial models extracting their corresponding meanings, not because some cosmic machine happens to be "running" partial models. It would be incoherent to talk about the existence of such a machine in any case, because existence is only a concept that has any meaning within the hierarchy.

Some readers will still want to know what is running the partial models and my answer of "nothing" will not be persuasive. To such readers I would point out that our current view of the world rests on what is, basically, the same idea. We think that atoms exist and we think that things like tables exist that are made out of atoms. Few people would claim that tables are somehow a delusion: we regard them as being "real" - as "existing" - yet at the same time we know that existence to be contingent on the arrangement of atoms in a certain way. When we say that atoms are arranged in a certain way, however, this only means that the atoms are arranged in a certain pattern - the arrangement of atoms obeys a certain rule - which corresponds to what we call a "table". Saying that there is a pattern is equivalent to saying that there is the logical capability to "extract" that pattern - which means that there is the logical capability for some algorithm to find it - because there is no situation in which a pattern could exist that cannot be found by an algorithm and there is no situation in which an algorithm could find a pattern where none exists.

Given all this, for any readers who want to know what is doing the implying in this article, I have this question:

What is it that causes a table to exist, given a suitable arrangement of atoms?

Most people would regard this as a useless question and this is my point: we are already taking the view that tables are implied by the logical capability of obtaining a corresponding meaning from the distribution of atoms. We do not think that anything has to imply tables or actually do anything to make a particular arrangement of atoms cause a table. Tables are simply implied by the existence of a certain type of pattern.

If I were wrong on this we would be in a strange position, because our experience of things is based on the ability of our brains to extract patterns and we would have to question whether those patterns meant anything at all. People cannot have it both ways: either existence is based on the logical capability for meanings to be extracted or we cannot use the ability of a brain to extract such meanings as evidence for anything.

All that we are doing, with a hierarchical system of partial models like that discussed in this article, is explicitly acknowledging and formalizing a position that we already informally accept and taking it to its logical conclusion.

If time is not fundamental when is the hierarchy?

If time is a feature of patterns in the hierarchy then some readers may ask how the hierarchy itself exists. Does the hierarchy exist at a single moment of time? Does the hierarchy exist in a single instant while the emergence of meanings describing temporal patterns deceives us into thinking that all of a human lifetime, instead of occurring all in the same instant, is spread out over many instants?

None of these questions make much sense. As has already been stated, a hierarchy of partial models does not really exist: it is just a conceptual device. Someone may point out, however, that there is the hierarchy of all the things that are supposed to be implied by these partial models. When do these things exist?

Some of these things of course, will exist in time. All the things around us are temporal because our small corner of the hierarchy of extractable meanings features time. They are only temporal, however, as a feature of partial models to which we have given the label "temporal", not in any "fundamental" sense. In any case, there will be lots of other things that are not temporal, time itself probably being an example (though there is nothing in principle to prevent our time from existing within some other temporal pattern, as Neo finds out in a rather extreme way in the film The Matrix). Some readers may then wonder when anything "outside" time can actually exist.

The answer is that there is no "when". "Time" and "when" are provincial concepts, only relevant within a temporal pattern and in such a situation any time which is mentioned is just another feature of the pattern.

Some readers may object that the very use of language itself is an admission of fundamental time. If I say "X exists atemporally" am I not using the present tense of the verb "to exist"? Even if I stop using the verb "to exist" and simply say "X is atemporal" I am using the present tense of the verb "to be". When we say something "is" we mean that it "is" now. This may be why people tend to think of atemporal ontologies like this as suggesting that everything exists in the same instant, which is not the case. There is no "instant" in which everything exists because the concept is incoherent for things outside the scope of time.

The limitations of human language could cause such misunderstanding of what is being discussed here. The problem is that all human verbs relate to time in some way - whether it is the past, present or future. When discussing atemporal situations we lack suitable vocabulary and usually end up using the present tense by default.

This does not just happen in situations like this. Mathematics, when expressed in casual human language is full of inappropriate temporal language; for example:

Two plus two equals four.

When is the "equalling" happening in the above statement? The present tense of the verb "to equal" was used and this means that the equalling is happening "now", but there is not really any temporal aspect to this statement. Of course, it can relate to temporal situations, but its true meaning is atemporal. Some people may take the statement as meaning that two plus two "always" equals four, as if this somehow justifies the use of temporal language, but the statement is really describing a relationship that has nothing to do with time.

The mathematical version of this statement does not have the same problem:

2+2=4

Does "=" mean "= in the past", "= in the future" or "equal now"? Some people would say that it means "always equals" but that is really just an attempt to translate the equation into informal human language, which is unsuited for expressing things like this. The "=" sign has nothing to do with time.

It may be this unsuitability of conventional human language - its inability to express things without reference to time - that causes this problem. If we only have temporal language for talking about the world then anything we say or read must be expressed in temporal terms. Even statements expressed in formal languages like mathematics are likely to be perceived temporally by many people because of the need to convert them into temporal language when translating them into casual terms. Even now, in an article like this in which I am talking directly about the subject I cannot say, using conventional language, what a statement like "2+2=4" really means. I could say "It means that two plus two is equal to four in an atemporal sense" but even that uses the present tense of a verb and wrongly injects time into the relationship. The best I could do might be to say "It implies an atemporal relationship of equality between two plus two and four" but even in this case I have been forced to use the present tense of the verb "to imply" - implying that this implying actually happens in time. All I can do is restrict myself to mathematical notation - which would hardly help in a discussion like this - or talk around the issue, which is what I am doing!

