Yet there remain huge chasms between artificial systems and forms of natural
intelligence in humans and other animals -- including weaver-birds, elephants,
squirrels, dolphins, orangutans, carnivorous mammals, and their prey.
(Sample weaver bird cognition here:
http://www.youtube.com/watch?v=6svAIgEnFvw.)
Fashionable "paradigms" offering definitive answers come and go (sometimes
reappearing with new labels). Yet no AI or robotic systems come close to
modelling
or replicating the development from helpless infant over a decade or two to
plumber, cook, trapeze artist, bricklayer, seamstress, dairy farmer,
shop-keeper, child-minder, professor of philosophy, concert pianist, mathematics
teacher, quantum physicist, waiter in a busy restaurant, etc. Human and animal
developmental trajectories vastly outstrip, in depth and breadth of achievement,
the products of artificial learning systems, although AI systems sometimes
produce super-human competences in restricted domains, such as proving
logical theorems,
winning at chess or Jeopardy, and perhaps playing table tennis at
championship level one
day in the distant future?
(http://www.youtube.com/watch?v=tIIJME8-au8).
I'll outline a very long-term multi-disciplinary research programme addressing
these and other inadequacies in current AI, robotics, psychology, neuroscience
and philosophy of mathematics and mind, in part by building on past work, and in
part by looking for very different clues and challenges: the
Meta-Morphogenesis project, partly inspired by Turing's work on morphogenesis.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
Figure 1:
What sequence of movements could get the shirt onto the child if the
shirt is
made of material that is flexible but does not stretch much? Why would it be a
mistake to start by pushing the head through the neck-hole? What difference
would it make if the material could be stretched arbitrarily without being
permanently changed?
In more obviously mathematical domains, where computers are commonly
assumed to excel, the achievements are narrowly focused on branches of
mathematics using inference methods based on arithmetic, algebra, logic,
probability and statistical theory.
However, mathematics is much broader than that, and we lack models of
the reasoning (for instance geometrical and topological reasoning) that
enabled humans to come up with the profoundly important and influential
mathematical discoveries reported in Euclid's
Elements 2.5 millennia
ago - arguably the single most important book ever written on this
planet. The early pioneers could not have learnt from mathematics
teachers. How did they teach themselves, and each other? What would be required
to enable robots to make similar discoveries without teachers?
Those mathematical capabilities seem to have deep, but mostly unnoticed,
connections with animal abilities to perceive practically important types of
affordance, including use of mechanisms that are concerned not only with the
perceiver's possibilities for immediate action but more generally with what is
and is not possible in a physical situation and how those possibilities and
impossibilities can change, for example if something is moved. A child could
learn that a shoelace threaded through a single hole can be removed from the
hole by pulling the left end of the lace or by pulling the right end. Why does
combining two successful actions fail in this case, whereas in other cases a
combination improves success (e.g. A pushing an object and B pushing the object
in the same direction)? Collecting examples of explanations of impossibilities
that humans understand but not yet current robots is one way to investigate gaps
in what has been achieved so far. It is also a route toward understanding the
nature of human mathematical competences, which I think start to develop in
children long before anyone notices.
Many animals, including pre-verbal humans, need to be able to perceive and think
about what is and is not possible in a situation, though in most cases without
having the ability to reflect on their thinking or to communicate the thoughts
to someone else. The meta-cognitive abilities evolve later in the history
of a species and develop later in individuals.
Thinking about what would be possible in various possible states of
affairs is totally different from abilities to make predictions about
what will happen, or to reason probabilistically. It's one thing to try
repeatedly to push a shirt on a child by pushing its hand and arm in
through the end of a sleeve and conclude from repeated failures that
success is improbable. It's quite another thing to understand that if
the shirt material cannot be stretched, then success is impossible (for
a normally shaped child and a well fitting shirt) though if the material
could be stretched as much as needed then it could be done. Additional
reasoning powers might enable the machine to work out that starting by
pushing the head in through the largest opening could require least
stretching, and to work this out without having to collect statistics
from repeated attempts.
