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Last updated: 11 Sep 2013
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This web page is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
Also accessible as:
http://tinyurl.com/TodTh
A messy automatically generated PDF version of this file is:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.pdf
or
http://tinyurl.com/BhamCog/misc/toddler-theorems.pdf
It is one of a set of documents on meta-morphogenesis, listed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
Also accessible as:
http://tinyurl.com/M-M-Gen
A partial index of discussion notes is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
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Two more papers based on the thesis work were published in 1965 and 1969:"Knowing and Understanding: Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truthhttp://www.cs.bham.ac.uk/research/projects/cogaff/07.html#706This argued (e.g. against Hume) that Immanuel Kant was right in claiming in 1781 that in
addition tothere are also truths that are neither empirical nor trivial but provide substantial
- empirical facts that can be refuted in experiments and observations with novel conditions
and- analytic, essentially trivial, truths that depend only on definitions and their logical
consequences, and do not extend knowledge
knowledge, namely truths of mathematics.Some of the concepts used here are explained in
"'NECESSARY', 'A PRIORI' AND 'ANALYTIC'" (1965)
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#701
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#714 Functions and Rogators (1965)
http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#712 Explaining Logical Necessity (1968-9)
Around 1970 Max Clowes introduced me to Artificial Intelligence, especially AI work on
Machine vision. That convinced me that a good way to make progress on my problems might be
to build a baby robot that could, after some initial learning about the world and what can
happen in it, notice the sorts of possibilities and necessities (constraints on
possibilities) that characterise mathematical discoveries. My first ever AI conference
paper distinguishing "Fregean" from "Analogical" forms of representation was a start on
that project, followed up in my 1978 book, especially Chapters 7 and 8.
From about 1973, I was increasingly involved in AI teaching and research and also had
- Interactions between philosophy and AI: The role of intuition and non-logical
reasoning in intelligence,
Proc 2nd IJCAI, 1971, London, pp. 209--226,
http://www.cs.bham.ac.uk/research/cogaff/04.html#200407
- CRP: The Computer Revolution in Philosophy: Philosophy, Science and
Models of Mind,
Harvester Press (and Humanities Press), 1978,
http://www.cs.bham.ac.uk/research/cogaff/crp
'micro-world' 'mini-world' 'micro-domain' 'micro-theory' 'theory' 'framework' 'framework-theory'
What is a domain?
I don't think there is any clear or simple answer to that question. But this document
presents several examples that differ widely in character, making it clear that domains
come in different shapes and sizes, with different levels of abstraction, different kinds
of complexity, different uses -- both in controlling visible behaviour and in various
internal cognitive functions --, different challenges for a learner, different ways of
being combined with other domains to form new domains, and conversely, different ways
of being divided into sub-domains, etc.
We might try to compare different sub-fields of academic knowledge to come up with an
analysis of the concept of domain, but there are many overlaps and many differences
between such domains as philosophy, logic, mathematics, physics, chemistry, biology,
biochemistry, zoology, botany, psychology, developmental psychology, gerontology,
linguistics, history, social geography, political geography, geography, meteorology,
astronomy, astrophysics, ....
Moreover within dynamic disciplines new domains or sub-domains often grow, or are
discovered or created, some of them found to have pre-existed waiting to be noticed by
researchers (e.g. planetary motions, Newtonian mechanics, chemistry, topology, the theory
of recursive functions) while others are creations of individual thinkers or groups of
thinkers, for example, art forms, professions (carpentry, weaving, knitting, dentistry,
physiotherapy, psychotherapy, architecture, various kinds of business management, divorce
law in a particular country, jewish theology, and many more). However, that distinction,
between pre-existing and human-created domains, is controversial with fuzzy boundaries.
Philosophers' concepts of "natural kinds" are attempts to make some sort of sense of this,
in my view largely unsatisfactory, in part because many of the examples are products of
biological evolution, and some are products of those products. I suspect the idea of
"naturalness" in this context is a red-herring, since the distinction between what is
created and what was waiting to be discovered is unclear and there are hybrids.
http://tinyurl.com/PopLog/teach/oop
More generally we can say that a domain involves relationships that can hold between types
of thing, and instances of those types can have various properties and can be combined in
various ways to produce new things whose properties, relationships, competences and
behaviours, depend on what they are composed of and how they are combined, and sometimes
the context. Often mathematicians specify such domain-types without knowing (or caring)
whether instances of those types actually existed in advance (e.g. David Hilbert's
infinite dimensional vector spaces?)
Formation of a new instance of a type in a domain can include assembling
pre-existing instances to create larger items (e.g. joining words, sentences, lego bricks,
meccano parts dance steps, building materials, mathematical derivations), or can include
inserting new entities within an existing structure, or changing properties, or
altering relationships. E.g. loosening a screw in a meccano crane can sometimes introduce
a new rotational degree of freedom for a part.
Some domains allow continuous change, e.g. growth, linear motion, rotation, bending,
twisting, moving closer, altering an angle, increasing or decreasing overlap, changing
alignment, getting louder, changing timbre, changing colour, and many more (e.g. try
watching clouds, fast running rivers, kittens playing, ...). Some allow only discrete
changes, e.g. construction of logical or algebraic formulae, or formal derivations,
operations in a digital computer, operations in most computational virtual machines (e.g.
a Java or lisp virtual machine), some social relations (e.g. being married to, being a
client of,), etc.
The world of a human child presents a huge variety of very different sorts of domains to
be explored, created, modified, disassembled, recombined, and used in many practical
applications. This is also true of many other animals. Some species come with a fixed,
genetically determined, collection of domain related competences, while others have fixed
frameworks that can be instantiated differently by individuals, according to what sorts of
instances are in the environment, whereas humans and others (often called "altricial"
species) have mechanisms for extending their frameworks as a result of what they encounter
in their individual lives -- examples being learning and inventing languages, games, art
forms, branches of mathematics, types of shelter, and many more. This diversity of
content, and the diversity of mixtures of interacting genetic, developmental and learning
mechanisms was discussed in more detail in two papers written with Jackie Chappell, one
published in 2005 and an elaborated version in 2007. There are complicated
relationships with the ideas of AK-S, which still need to be sorted out.