If you get the idea of what this problem is then talking about an atemporal situation will feel frustrating in the same way as a game called Taboo in which you have to describe something without using certain words. In Taboo the difficulty comes from certain words on a list of banned words. In this situation the difficulty comes from certain words being effectively banned by virtue of not existing.

If you still need an idea of how this situation can all be caused by deficiencies in a language, try imagining that you are banned from using a particular tense - past, present, or future - and that you have to express things for which you would normally use that tense in one of the other tenses. For example, you may be banned from using the future tense and you would have to do something else instead - such as using the present tense - to talk about what you were going to do. You could still express yourself, but only by contorting language, and you could not express yourself properly. If you used the present tense you would be acting as if the future was some part of the present. Maybe you can even imagine meeting someone who does use the present tense to describe future situations and sees nothing wrong with this! He/she may ask you to explain this mysterious "future tense" by using the other tenses and you would find this difficult. This is what is going on when we make the mistake of thinking that language must be temporal in nature or when we make the mistake of expecting someone to describe an atemporal situation, or the meaning of an atemporal tense, in temporal terms.

If we really wanted we could deal with this situation by making up a special time-free or atemporal tense for verbs in casual human language. When it was used it would not imply anything happening in time at all. As a crude example we could put the suffix "zog" onto the infinitive of the verb to get our atemporal form.

e.g. two plus two equalzog four.

and "When is the equalzogging happening?" would be an incoherent question.

Most verbs would, of course, not be usable in an atemporal context because they implicitly assume a temporal situation; for example, it would not make sense to talk about atemporal "kicking" or "pushing" as these words relate to events that happen in space and time. Some readers may feel that the very concept of a verb is temporal in nature and that the idea of an atemporal tense is incoherent. This is semantics. We can either have an atemporal tense and allow some verbs to have an extended atemporal meaning, or we can regard verbs as being temporal and have a new class of words that include both the temporal and atemporal: verbs would be included in this class as a special case. The approach that we take does not really matter - particularly as I am just making a philosophical point and not really saying that we need to modify human language!

There is nothing wrong with temporal language when we use it for things in that part of reality which we experience. That part of reality is temporal. We should just not deceive ourselves that there is anything fundamental about this.

Practical Considerations

The need to assume any bottom level of "reality" or sequence of observations, the assumption of algorithmic description, time or causality has been removed, but this does not mean that we will never assume these. There are two ways in which we could use the idea of hierarchies of partial models. One is in an ontologically profound way - when we are thinking about things in a philosophically deep way, and in such situations we will generally regard any observers as being "embedded" in the system. The other is in a less deep way - when we are dealing with practical situations, probably from the point of view of a particular observer.

When we are using the ontolology in this second way we may choose to assume these things. We may assume that any partial models will be generated from some "fundamental" sequence of observations because when we are attempting to apply ideas like these we will generally will have a sequence of observations - for example, the sensory inputs of a computer which we want to model reality. Even though we would not really think that these sensory inputs are "fundamental" it may be adequate to proceed as if this is the case in a universe in which causality, time and the assumption of algorithmic description are relevant. Assuming time and causality, things that will be common to any model used by a machine made by us to function in the "real" world, could save time and processing resources that would otherwise be needed to derive them as "things" from an atemporal ontology. Time and causality will also cause the assumption of algorithmic description to be locally correct, also allowing time and processing resources to be saved.

What About Occam's Razor?

Although this series of articles is about Occam's razor, we have not been discussing it directly in this article. As I stated in a previous article, we need an adequate way of representing a worldview, or model of the world, before we can make much progress with Occam's razor and evaluating worldviews or parts of worldviews. This article has continued the process of trying to define a suitable way of representing a worldview.

We will need to give more consideration to the issue of how to apply Occam's razor in a hierarchy like this later. For now, this at least gives us a structure in which Occam's razor can work. As I stated earlier in this article, Occam's razor would really need to be a result of measure - the frequency of occurrence - of certain patterns in the hierarchy.

It is not unreasonable to expect Occam's razor to follow from what we have. Previously, Occam's razor followed from our assumption that reality was describable by an algorithm. When an algorithm is involved like this, measure favours certain futures. Although we have discarded this idea, at least in any fundamental sense, we still have a set of algorithms. Rather than act as if an algorithmic machine is churning our the universe we have incorporated algorithms into the very definitions of objects, and temporal situations have now become objects. These algorithms can be considered a replacement for the actual algorithm in our earlier consideration: we should expect the same issues of measure to apply and for certain types of world to be favoured.