3 Shallow statistical vs deep knowledge
It is possible to have a shallow (statistical) predictive capability based on
observed regularities while lacking deeper knowledge about the set of
possibilities sampled in those observations. An example is the difference
between (a) having heard and remembered a set of sentences and noticed some
regular associations between pairs of words in those sentences and (b) being
aware of the generative grammar used by the speakers, or having acquired such a
grammar unconsciously. The grasp of the grammar, using recursive modes of
composition, permits a much richer and more varied collection of utterances to
be produced or understood. Something similar is required for visual perception
of spatial configurations and spatial processes that are even richer and more
varied than sentences can be. Yet it seems that we share that more powerful
competence with more species, including squirrels and nest-building birds.
This suggests that abilities to acquire, process, store, manipulate, and use
information about spatial structures evolved before capabilities that are unique
to humans, such as use of spoken language. But the spatial information requires
use of something like grammatical structures to cope with scenes of varying
complexity, varying structural detail, and varying collections of possibilities
for change. In other words visual perception, along with planning and acting on
the basis of what is scene, requires the use of internal languages that have
many of the properties previously thought unique to human communicative
languages. Finding out what those languages are, how they evolved, how they can
vary across species, across individuals, and within an individual during
development is a long term research programme, with potential implications for
many aspects of AI/Robotics and Cognitive Science - discussed further in
[
8].
Conceivably a robot could be programmed to explore making various
movements combining a shirt and a flexible, child-shaped doll. It might
discover one or more sequences of moves that successfully get the shirt
on, provided that the shirt and doll are initially in one of the robot's
previously encountered starting states. This could be done by exploring
the space of sequences of possible moves, whose size would depend on the
degree of precision of its motion and control parameters. For example,
if from every position of the hands there are 50 possible 3-D directions
of movement and the robot tries 20 steps after each starting direction,
then the number of physical trajectories from the initial state to be
explored is
5020 = 9536743164062500000000000000000000 |
and if it tries a million new moves every second, then it could explore
that space in about 302408000000000000 millennia. Clearly animals do
something different when they learn to do things, but exactly how they
choose things to try at each moment is not known.
The "generative grammar" of spatial structures and processes is rich and deep,
and is not concerned only with linear sequences or discrete sequences. In fact
there are multiple overlapping space-time grammars, involving different
collections of objects assembled, disassembled, moved, repaired, etc. and used,
often for many purposes and in many ways. Think of what structures and
processes are made possible by different sorts of children's play materials and
construction kits, including plasticine, paper and scissors, meccano, lego,
tinkertoys, etc. The sort of deep knowledge I am referring to involves grasp of
the structure of a construction-kit with generative powers, and the ability to
make inferences about what can and cannot be built with that kit, by assembling
more and more parts, subject to the possibilities and constraints inherent in
the kit.
5
There are different overlapping subsets of spatio-temporal possibilities, with
different mathematical structures, including Euclidean and non-Euclidean
geometries (e.g. the geometry of the surface of a hand, or face is
non-euclidean) and various subsets of topology. Mechanisms for acquiring and
using these "possibility subsets", i.e. possible action sequences and
trajectories, seem to be used by pre-verbal children and other animals. That
suggests that those abilities, must have evolved before linguistic capabilities.
They seem to be at work in young children playing with toys before they can
understand or speak a human language. The starting capabilities extended through
much spatial exploration, provide much of the subject matter (semantic content)
for many linguistic communications.
Some of the early forms of reasoning and learning in young humans, and
corresponding subsets in other animals, are beyond the scope of current AI
theorem provers, planners, reasoners, or learning systems that I know of. Some
of those forms seem to be used by non-human intelligent animals that are able to
perceive
both possibilities and constraints on possibilities in spatial configurations.
Betty, a New Caledonian crow, made headline news in 2002 when she
surprised Oxford researchers by making a hook from a straight piece of wire, in
order to lift a bucket of food out of a vertical glass tube. Moreover, in a
series of repeated challenges she made multiple hooks, using at least four very
different strategies, taking advantage of different parts of the environment,
all apparently in full knowledge of what she was doing and why - as there was
no evidence of random trial and error behaviour. Why did she not go on using the
earlier methods, which all worked?
Several of the videos showing the diversity
of techniques are still available here:
http://users.ox.ac.uk/~kgroup/tools/movies.shtml.
The absence of
trial-and-error processes in the successful episodes suggests that Betty had a
deep understanding of the range of possibilities and constraints on
possibilities in her problem solving situations.
It is very unlikely that you have previously encountered and solved the
problem posed below the following image, yet many people very quickly
think of a solution.