Tarskian model theory http://plato.stanford.edu/entries/model-theory/ is also relevant.
Several computer scientists have developed theories about theories that should be relevant
to clarifying some of these issues, e.g. Goguen, Burstall and others (for example, see
http://en.wikipedia.org/wiki/Institution_(computer_science).
At some future time I need to investigate the relationships. However, I don't know whether
they include domains that allow (continuous representations of) continuous changes,
essential in Euclidean geometry, Newtonian mechanics, and some aspects of biology.
I don't know if anyone has good theories about discovery, creation, combination, and uses
of domains in more or less intelligent agents, including a distinction between having
behavioural competence within a domain, having a generative grasp of the domain, and
having meta-cognitive knowledge about that competence. These distinctions are important in
the work of AK-S, though she doesn't always use the same terminology.
The rest of this discussion note presents a scruffy collection of examples of domains
relevant to what human toddlers (and some other animals and older humans) are capable of
learning and doing in various sorts of domains whose instances they interact with, either
physically or intellectually. The section on
Learning about numbers (Numerosity, cardinality, order, etc.) includes examples of
interconnected domains, though not all the relationships are spelled out here.
Theorems about domains are of many kinds. Often they are about invariants of a set
of possible configurations or processes within a domain (e.g. "the motion at the far end
of a lever is always smaller than the motion at the near end if the pivot is nearer the
far end", "moving towards an open doorway increases what is visible through the doorway,
and moving away decreases what is visible"). (See the section on epistemic affordances, below.)
We need a more developed theory about the types of theorems available to toddlers and
others to discover, when exploring various kinds of environment, and about the
information-processing mechanisms that produce what AK-S calls "representational
redescription" allowing the theorems to be discovered and deployed.
(I think architectural changes are needed in many cases.)
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There are also transitions in information-processing capabilities and
mechanisms that are much harder to detect, though their consequences
may include observable behaviours.
A draft (incomplete, messy and growing) list of transitions in biological information
processing is here.
The transitions producing new capabilities and mechanisms are examples of a generalised
concept of morphogenesis, originally restricted to transitions producing physical
structures and properties.
Among the transitions are changes in the mechanisms for producing morphogenesis. These are
examples of meta-morphogenesis (MM). The examples of information processing competence
described here may occur at various stages during the lives of individuals. The mechanisms
that produce new ways of acquiring or extending competences are mechanisms of
meta-morphogenesis, about which little is known. Piaget identified many of the transitions
in children he observed, and thought that qualitative changes in competence producing
competences were global, occurring in succession, at different ages, during the
development of a child. Karmiloff-Smith, in Beyond Modularity suggests that
transitions between stages may occur within different domains of competence, and will
often be more a function of the nature of the domain than the age of the child, though she
allowed that there are also some age-related changes.
See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/beyond-modularity.html
Transitions occur across species, within a species, within an individual, concurrently in
different species, and in some cases in eco-systems or sub-systems involving more than
one species.
A draft (growing) list of significant transitions in types of information-processing in
organisms is here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/evolution-info-transitions.html
People who have not designed, tested or debugged working systems may lack the concepts and
theories required.
Exploration here does not necessarily refer to geographical exploration. It can including investigating the space of possible actions on some object or type of object, e.g. things that can be done with sand, with water, with wooden blocks, with string, with paper, with diagrams, etc. [See Sauvy and Sauvy]
I am not saying that that's a model of human or animal motive-generation, but that
something with those features could usefully be an important part of a motive
generation mechanisms if the genetically determined motive generating reflexes are
selected (by evolution) for their later usefulness in ways that the individual cannot
understand. This idea was independently developed and tested in a working computer
model, reported by Emre Ugur (2010).
As an individual's competence grows the amount of stored information about each domain
grows, extending the variety and complexity of situations they can cope with (e.g.
predicting what will happen, deciding what to do to achieve a goal, understanding why
something happens, preventing unwanted side-effects, reducing the difficulty of the task,
etc.)
N.B. This is totally different from building something like a Bayes Net
storing learnt correlations and allowing probability inferences to be made.
Bayesian inference produces probabilities for various already known possibilities. What I
am talking about allows new possibilities and impossibilities to be derived, but often
without any associated probability information: if a polygon has three sides then its
angles must add up to half a rotation.
Compare using a grammar to prove that certain sentences are possible and others
impossible. That provides no probabilistic information. In fact a very high proportion of
linguistic utterances had zero or close to zero probability before they were produced. But
that does not prevent them being constructed if needed, or understood if constructed.
The same can be said about possible physical structures and processes. Before the first
bicycle was constructed by a clever designer, the probability of it being constructed was
approximately zero.
For non-logical reasoning, e.g. reasoning about transformations of a set of topological orSuggested by Kenneth Craik, Phil Johnson-Laird and others http://en.wikipedia.org/wiki/Mental_model
The key idea is that under some conditions it is possible to discover that properties of a
schematic structure or schematic process are invariant -- i.e. the properties do
not depend on the precise instantiation of the abstraction, though sometimes it is
necessary to add previously unnoticed conditions (e.g. no larger object is between the
grasping surfaces) for a generalisation to be true.
This idea will have to be fleshed out very differently for different domains of structures
and processes, or for different sub-domains of rich domains -- e.g. Euclidean geometry,
operations on the natural numbers. (See examples about counting below.)
The kinds of discoveries discussed here are not empirical discoveries, but that does not
mean that the reasoning processes are infallible. The history of mathematics (e.g.
the work of Lakatos below) shows that even brilliant mathematicians can fail to
notice special cases, or implicit assumptions. Nevertheless I think these ideas if fleshed
out would support Kant's ideas about the nature of mathematical discoveries, as discoveries
of synthetic necessary truths. (As far as I know, he did not notice that the discovery
processes could be fallible.)