Conclusion

A way of using a hierarchy of partial models to model reality has been described. A partial model is applied to a matrix of observations, an index being used as a reference point within that matrix and different meanings extracted by using different indices. Different indices can be used to assemble a matrix of meanings extracted by a partial model and, as such a result is in the form of matrix, it can be used as the "observations" or "reality" for another partial model. This means that, rather than operating on reality directly, partial models can operate on a "processed" reality already containing some meaning extracted by a lower level partial model.

There are some ontological implications of this - at least if we take the position that I call ontologically strong hierarchy. A definition of the word "real" follows from such a view. A thing is viewed as "real" if there is the logical capability for a partial model to extract a corresponding meaning from the hierarchy.

It is easy to misinterpret this concept of reality as meaning that a thing is real when someone or something performs some thinking or information processing to extract its meaning, but this would be incorrect: a thing is real because of the logical capability of an algorithm (partial model) to extract its corresponding meaning from the hierarchy, regardless of whether or not anything actually "runs" such an algorithm.

This view may seem extreme, but human worldviews are based on the hierarchical extraction of meaning from information. This is therefore merely a formalized view of what we already do, taken to its logical conclusion in which everything that can be logically obtained by a partial model from any layer in the hierarchy exists in some way.

In previous discussions of Occam's razor we have assumed that we are generating models based on some sequence of observations. No such thing is assumed to have any fundamental existence now. All partial models extract meaning from meaning that is implied by other partial models. There is an infinite hierarchy of implied meaning upon implied meaning with no bottom to it and no "fundamental" reality beyond this system.

Such a view dispenses with the assumption of algorithmic description, causality and time as basic things. These are now simply features of certain partial models. The hierarchy itself is atemporal. Time is simply a "thing" in some parts of the hierarchy- just as any "physical" object is. Thinking that time is fundamental physics because things are contained in time and things seem to "happen" in time is like thinking that the economy of Britain is fundamental physics because it contains lots of people buying things or that World War II was fundamental physics because lots of things happened in it. In reality there is no fundamental physics, although we may find something that has the appearance of it.

Although we can dispense with the assumption of algorithmic description and related things, this does not mean that we will never assume them. When dealing with practical modelling situations we may be more concerned with the issue of making partial models to extract meaning from a sequence of observations (for example, sensory inputs in a computer) and we may just assume that this sequence of observations is fundamental, rather than being too concerned with deep ontological issues. Likewise, it may be convenient to make the assumption of algorithmic description in such situations.

The ontology proposed in this article is probably incomplete and will need more work on it later: this is just a starting point. One capability that will probably need adding to the ontology is what I will call dimensionalization. A partial model can only extract meaning from meaning created by lower level partial models by sharing an index value with these lower level partial models, and this requires the partial models to have the same number of dimensions. The ontology should be modified to remove this restriction.

The concepts discussed in this article may be helpful in understanding another of my articles Many Worlds Assisted Mind Uploading: A Thought Experiment [13].

References

[1] Web Reference: Almond, P. (2005). Occam's Razor Part 1: What Is Occam's Razor? Retrieved 22 August 2005 from http://www.paul-almond.com/OccamsRazorPart01.htm.

[2] Web Reference: Almond, P. (2005). Occam's Razor Part 2: Principles of Language. Retrieved 9 October 2005 from http://www.paul-almond.com/OccamsRazorPart02.htm.

[3] Web Reference: Almond, P. (2005). Occam's Razor Part 3: Assumptions About Reality. Retrieved 13 November 2005 from http://www.paul-almond.com/OccamsRazorPart03.htm.

[4] Web Reference: Almond, P. (2005). Occam's Razor Part 4: An Overview of How Occam's Razor Works. Retrieved 24 December 2005 from http://www.paul-almond.com/OccamsRazorPart04.htm.

[5] Web Reference: Almond, P. (2006). Occam's Razor Part 5: How Mapping Can Work. Retrieved 14 January 2006 from http://www.paul-almond.com/OccamsRazorPart05.htm.

[6] Web Reference: Almond, P. (2006). Occam's Razor Part 6: Partial Models as "Envelopes". Retrieved 1 March 2006 from http://www.paul-almond.com/OccamsRazorPart06.htm.

[7] Web Reference: Almond, P. (2005). What is a Low Level Language? Retrieved 17 July 2005 from http://www.paul-almond.com/WhatIsALowLevelLanguage.htm.

[8] Hawkins, J., Blakeslee, S. (2004). On Intelligence. New York: Henry Holt.

[9] Web Reference: George, D., Hawkins, J. (?). Belief Propagation and Wiring Length Optimization as Organizing Principles for Cortical Microcircuits. Retrieved 24 April 2006 from http://www.stanford.edu/~dil/invariance/Download/CorticalCircuits.pdf.

[10] Schmidhuber, J. (1997). A Computer Scientist's View of Life, the Universe and Everything. Foundations of Computer Science: Potential-Theory-Cognition, Volume 1337, Lecture Notes in Computer Science, pp201-208. (Edited by Freksa, C., Jantzen, M, Valk. R.). Berlin: Springer Verlag.

[13] Web Reference: Almond, P. (2006). Many Worlds Assisted Mind Uploading: A Thought Experiment. Retrieved 6 April 2006 from http://www.paul-almond.com/ManyWorldsAssistedMindUploading.htm.