Figure 2:
Suppose you wanted to use one hand to lift the mug to a nearby table without
any part of your skin coming into contact with
the mug, and without moving the
book on which the mug is resting,
what could you do, using only one hand?
In order to think of a strategy you do not need to know the exact, or
even the approximate, sizes of the objects in the scene, how far away
they are from you, exactly what force will be required to lift the mug,
and so on. It may occur to you that if the mug is full of liquid and you
don't want to spill any of it, then a quite different solution is
required. (Why? Is there a solution?).
--
Figure 3:
Consider one or more sequences of actions that would enable
a person or robot to change the physical configuration depicted on the left
into the one depicted on the right - not
necessarily in exactly the same locations as the objects
depicted. Then do the same for the actions required to
transform the right configuration to the left one.
The two pictures in Figure
3 present another set of example action
strategies for changing a situation from one configuration to another. At how
many different levels of abstraction can you think of the process, where the
levels differ in the amount of detail (e.g. metrical detail) of each
intermediate stage. For example, when you first thought about the problem did
you specify which hands or which fingers would be used at every stage, or at
which location you would need to grasp each item? If you specified the locations
used to grasp the cup, the saucer and the spoon, what else would have to change
to permit those grasps? The point about all this is that although you do not
normally think of using mathematics for tasks like this, if you choose a
location at which to grasp the cup using finger and thumb of your left hand,
that will mathematically constrain the 3-D orientation of the gap between
between finger and thumb, if you don't want the cup to be rotated by the fact of
bringing finger and thumb together. A human can think about the possible
movements and the orientations required, and why those orientations are
required, without actually performing the action, and can answer questions about
why certain actions will fail, again without doing anything.
These are examples of "offline intelligence", contrasted with the "online
intelligence" used in actually manipulating objects, where information required
for servo-control may be used transiently then discarded and replaced by new
information. My impression is that a vast amount of recent AI/Robotic research
has aimed at providing online intelligence with complete disregard for the
requirements of offline intelligence. Offline intelligence is necessary for
achieving complex goals by performing actions extended over space and time,
including the use of machines that have to be built to support the process, and
in some cases delegating portions of the task to others. The designer or builder
of a skyscraper will not think in terms of his/her own actions, but in terms of
what motions of what parts and materials are required.
3.1 Limitations of sensorymotor intelligence
When you think about such things even with fairly detailed constraints
on the possible motions, you will not be thinking about either the
nervous signals sent to the muscles involved, nor the patterns of
retinal stimulation that will be provided - and in fact the same
actions can produce different retinal processes depending on the precise
position of the head, and the direction of gaze of the eyes, and whether
and how the fixation changes during the process. Probably the fixation
requirements will be more constrained for a novice at this task than for
an expert.
However, humans, other animals, and intelligent robots do not
need to reason about sensory-motor details if they use an ontology of 3-D
structures and processes, rather than an ontology of sensory and motor
nerve signals.
Contrast this with the sorts of assumptions discussed in
[
2], and many others who attempt to build
theories of cognition on the basis of sensory-motor control loops.
As John McCarthy pointed out in [
4] it would be surprising if
billions of years of evolution failed to provide intelligent organisms with the
information that they are in a world of persisting 3-D locations, relationships,
objects and processes - a discovery that, in a good design, could be shared
across many types of individuals with very different sensors and motors, and
sensory motor patterns. Trying to make a living on a planet like this, whose
contents extend far beyond the skin of any individual, would be messy and highly
inefficient if expressed entirely in terms of possible sensory-motor sequences,
compared with using unchanging representations for things that don't change
whenever sensory or motor signals change. Planning a short cut home, with
reference to roads, junctions, bus routes, etc. is far more sensible than
attempting to deal, at any level of abstraction, with the potentially infinite
variety of sensory-motor patterns that might be relevant.
This ability to think about sequences of possible alterations in a physical
configuration without actually doing anything, and without having full metrical
information, inspired much early work in AI, including the sorts of symbolic
planning used by Shakey, the Stanford robot, and Freddy, the Edinburgh robot,
over four decades ago, though at the time the technology available (including
available computer power) was grossly inadequate for the task, including ruling
out visual servo-control of actions.
Any researcher claiming that intelligent robots require only the right physical
mode of interaction with the environment, along with mechanisms for finding
patterns in sensory-motor signals, must disregard the capabilities and
information-processing requirements that I have been discussing.