The ideas in this section are elaborations of some of the ideas in
___________________________________________________________________________________Jackie Chappell and Aaron Sloman, Natural and artificial meta-configured altricial information-processing systems, International Journal of Unconventional Computing 2007, pp. 211--239, http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0609,
[Added 27 Oct 2011]
It is also connected with our discussion of precursors to the use of language for
communication -- in pre-verbal humans, in pre-human ancestors and in other species.
E.g. see
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#glang
Evolution of minds and languages.
What evolved first and develops first in children:
Languages for communicating, or languages for thinking (Generalised Languages: GLs)?
Somatic and exo-somatic ontologies/forms of representation
An example, going from sensory information in a 2-D discrete retina to assumed
continuously moving lines sampled by the retina, or even a 3-D structure (e.g. rotating
wire-frame cube) projecting onto the retina, is discussed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/simplicity-ontology.html
Ontologically non-conservative transitions refute the philosophical theory of concept
empiricism (previously refuted by Immanuel Kant), and also demolish symbol-grounding theory,
despite its popularity among researchers in AI and cognitive science.
They also defeat forms of data-mining that look for useful new concepts (or features) that
are defined in terms of the pre-existing concepts or features used in presenting
the data to be learnt from. (Some work by Stephen Muggleton, using Inductive Logic
Programming may be an exception to this, if some of the concepts used to express new
abduced hypotheses, are neither included in nor definable in terms of some initial subset
of symbols.)
It is easy to see that integers (though not just positive integers) with addition, and
also rational numbers, both form groups.
Many mathematical abstractions go beyond the exemplars that led to their discovery.
In fact the discovery may be triggered by relatively simple cases that are much less
interesting than cases discovered later. The initial cases that inspired the abstraction
may be completely forgotten and perhaps not even mentioned in future teaching
of mathematics.
This use of abstraction in mathematics is often confused with use of metaphor.
Unlike use of abstraction, use of metaphor does require the original cases to be
retained and constantly referred to when referring to new cases.
Some of the problems are discussed in more detail in
That is a result of very bad philosophy of science.
I'll outline some alternatives.
How to discover relevant possibilities:
First try to find situations where you can watch infants, toddlers, or older children
play, interact with toys, machines, furniture, clothing, doors, door-handles, tools,
eating utensils, sand, water, mud, plasticine or anything else.
Similar observations of other animals can be useful, though for non-domesticated animals
it can be very difficult to find examples of varied and natural forms of behaviour. TV
documentaries available on Cable Television and the like are a rich source, but it is not
always possible to tell when scenarios are faked.
Some videos that I use to present examples are here:
http://www.cs.bham.ac.uk/research/projects/cogaff/movies/vid
More examples are presented or referenced below. Some are still in need of development:
more empirical detail and more theoretical analysis of possible mechanisms.
Compare Robert Lawler's video archive described below.
[To be continued.]
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To avoid shallow questions, learn to think like a designer:Which animals can do X? At what age can a human child first do X? What proportions of children at ages N1, N2, N3, ... can do X? Under what conditions will doing X happen earlier? What features of the situation make it more likely that a child, or animal, will do X? Which aspects of ability, or behaviour, or temperament are innate?
Sometimes that requires thinking like a mathematician, as illustrated below in several
examples -- a designer needs to be able to reason about the consequences of various design
options, in a way that covers non-trivial classes of cases (as opposed to having to
consider every instance separately).
That normally involves thinking like a mathematician -- especially discovery of, and
reasoning about, invariants of a class of cases. For example, an invariant can be a
feature of a diagram that represents reasoning about all possible circles or all possible
triangles, in Euclidean geometry. Usually that does not require the diagram to be
accurate. When schoolkids are taught to measure angles of a collection of triangles to
check the sums of the angles, they are NOT being taught to think like a mathematician.
Sometimes people who are not able think like a designer or a mathematician resort to doing
experiments (often on very small and unrepresentative groups of subjects). I have compared
that with doing Alchemy, here:
http://tinyurl.com/BhamCog/misc/alchemy/
(Is education research a form of alchemy?)
Unfortunately, the educational experience of many researchers includes neither learning to
think like a mathematician nor learning to think like a designer.
E.g. many people who can state Pythagoras' theorem, or the triangle sum theorem have no
idea how to prove either, and in some cases don't even know that proofs exist, as opposed
to empirical evidence obtained by measuring angles, areas, etc.
I suspect that is a deep error, and that for many biological organisms instead of
probabilities the ontology includes
This list of examples is a tiny sample. I shall go on
extending it.
(Contributions welcome.)
NOTE: The order of the examples presented here is provisional.______________________________________________________________________________________
Later I'll try to impose a more helpful structure.
Some of the examples were inspired by this wonderful little book:J. Sauvy and S. Sauvy, The Child's Discovery of Space: From hopscotch to mazes -- an introduction to intuitive topology, Penguin Education, 1974, Translated from the French by Pam Wells, http://www.amazon.co.uk/The-Childs-Discovery-Space-Hopscotch/dp/014080384XA provisional collection of examples follows.
(To be extended and re-organised.)
(Needs to be re-ordered):
At first very young children playing with 'lift out' toys like these find it difficult to
insert a cut-out picture into its recess, even if they remember which recess it came from.
E.g. They put the picture down in approximately the right place and if it doesn't go in
they may press hard, but not attempt any motion parallel to the picture surface.
After a while they seem to learn that both the recesses and movable objects have
boundaries, and that when flat objects are brought together the boundaries may or
may not be merged.
At this more advanced stage, a child may place the picture object in roughly the right
place and then try sliding and rotating until it falls into the recess.
Still later, the child realises that boundaries can be divided into segments and that
segment of the recess boundary may match a segment of the object boundary, and then try to
insert the object by first ensuring that matching segments are adjacent and then slightly
varying the location and orientation of the piece until it falls into the recess.