4 Inflating what "passive walkers" can do
Some (whom I'll not mention to avoid embarrassing them) have attempted to
support claims that only interactions with the immediate environment are needed
for intelligence by referring to or demonstrating "passive
walkers",
6
without saying what will happen if a brick is in the way of a passive walker, or
if part of the walking route starts to slope uphill. Such toys are interesting
and entertaining but do not indicate any need for a "New artificial
intelligence", using labels such as "embodied", "enactivist", "behaviour
based", and "situated", to characterise their new paradigm. Those new
approaches are at least as selective as the older reasoning based approaches
that they criticised, though in different ways. (Some of that history is
presented in Boden's survey [
1].)
The requirements for perception and action mechanisms differ according
to which "central" layers the organism has. For instance, for an
organism able to use deliberative capabilities to think of, evaluate, and
select multi-step plans, where most of the actions will occur in
situations that do not exist yet, it is not enough to identify objects
and their relationships (pencil, mug, handle of mug, book, window-frame,
etc.) in a current visual percept. It is also necessary to be able to
"think ahead" about possible actions at a suitable level of abstraction,
including consideration of objects not yet known, requiring a potentially
infinite variety of possible sensory and motor patterns.
5 The birth of mathematics
The ability to reason about possible actions at a level of generality that
abstracts from metrical details seems to be closely related to the abilities of
ancient Greeks, and others, to make mathematical discoveries about possible
configurations of lines and circles and the consequences of changing those
configurations, without being tied to particular lengths, angles, curvatures,
etc., in Euclidean geometry or topology. As far as I know, no current robot can
do this, and neuroscientists don't know how brains do it. Some examples of
mathematical reasoning that could be related to reasoning about practical tasks
and which are currently beyond what AI reasoners can do, are presented on
my web site.
7,8
In 1971 I presented a paper at IJCAI, arguing that the
focus solely on logic-based reasoning, recommended by McCarthy and Hayes in
[
5]
could hold up progress in AI, because it ignored forms of spatial reasoning that
had proved powerful in mathematics and practical problem solving. I did not
realise then how difficult it would be to explain exactly what the alternatives
were and how they worked - despite many conferences and journal papers on
diagrammatic reasoning since then.
There have also been several changes of fashion promoted by various AI
researchers (or their critics) including use of neural nets, constraint nets,
evolutionary algorithms, dynamical systems, behaviour-based systems, embodied
cognition, situated cognition, enactive cognition, autopoesis, morphological
computation, statistical learning, bayesian nets, and probably others that I
have not encountered, often accompanied by hand-waving and hyperbole without
much science or engineering. In parallel with this there has been continued
research advancing older paradigms for symbolic and logic based, theorem
proving, planning, and grammar based language processing. Several of the debates
are analysed in [
1],
6 Other inadequacies
There are many other inadequacies in current AI, including, for example the lack
of an agreed framework for relating information-processing architectures to
requirements in engineering contexts or to explanatory models in scientific
contexts. For example attempts to model emotions or learning capabilities, in
humans or other animals, are often based on inadequate descriptions of what
needs to be explained, for instance poor theories of emotions that focus only on
emotions with characteristic behavioural expressions: a small subset of
phenomena requiring explanation, or poor theories of learning that focus only on
a small subset of types of learning (e.g. reinforcement learning where learners
have no understanding of what's going on). That would exclude the kind of
learning that goes on when people make mathematical discoveries or learn to
program computers or learn to compose music.
Moreover, much AI research uses a seriously restricted set of forms
of representation (means of encoding information) partly because of the
educational backgrounds of researchers - as
a result of which many of them assume
that spatial structures must be represented using mechanisms based on Cartesian
coordinates -
and partly because of a failure to analyse in sufficient detail the
variety of problems overcome by many animals in their natural environments.
Standard psychological research techniques are not applicable to the study of
learning capabilities in young children and other animals because there is so
much individual variation, but the widespread availability of cheap video
cameras has led to a large and growing collection of freely available examples.
7 More on offline and online intelligence
Researchers have to learn what to look for. For example,
online intelligence requires highly trained precisely controlled
responses matched to fine details of the physical environment, e.g.
catching a ball, playing table tennis, picking up a box and putting it
on another. In contrast
offline intelligence involves understanding
not just existing spatial configurations but also the possibilities for
change and constraints on change, and for some tasks the ability to find
sequences of possible changes to achieve a goal, where some of the
possibilities are not specified in metrical detail because they do not
yet exist, but will exist after part of the plan has been carried out.