Long before they can do this, I suspect they can insert a circular disc into a recess,
since there is no problem of alignment. If there are different discs and recesses of
different sizes the insertion requires size and location to be perceived and used in
controlling the insertion process. When the items are not symmetrical, inserting requires
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See the short, tentative, discussion in this PDF presentation:
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/math-order-stacking-sloman.pdf
Fiona McNeill provided this example of a domain still being
explored
and only
partially understood, in March 2009:
"One interesting aspect of Eilidh's ontology that I noticed over the weekend:NOTE: there is research reported elsewhere (Spelke's lab?) that shows a
She has stacking cups that go inside one another that she loves to play
with. Until recently, getting them to go in the right order was more or
less a case of trial and error, but she has just made a big step forward.
She is now very good at noticing 'holes' - so if she has, say, cups 2,3,5,6
all stacked, and 1,4,7 loose, she will immediately remove cups 2 and 3,
recognising with no apparent effort that something needs to go between them
and the bigger cups 5 and 6.
However, she seems to have no concept of relative size and will, seemingly,
pick up either 1, 4 or 7 with equal probability to put them in this hole,
not perceiving that 7 is clearly too big to go into 5 or that 1 is clearly
too small to fill the hole.
I would have thought that judging relative size, when there is a fairly
large difference in the sizes, would be far more instinctive than noticing
that cup 3 is a little loose in cup 5, which is not immediately obvious to
the eye. Apparently not!
She has also does not have the concept of 'largest object'. If she starts
off by picking up the biggest cup (cup 10), she will try to fit it into all
the others, and when it will not, instead of trying to fit something into
it, she tries again and again to fit it into another one, getting
increasingly frustrated. I usually put it down for her and put another in
it, and then she is happy to go on putting cups into it, but she has not
got this for herself yet."
Walking
(To be added)
Falling backwards
(Reported by
Michael Zillich
April 2009. Name of toddler changed.)
"LLLL last week suddenly learned to walk. It seems she figured that handlingTrampolines
her little suitcase while crawling was too cumbersome and so just stood up
and walked, carrying the suitcase around for hours :)
Now she also walks on quite uneven ground outside.
One really nice detail: She is quite good at maintaining balance (briefly
stopping to regain it when necessary) and at using her hands (and bottom)
to cushion falls, in case balance is truly lost.
But when she is in our bed, with soft cushions and blankets, she loves to
stand up straight and simply let herself fall backwards, with a relaxed
sigh. She knows she can only do this in bed. We did not teach or show to
her (I am too tall to do that) so she had to figure that out herself. And
she seems to enjoy the "thrill" of losing control."
Three children on a trampoline
I watched three children on a trampoline. The youngest seem to be pre-verbal
though he could walk and climb. The oldest was a boy who might have been four or
five years old. In between, was a girl who seemed to be at an intermediate age
(and size).
At one stage the girl started going head over heels on the trampoline: jumping
in such a way that her hands and head hit the trampoline with her trunk going
over. The other two were intrigued.
The little one seemed to want to do something inspired by her tumbling, but did
not seem to know what to do. He jumped around a bit stepping with alternate feet
on the trampoline then seemed to give up.
The older boy seemed to know that he had to do something about getting his head
down, but at first merely made clumsy and ineffectual movements. (I wish I had
had a video recorder.) After a few attempts he seemed to realise what was
necessary, and managed to go head over heels several times, rather clumsily at
first and then apparently with greater understanding of the combination of
movements needed to initiate the tumble, after which momentum and gravity could
complete the process.
I don't think any of them could express in a human communicative language what
they had learnt but clearly there was something in the information structures
they created internally, to function as a goal specification, as a control
strategy for actions to achieve the goal, as a critical evaluation of early
attempts, as a debugging process to modify the details of the action so as to
complete and "clean up" the final desired action.
Modelling this on a robot (possibly simulated -- to reduce the risk of damaging
expensive equipment!) would not be trivial. The process involves a mixture of
fine control with ballistic action and requires sufficient understanding to
manage the initial controlled movements in such a way as to launch the right
kind of ballistic action.
This is nothing like the standard statistical AI approach to learning which
requires a space of motor (or sensory-motor) signals to be sampled using
statistics (and perhaps hill-climbing) to direct the search, possibly using a
numerical evaluation/reward function.
Karen Adolph's work is also relevant:Playing on a slideKaren E. Adolph, Learning to learn in the development of action, in Action as an organizer of perception and cognition during learning and development: Minnesota Symposium on Child Development, 33, Eds. J. Lockman, J. Reiser and C. A. Nelson, Erlbaum, 2005, pp. 91--122, http://www.psych.nyu.edu/adolph/PDFs/MinnSymp2005.pdf However it is important to distinguish the acquisition of 'on-line' intelligence, investigated by Adolph, which involves learning to control actions as they are being performed (e.g. catching a ball, falling in a way that prevents injury) and 'off-line' or 'deliberative' intelligence which involves being able to represent and reason about classes of processes, including some of their invariant properties -- discovered in toddler theorems and later on in more sophisticated theories. Various kinds of deliberative competence are discussed in (Sloman 2006)
walking up a slope while holding onto a rope attached near the top.
[To be continued]
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This is an example of matter manipulation, a type of competence that
subsumes tool-use and many other things that have been studied in children and
other animals.
A broom can be thought of as a "tool for shifting dirt on a floor", but in the
video is not being used in that way. Rather the child appears to be moving the
broom around for its own sake, rather than for the sake of some other effect.
Such matter-manipulation sometimes has utilitarian functions (e.g. obtaining
food, putting on clothes, getting hold of some object that is out of reach) but
need not have. With or without serving an explicit goal of the manipulator the
processes seem to be a pervasive type of activity in very young children and
also some other animals.
Presumably this is because playful, exploratory, manipulation can provide much
information about, for example:
Suppose it is formed from a stretched rubber band held in place by pins.
There are many ways the shape, size, orientation and location of the triangle
could be transformed, by moving the pins.
Think of some possible changes do-able by moving one, or two or all three pins,
and for each change try to work out its consequences.
That is an easy task for a mathematician since much of mathematics is a result
of the human (animal?) ability to look at something and think about how it could
be changed, and what the consequences would be.
Most humans do it often in everyday life, e.g. when considering rearrangements
of furniture.
The ability to do this develops slowly and erratically in children -- and in
cultures! See also (Piaget & others, 1981, 1983)
Among the many possible ways you could alter the triangle, e.g. moving, or
rotating the whole thing there is one that involves moving only one pin,
parallel to the opposite side, in either direction, e.g. moving the top
pin here, parallel to the opposite side (the "base").
Another possibility involves moving the top pin up or down in either direction
perpendicular to the opposite side.
Can you see any interesting difference between those two sets of possible
changes to the configuration?
One set of changes will increase or decrease the total area of the interior of
the triangle.
The other set of changes will leave the area of the triangle unchanged.
Can you see why that must be so? Here's the explanation:
If you don't recognize what's going on, try reading this introduction to
thinking about triangles and their areas:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
The crucial point about such a diagram is that (like all diagrams used in proofs
in Euclidean geometry) the relationships perceived in the diagram do not depend
on the specific size, shape, colour, location, orientation, etc.
They don't even depend on the diagram being drawn accurately (with perfectly
thin, perfectly straight lines). That's because once the proof is understood
correctly its scope covers a very large class of abstraction. It's not clear
that people not trained in mathematics can easily think that way.
There's an interesting 'bug' in the proof-sketch as shown in the diagram
which is related to the need to do proper case analysis. It's a simple
example of the sort of phenomenon discussed by Imre Lakatos in Proofs
and Refutations, mentioned below. The bug in the 'chocolate' theorem,
discussed below, is another example. Identifying the bug is, for now, left
as an exercise for the reader, though mathematicians will find it obvious.
Max Wertheimer discussed an analogous bug in a proof given by a school
teacher regarding the area of a parallelogram, described in his book
Productive Thinking. More examples of buggy, but fixable, proofs are
given below.
[The relationship between this sort of bug and the problems a child has in
handling exceptions to grammatical rules in language may be illuminating,
as regards information processing architectures and mechanisms required.]
This human ability to reason about necessary consequences of alterations to
configurations in the environment may be closely related to Kenneth Craik's
hypothesis that some animals can use internal models of the environment to work
out consequences of possible actions. (Craik, 1943)
Compare also (Karmiloff-Smith, 1992), and Piaget's work on possibility and
necessity, and also Kant's philosophy of mathematics (Kant 1781).
Work that remains to be done includes finding out how a child, or non-human
animal, or future robot, could notice that some collection of structures and
processes forms a domain that has interesting properties, including
invariants that are discoverable by reasoning about the structures and
relationships, how the relationships can be discovered and supported by a
non-empirical argument, how different domains can be combined to form new
domains of expertise, and how all of this can lead to the phenomena of
Representational Redescription discussed by K-S.
We also still need to understand how to get robots and other learning machines
to go through similar procedures. See also:
http://www.cs.bham.ac.uk/research/cogaff/96-99.html#15 A. Sloman, Actual Possibilities, in Principles of Knowledge Representation and Reasoning: Proc. 5th Int. Conf. (KR `96), Eds. L.C. Aiello and S.C. Shapiro, Morgan Kaufmann, Boston, MA, 1996, pp. 627--638, Added 11 Sep 2013
Based partly on ideas by Mary Pardoe developed while she was teaching children
mathematics.
Here's an extract from that discussion:
ADDED 10 Sep 2012, Updated 9 Apr 2013: I now have a more detailed analysis of
requirements for discovering theorems in geometry here: "Hidden Depths of Triangle Qualia"
http://tinyurl.com/BhamCog/misc/triangle-theorem.html
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If you have a rubber band (elastic band), some pins, and a board into which the
pins can be stuck, you can make figures by using the pins to hold the band
stretched into a shape bounded by straight lines (if the band is stretched
between the pins).
The following are sample questions about what is possible, what is impossible,
and how many pins or rubber bands are needed to make something possible.
For example, you can make a triangle, a square, an outline capital "T" with one
rubber band and a set of pins?
Is it possible to make an outline capital "A" ?
Is it possible to make a circle?
Is it possible to make a star-shaped figure, with alternating convex
and concave corners?
What's the minimum number of pins required for that?
How can you be sure?
For more examples see
http://tinyurl.com/BhamCog/talks/#toddler
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There are more videos with very short comments that need to be
expanded, here:
http://www.cs.bham.ac.uk/research/projects/cosy/conferences/mofm-paris-07/sloman/vid/
For a PDF presentation on learning about different kinds of 'stuff' see
http://tinyurl.com/BhamCog/talks/#brown From "baby stuff" to the world of adult science: Developmental AI from a Kantian viewpoint.___________________________________________________________________________________
"Today might be much more hotter than it usually bees"More generally, the phenomena of "U-shaped" language learning provide many clues
Consider a slow moving van and a fast moving racing car. They start moving towards
each other at the same time.
The racing car on the left moves much faster than the van on the right:
Whereabouts will they meet -- more to the left or to the right, or in the middle?
One five year old answered by pointing to a location on the left, somewhere
near "b" or "c".
Me: Why?
Child: It's going faster so it will get there sooner.
What produces this answer? Could it be:
The first premiss is a buggy generalisation: it does not allow for different
kinds of "race".
The others have conditions of applicability that need to be checked.
Perhaps the child had not taken in the fact that the problem required the racing
car and the van to be travelling for the same length of time, or had not
remembered to make use of that information.
Perhaps the child had the information (as could be tested by probing), but
lacked the information-processing architecture required to make full and
consistent use of it, and to control the derivation of consequences properly?
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Is Vygotsky's work relevant?
Some parts of Piaget's theory of "formal operations"?
Compare Karmiloff-Smith on "Representational Redescription",
discussed in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/beyond-modularity
Could the child's reasoning be evidence for a process of representational
redescription that is still incomplete: i.e. generally useful items of
information that can be recombined in different contexts have been extracted
from the collection of empirically learnt associations. But the conditions for
recombination, and the constraints on applicability of inferences, have not yet
been discovered. In principle, this looks like a type of learning that could be
modelled in terms of construction of a rule-set capable of supporting deductive
inference.