This requires the ability to construct relatively abstract
forms of representation of perceived or remembered situations to allow
plans to be constructed with missing details that can be acquired later
during execution. You can think about making a train trip to another
town without having information about where you will stand when
purchasing your ticket or which coach you will board when the train
arrives. You can think about how to rotate a chair to get it through a
doorway without needing information about the precise 3-D coordinates of
parts of the chair or knowing exactly where you will grasp it, or how
much force you will need to apply at various stages of the move.
There is no reason to believe that humans and other animals have to use
probability distributions over possible precise metrical values, in all planning
contexts where precise measurements are not available. Even thinking about such
precise values probabilistically is highly unintelligent when reasoning about
topological relationships or partial orderings (nearer, thinner, a bigger angle,
etc.) is all that's needed
9
Unfortunately, the mathematically sophisticated, but nevertheless
unintelligent, modes of thinking are used in many robots, after much
statistical learning (to acquire probability distributions) and complex
probabilistic reasoning, that is potentially explosive. That is in part
a consequence of the unjustified assumption that all spatial properties
and relations have to be expressed in Cartesian coordinate systems.
Human mathematicians did not know about them when they proved their
first theorems about Euclidean geometry, built their first shelters.
8 Speculations about early forms of cognition
It is clear that the earliest spatial cognition could not have used full
Euclidean geometry, including its uniform metric. I suspect that the
metrical version of geometry was a result of a collection of transitions
adding richer and richer non-metrical relationships, including networks
of partial orderings of size, distance, angle, speed, curvature, etc.
Later, indefinitely extendable partial metrics were added: distance
between X and Y is at least three times the distance between P and Q and
at most five times that distance. Such procedures could allow previously
used standards to be sub-divided with arbitrarily increasing precision. At
first this must have been applied only to special cases, then later
somehow (using what cognitive mechanisms?) extrapolated indefinitely,
implicitly using a Kantian form of potential infinity (long before
Kant realised the need for this).
Filling in the details of such a story, and relating it to varieties of
cognition not only in the ancestors of humans but also many other existing
species will be a long term multi-disciplinary collaborative task, with deep
implications for neuroscience, robotics, psychology, philosophy of mathematics
and philosophy of mind. (Among others.)
Moreover, human toddlers appear to be capable of making
proto-mathematical discoveries ("toddler theorems") even if they are
unaware of what they have done. The learning process starts in infancy, but
seems to involve different kinds of advance at different stages of development,
involving different domains as suggested by Karmiloff-Smith
in [
3].
For example, I recently saw an 11 month old infant discover, apparently with
great delight, that she could hold a ball between her upturned foot and the
palm of her hand. That sort of discovery could not have been made by a one month
old child. Why not?
10
Animal abilities to perceive and use complex novel affordances appear to be
closely related to the ability to make mathematical discoveries. Compare the
abilities to think about changes of configurations involving ropes or strings
and the mathematical ability to think about continuous deformation of closed
curves in various kinds of surface.
Not only computational models, but also current psychology and neuroscience,
don't seem to come close to describing these competences accurately or producing
explanations - especially if we consider not only simple numerical mathematics,
on which many psychological studies of mathematics seem to focus, but also
topological and geometrical reasoning, and the essentially mathematical ability
to discover a generative grammar closely related to the verbal patterns a child
has experienced in her locality, where the grammar is very different from those
discovered by children exposed to thousands of other languages.
There seem to be key features of some of those developmental trajectories that
could provide clues, including some noticed by Piaget in his last two books on
Possibility and Necessity, and his former colleague, Annette Karmiloff-Smith
[
3].
9 The Meta-Morphogenesis project
Identifying gaps in our knowledge requires a great deal of careful observation
of many forms of behaviour in humans at various stages of development and many
other species, always asking: "what sort of information-processing mechanism
(or mechanisms) could account for that?"
Partly inspired by one of Alan Turing's last papers on Morphogenesis
[
10], I proposed the Meta-Morphogenesis (M-M) project
in [
9], a very long term
collaborative project for building up an agreed collection of
explanatory tasks, and present some ideas about what has been missed in
most proposed explanatory theories.