(I think Richard Young's PhD thesis around 1972 was concerned with a process
something like this, but involving ordering of objects by height.)
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Today, our daughter Ada (named for Lovelace), who turned 2 earlier this month, said "Kitties have tails. I do not have a tail. I'm not a Kitty."Is it possible that a two year old has grasped the general principle that from
Xs have Ys A doesn't have Yit follows that
A is not an X ?Some initial thoughts about this:
P or Q not-P Therefore QHe was assembling a jigsaw puzzle with help from an adult. Together they had reached
Does starting from a different configuration change what is possible?
Can you get from configuration (a) below to configuration (b), using
only diagonal moves?
The next one is harder:
How people work on such problems differs according to prior knowledge and
experience.
Sometimes proving that something is impossible can be done by exhaustive search
(though understanding the need to ensure that the search is exhaustive is an
achievement, as is organising the search so as to ensure exhaustiveness.
A different kind of competence can lead to a much more economical explanation of
why the task is impossible.
The core characteristic of mathematical thinking, which frequently motivates new
developments in mathematics is productive laziness, which I suspect begins
to develop between ages 1 and 3 years.
This is a case where the advance of knowledge involves noticing that a
particular problem is a special case of a general type of problem.
(If a problem is too hard to solve, trying a harder one sometimes gives new
insights.)
If you have not noticed the easy way to solve the above problems consider what
difference it would make if the squares were black and white, as on a chess
board. Mathematicians can use the notion of "parity" here. E.g. giving squares
coordinates, they can be divided into two classes: those whose coordinates sum
to an even number and those whose coordinates sum to an odd number. The squares
in a horizontal or vertical line will have alternating parity.
Squares in a diagonal line will have the same parity. This makes it very easy to
check whether a start configuration can be transformed to a target
configuration.
Normally such discoveries are made only by adult or bright mathematical
learners. My point is that a young child could learn some of the generative
facts about the diagonal moving coin domain by playing. Using a two-colour grid
will make some things easier to learn. (Why?)
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Jump to CONTENTS List
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See
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/orthogonal-competences.html#blanket
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Starting from the configuration on the left the aim is to get to the
configuration on the right, without disconnecting the rope from the two disks
at its ends.
(This picture is from the very interesting paper by Cabalar and Santos, below.)
There are many more puzzles shown and offered for sale at the "MrPuzzle"
web site, e.g.
See also http://www.mrpuzzle.com.au/
Dealing with such puzzles requires the ability to think about topology-preserving
transformations of physical objects involving flexible inelastic strings, beads,
discs, and various rigid objects with holes and slots through which string and
other things can pass.
In many cases it is also important to make use of non-topological relationships
such as relative size (e.g. a bead is too large to pass through a hole, and a
string loop is too short to pass over the far edge of an object).
In such cases, an important kind of discovery is how an alteration that does not
transform the topology can transform a metrical relationship. E.g. pulling part
of a string from one portion of the puzzle to another portion can increase the
size of a loop until some object can pass through it that previously could not.
For each class of puzzle there can be a wide range of possible actions to
consider. In particular the learner may need to learn:
There seem to be many different domains/microdomains a learner can explore:NB
including the possible processes associated with a particular puzzle, the
possible processes associated with a class of puzzles, and the
possibilities created by combining features of different puzzles.For more on such puzzles and formal reasoning about them see
Pedro Cabalar and Paulo E. Santos, Formalising the Fisherman's Folly puzzle, AIJ, 175,1,pp 346--377, 2011 http://www.sciencedirect.com/science/article/pii/S0004370210000408
"If I push that disk through the slot, I shall then be able to slide the ring up over the top of the post, but..."Such thoughts seem to make intrinsic use of the structure of the perceived
(The 1971 paper made a distinction between "Fregean" representations, where all
syntactic complexity represents application of functions to arguments, and
"Analogical" representations in which parts of the representation represent
parts of what is represented, and properties and relations within the
representation represent properties and relations within the thing represented.
It is often assumed that analogical representations must make use of
isomorphisms, but the paper showed that that is not true. In particular a
particular syntactic property or relation (in the representation) can have
different semantic functions in different contexts, representing different
properties and relations in the scene depicted. That's trivially obvious for 2-D
representations of 3-D scenes, since isomorphism is impossible in that case.)
These questions are all related to the question: what sort of understanding of
the puzzle (and what form of representation of that understanding) allowed the
authors to discover the axioms that characterise it well enough to be used by an
AI system? This is also related to the problem of how our ancestors perceived,
thought and reasoned about spatial structures and relationships before Euclidean
geometry had been codified, and even longer before cartesian coordinates were
used to represent geometry arithmetically and algebraically.
It seems very likely that those pre-Euclidean and pre-Logical forms of
representation and reasoning are still used, unwittingly, by young children and
by other animals with spatial intelligence, e.g. nest-building birds and hunting
animals.
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One problem with a concept of numerosity based on combining (a) an ability to
detect and estimate density and (b) an ability to detect and estimate some sort
of spatial or temporal extent (of a linear interval, an area, a volume, a
temporal interval, etc.) is that when the density varies across the items, then
an average density has to be computed to get a measure for the whole set. Since
density is already an average, that requires averaging a spatially varying
average -- a non-trivial computation. Another problem is detecting whether two
densities or two areas are the same. The larger the areas the harder it may be
to compare densities accurately. In particular, the harder it is to tell if A's
numerosity is greater than B's. So more dots may need to be added to a large
collection to make the size difference noticeable. This means that the graph of
perceived numerosity against actual cardinality flattens out as cardinality
increases. This may take a logarithmic form.
(I have no idea whether anyone has actually investigated which of these
computations brains are capable of, for which modes of sensory input.)
If a child (or animal) with an ability to estimate numerosity as described
above, perceives two groups G1 and G2 which have both different sizes and
different densities comparing numerosity is much harder than where G1 and G2
have the same density, or the same extent. If the density is roughly uniform
within each group, and if the perceiver can compute numerical values for both
density and area or volume, then the two numbers can be multiplied to provide an
estimate of numerosity. The ability to multiply seems to require a prior grasp
of numbers, but that can be avoided if the multiplication is done by dedicated,
domain specific machinery. In that case, there can be no comparison of
numerosity of a sequence of heard sounds and numerosity of dots scattered around
an area.
However when both numbers are small they can be compared directly by some form
of counting, or setting up a one to one correspondence between the sounds and
the dots. That will show if there are more of one than the other. So in that
case the ability to estimate cardinality directly removes the need to compute
numerosity by performing a multiplication of density and extent.
It seems that humans can compute and compare numerosities from quite an early
age (e.g. before being able to count), but they get better as they grow older
(and presumably have more experiences of numerosity judgements), and also
gradually get a better meta-cognitive understanding of what they are doing.
Before that, as Piaget showed, they can display extraordinary confusions because
they don't yet have a concept of cardinality as something that is conserved as
objects are packed closer together or spread out more.
If the distribution of items in the space is highly irregular the task of
comparing numerosities can become very difficult, and in some cases deceptive.
There's a lot more to be said about numerosity, but for now the main point is
that it is a totally different concept from cardinality, which is fundamentally
connected with the notion of a one to one mapping, and researchers who don't
make this distinction often write as if there were just one concept of number.
The following seems to be a fairly standard (but mostly unnoticed by researchers??)
way of acquiring cardinality competences, though these components are not learnt
in sequence, but interleaved:
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A child given a set of wooden cube-shaped blocks can do all sorts of experiments --
exploring the space of processes involving the blocks.
Then the child may notice that attempts to rearrange a configuration
into a rectangle always fail:
What kind of experimentation can that provoke, and what sorts of
discoveries can be made?
How could one be sure that there is NO way of arranging the last collection into
a rectangular array, apart from the straight line shown?
Could such a child discover the concept of a prime number?
When I discussed this collection of examples (discovering theorems about
prime numbers and factorisation by playing with blocks) with some people at
a conference, one of them told me he had encountered a person at a
conference registration desk who liked to keep all the unclaimed name cards
in a rectangular array. However she had discovered that sometimes this was
not possible, which she found frustrating. She had discovered that some
numbers of prime, though I assume she had not worked out any of the
implications.
Could the child rearranging blocks discover and articulate the fundamental
theorem of arithmetic?
(The unique factorization theorem.)
Are some forms of mathematical discovery impossible without a social
environment?
Don't assume a teacher with prior knowledge of the theorems has to be involved:
someone must have made some of these discoveries without being told them by a
teacher.
NOTE 1
One of the fundamental requirements for mathematical thinking is being able to
organise collections of possibilities and making sure that you have checked them all.
If you can't do that you don't have a mathematical result, only a guess.
NOTE 2 (9 Aug 2012)
I have just discovered that this kind of discovery of primeness by a computer
program was discussed in
Alison Pease, Simon Colton, Ramin Ramezani, Alan Smaill and Markus Guhe, Using Analogical Representations for Mathematical Concept Formation, in Eds. L. Magnani et al, Model-Based Reasoning in Science & Technology, Springer-Verlag, pp. 301-314, 2010, http://homepages.inf.ed.ac.uk/apease/papers/pease_mbr09.pdf
Someone who can accurately visualise the effect of moving one bounded area while
another remains fixed, or who can cut out the area and move it onto another, may
discover that area A can fit entirely inside B. So the area of A is less than
the area of B.
However, the shape A cannot be contained in C, and C cannot be contained in A.
Moreover, C cannot be contained in B, and B cannot be contained in C.
This means it is impossible to rank shapes A, B and C in area on that criterion.
They form only a partial ordering relative to the containment criterion.
Someone who has (somehow -- this is non-trivial) discovered a way of assigning
measures of area to rectangular shapes, and then has discovered that that can be
extended to a way of assigning measures to triangles:
area = half(base x height)could realise (how) that any area bounded by straight edges (i.e. any polygon)
http://www.cs.bham.ac.uk/research/projects/cogaff/10.html#1001
If Learning Maths Requires a Teacher,
Where did the First Teachers Come From?
Alan Bundy has reminded me that some children learn from clock faces and other
structures that it is possible to do a kind of counting that goes up to a
certain number and then re-starts from 1, for instance reciting the numbers on
an old-fashioned clock face.
For mathematicians, this is a special case of 'modulo' arithmetic, namely
arithmetic in which there is only a finite set of numbers and counting beyond
the largest number always starts again from 1.
For example, 3+4 modulo 5 is 2, 3+4 modulo 6 is 1, 3+9 modulo 6 is 0.
If we assign numerical coordinates to rows and columns of a chess board, then
associate each square on the board with the sum of its coordinates, then the
bottom left 3x3 corner would have these numbers:
456 345 234However, if each square is associate with number sum of coordinates
010 101 010
You have a slab of chocolate in the form of a 7 by 7 square of pieces dividedThe puzzle draws attention to a domain of processes of subdivision of a rectangular
by grooves, and you want to give 49 friends, each one piece.
You have a knife that can cut along a groove.
What is the minimum number of groove cuts that will divide the bar into
49 pieces?
RULES FOR CHOCOLATE CHOPPING:
Stacking or overlaying two or more pieces, or abutting two pieces, to divide
them both with one cut is not allowed: each cut is applied to exactly one of the
pieces of chocolate.
It's an exception because the original argument assumed that every
cut divides one piece into two pieces.
With holes, is it a slab or isn't it?
Often a proof in mathematics that seemed valid works for a range of
cases, but has counter-examples not thought of when the proof was
constructed, or when it was checked.
Many such examples connected with the history of Euler's theorem about
plane polyhedra were discussed in this famous book.
Imre Lakatos:One of the consequences of our ability to perceive, imagine, or create instances
Proofs and refutations: The Logic of Mathematical Discovery
Cambridge University Press, 1976
This is different from the empirical refutation of "All swans are white",
which turned up in Australia.
It is important not to inflate Lakatos' argument in Proofs and Refutations
as demonstrating that there is never any real progress in mathematics, or
that mathematics is empirical.
On the contrary, every mistake that leads to a revision of a definition, or a
statement of a theorem, or a proof adds to our mathematical knowledge:
mathematicians can make non-empirical discoveries without being infallible.
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Things you probably know, but did not always know:
The idea of an "aspect graph" can be viewed as a special case of a domain
of actions related to changing epistemic affordances (as defined above).
That's not normally how aspect graphs are presented. Normally the aspect
graph of an object is thought of as a graph of topologically distinct views
of the object linked by minimal transitions. For example as you move round
a cube some changes in appearance will merely be continuous changes in
apparent angles and apparent lengths of edges, but there will be
discontinuities when one or more edges, vertices or faces goes in or out of
view. In the aspect graph all the topologically equivalent views are
treated as one node, linked to neighbouring nodes according to which
movements produce new views, e.g. move up, move down, move left, move
diagonally up to the right, etc. For a non-convex object, e.g. an L-shaped
polyhedron the aspect graph will be much more complex than for a cube, as
some parts may be visible from some viewpoints that are not connected by
visible portions.
Here's a useful introduction By Barb Cutler:
http://people.csail.mit.edu/bmcutler/6.838/project/aspect_graph.html
Some vision researchers have considered using aspect graphs for recognition
purposes: a suitably trained robot could see how views of an object change
as it moves, and in some cases use that to identify the relevant aspect
graph, and the object. (Related ideas, without using the label "aspect
graph" were used by Roberts, Guzman and Grape for perception of polyhedral
scenes in the 1960s and early 1970s, though the scenes perceived were
static.)
However, for complex objects aspect graphs can explode, and in any case, we
are not concerned with vision but with understanding perceived structures.
A perceiver with the right kind of understanding should be able to derive
the aspect graph, or fragments of it, from knowledge of its shape, and use
that to decide which way to move to get information about occluded
surfaces.
In 1973 Minsky introduced a similar idea for which he used the label "Frame
system".
http://web.media.mit.edu/~minsky/papers/Frames/frames.html
A few years ago, in discussion of plans for the EU CoSy project
http://www.cognitivesystems.org/,
Jeremy Wyatt suggested an important generalisation. Instead of considering
only the effects of movements of the viewer on changing views of an object
we could enhance our knowledge of particular shapes with information about
how things would change if other actions were performed, e.g. if an object
resting on a horizontal surface is touched in a particular place and a
particular force applied, then the object may rotate or slide or both, or
if there is a vertical surface resisting movement it may do neither.
This suggested a way of representing knowledge about the structure of an
object and its relationships to other surfaces in its immediate
environment, in terms of how the appearance of the object would change if
various forces were applied in various directions at various points on the
surface, including rotational forces.
This large set of possibilities for perceived change, grouped according to
how the change was produced, we labelled a "Generalised Aspect graph". This
would be even more explosive than the aspect graph as more complex objects
are investigated. For various reasons, we were not able to pursue that idea
in the CoSy project (though a subset of it re-emerged in connection with
learning about the motion of a simply polyflap in work done by Marek
Kopicki).
In currently favoured AI approaches to perception and action the standard
approach to use of generalised aspect graphs would require a robot to be
taught about them in some very laborious training process.
In the context of an investigation of "toddler theorems" the problem is
altered: how can we give a robot the ability to understand spatial
structures and the effects of forces on them so that instead of having to
learn aspect graphs, or generalised aspect graphs, it can derive
them, or fragments of them, on demand, as part of its understanding of
affordances.
That, after all, is what a designer of novel objects to serve some purpose
needs to be able to do.
However, in order to reduce the combinatorics of such a derivation process
I suggest that the representation of objects used to work out how the would
move, should not be in terms of sensory-motor patterns (not eve multi-modal
sensory-motor patterns including haptic feedback and vision), but in terms
of exosomatic concepts referring to 3-D structures in the environment and
their surfaces and relationships, independently of how they are perceived.
Prediction of how a perceived scene would change if an action were applied
would take two major steps: first of all deriving the change in the
environment, and secondly deriving the effect of that change on the visual
and tactile experiences of the perceiver. Among other things that would
allow reasoning to be done about objects that are moved using other held
objects, e.g. rakes, hammers, and also reasoning to be done about what
other perceivers might experience: a necessary condition for empathy.
This is a complex and difficult topic requiring more discussion, but I
think the implications for much current AI are deep, and highly critical,
since so much work on perceiving and producing behaviour in the environment
does not yield the kind of understanding provided by toddler
theorems, an understanding that, later on, can grow into mathematical
competence, when generalised and articulated.
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Jump to CONTENTS List
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What sorts of representational, architectural, and reasoning (information
manipulation) capabilities could enable a child to work out
The answer seems to have two main aspects, one non-empirical, to do with
consequences of surfaces moving towards each other with and without some
object between them, and the other an empirical discovery about
relationships between compression of, or impact on, a body part and pain.
A sign that the child has discovered a theorem derived in a generative
system, may be the ability to avoid shutting a door by grasping its
vertical edge without first having to try it out and discover the painful
consequence.
Perceiving the commonality between what happens to the edge of a door as it
is shut (a rotation) and what happens to the edge of a drawer when it is
shut (a translation) seems to require the ability to use an ontology that
goes beyond sensory-motor patterns.
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I noticed a very young child (age unknown, though he could stand, walk, and
manipulate a hoop, but looked too young to be talking) playing with a hoop
on a trampoline in the garden next door.
He seemed to have learned a number of things about hoops, including
Why it is easier to carry a tray full of cups and saucers using a hand at
each side than using both hands on the same side?
Why is it easier with two grasp points than with only one?
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It is very unlikely that Sofya has had to learn every possible combinationhttp://www.youtube.com/watch?v=cij-cT5ZkHo Early, partially successful attempts. http://www.youtube.com/watch?v=FmH8jFLrwDU Fairly expert performance.
http://nlcsa.net/lc1a-nls/lc1a-video/ "Under Arrest"illustrated many different things simultaneously, including how two part-built
PRESENTATIONS (PDF)
PAPERS, CHAPTERS, BOOKS
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham
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