Perhaps researchers who disagree, often fruitlessly, about what the answers are
can collaborate fruitfully on finding out what the questions are, since much of
what needs to be explained is far from obvious. There are unanswered questions
about uses of
vision, varieties of motivation and affect, human and animal mathematical
competences, information-processing architectures required for all the different
sorts of biological competences to be combined, and questions about how all
these phenomena evolved across species, and develop in individuals. This leads
to questions about what the universe had to be like to support the forms of
evolution and the products of evolution that have existed on this planet. The
Meta-Morphogenesis project is concerned with trying to understand what varieties
of information processing biological evolution has achieved, not only in humans
but across the spectrum of life. Many of the achievements are far from
obvious.
11
Unfortunately, researchers all too often mistake impressive new developments for
steps in the right direction. I am not sure there is any way to change this
without radical changes in our educational systems and research funding systems.
But those are topics for another time. In the meantime I hope many more
researchers will join the attempts to identify gaps in our knowledge, including
things we know happen but which we do not know how to explain, and in the longer
term by finding gaps we had not previously noticed. I think one way to do that
is to try to investigate transitions in biological information processing across
evolutionary time-scales, since its clear that types of information used, the
types of uses of information, and the purposes for which information is used
have changed enormously since the simplest organisms floating in a sea of
chemicals.
Perhaps some of the undiscovered intermediate states in evolution
will turn out to be keys to unnoticed features of the current most sophisticated
biological information processors, including humans.
References
- [1]
-
Boden, M.A.: Mind As Machine: A history of Cognitive Science (Vols 1-2).
Oxford University Press, Oxford (2006)
- [2]
-
Clark, A.: Whatever next? Predictive brains, situated agents, and the future
of cognitive science. Behavioral and Brain Sciences 36(3), 1-24 (2013)
- [3]
-
Karmiloff-Smith, A.: Beyond Modularity: A Developmental Perspective on
Cognitive Science. MIT Press, Cambridge, MA (1992)
- [4]
-
McCarthy, J.: The well-designed child. Artificial Intelligence 172(18),
2003-2014 (2008), http://www-formal.stanford.edu/jmc/child.html
- [5]
-
McCarthy, J., Hayes, P.: Some philosophical problems from the standpoint of
AI. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence 4, pp.
463-502. Edinburgh University Press, Edinburgh, Scotland (1969),
http://www-formal.stanford.edu/jmc/mcchay69/mcchay69.html
- [6]
-
Minsky, M.L.: Steps toward artificial intelligence. In: Feigenbaum, E.,
Feldman, J. (eds.) Computers and Thought, pp. 406-450. McGraw-Hill, New York
(1963)
- [7]
-
Sauvy, J., Sauvy, S.: The Child's Discovery of Space: From hopscotch to mazes
- an introduction to intuitive topology. Penguin Education, Harmondsworth
(1974), translated from the French by Pam Wells
- [8]
-
Sloman, A.: Evolution of minds and languages. What evolved first and develops
first in children: Languages for communicating, or languages for thinking
(Generalised Languages: GLs)? (2008),
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#pr0702
- [9]
-
Sloman, A.: Virtual machinery and evolution of mind (part 3)
meta-morphogenesis: Evolution of information-processing machinery. In:
Cooper, S.B., van Leeuwen, J. (eds.) Alan Turing - His Work and Impact, pp.
849-856. Elsevier, Amsterdam (2013),
http://www.cs.bham.ac.uk/research/projects/cogaff/11.html#1106d
- [10]
-
Turing, A.M.: The Chemical Basis Of Morphogenesis. Phil. Trans. R. Soc.
London B 237 237, 37-72 (1952)
Footnotes:
1Nest building cognition of a weaver bird can be sampled
here:
http://www.youtube.com/watch?v=6svAIgEnFvw
2Though it's best not to believe
everything you see in advertisements
http://www.youtube.com/watch?v=tIIJME8-au8
3
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
This project is unfunded and I have no plans to apply
for funding, though others may do so if they wish.
4As illustrated in this video.
http://www.youtube.com/watch?v=WWNlgvtYcEs
5An evolving discussion note on this topic can be
found here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/explaining-possibility.html
Under construction:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.html
6E.g.
http://www.youtube.com/watch?v=N64KOQkbyiI
7
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html
8
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html
9As I have tried to illustrate in:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/changing-affordances.html
10A growing list of toddler theorems and discussions of
their requirements can be found in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
11A more detailed, but still evolving, introduction to the
project can be found here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
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Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham