KNOTS OVERVIEW
Knotty Reflections
(DRAFT: Liable to change)

Aaron Sloman
School of Computer Science, University of Birmingham.
(Philosopher in a Computer Science department)

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Installed: 26 May 2014
Last updated: 28 May 2014, 4 Jun 2014; 5 Jun 2014

This web site (with four subsidiary sites listed below) is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/knots/
A PDF version may be added later.
A partial index of discussion notes is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
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This is part of a collection of examples of abilities of humans to reason in a mathematical way even if they don't know they are doing it.

For more examples, see the 'Toddler Theorems' Web page:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
including examples of reasoning about triangles, e.g.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html
and this discussion of blankets, strings, and other things for pulling objects nearer.

The processes involved in detecting and understanding knots are also relevant to problems in designing perceptual (especially visual) systems. It seems that "knottedness", like "connectedness" can be computationally hard to detect, and I suspect that like connectedness it is highly resistant to parallel visual processing in a fixed number of steps. For an incomplete discussion of limitations of current AI/Robotics visual mechanisms and theories of vision in psychology/neuroscience (work in progress) see:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vision

Knotty and not-knotty perceptual problems

This directory contains several sets of images of strings with and without knots. Each set of images is associated with a problem about possibilities for motion of a string or rope lying on a flat surface, and curled or looped in various ways, some with some and without knots. In some cases, it is hard to tell whether a knot exists without attempting to pull the ends of the string apart.

This is related to an earlier collection of examples (here) (loosely inspired by some of Piaget's experiments with young children, involving use of blankets, string, and other things to bring an object within reach. [REF needed]

Four sets of example problems: in pictures

The photographs are not very good, and may later be replaced by clearer images. In some cases the thicker string (rope) was a little too stiff for the purposes of these displays, so I may try again with a more flexible rope.

  1. Distinguishing knotted and knot-free (1)

  2. Distinguishing knotted and knot-free (2)

  3. Which configurations are simple modifications of others?

  4. Recognizing a configuration from different viewpoints Some of these are easy others much more difficult. (Why?)

Some Questions

For each set of images there is a set of questions which you can try to answer by examining and thinking about possible changes in the string configurations, or effects of changes of viewpoint. Some of the examples are very simple, others more challenging. (Answers to the questions will be added later.)

Here are some additional questions:

  1. Can any existing machine vision system perform any of these tasks?
  2. Can any known machine learning mechanism enable a robot vision system to perform these tasks?
  3. At what ages can young humans perform tasks like these?
  4. What changes in an individual between not being able and being able to perform the tasks?
    • E.g. are new forms of learning needed? Are new forms of representation needed?
    • Are new ontological extensions (adding types of structure, adding types of process) needed?
    • Are new forms of reasoning required?
    • Do new brain mechanisms have to be grown, or activated? If so, will the developments be genetically programmed, or based partly on the genome and partly on what has been learnt earlier, as suggested in Chappell & Sloman (2007)?
  5. Can logical/algebraic/symbolic-computational forms of representation suffice to implement these abilities?
  6. Can individuals have and use such abilities without being aware of the fact?
  7. What sort of knowledge about strings, space, motion, and continuity do you need in order to understand why the potential to be straightened without a knot is preserved by moving only parts of a string between its two fixed ends. Could you prove this using logic?
  8. It is possible in principle to program a machine to use an algorithm that traverses its percept of a string from one end to the other recording the crossing points as 'going over' or 'going under' to build a 'characteristic' abstract description of the configuration. This description will be preserved if portions of the string are moved around on the surface while the ends of the string are held in place. [Problems about this sort of algorithm.....]
  9. Mathematicians have studied knots for some time, as indicated in Braungardt's introduction (REF below). One of the interesting facts about knots, illustrated by the difficulty of some of the above tasks, is that there are mathematically specifiable types of knots that can have many different appearances, and which are very difficult for a human visual system to distinguish. This is one of many ways in which the biological mechanisms for perceiving and representing types of physical structure are limited in the complexity that they can handle, despite being extremely fast and extremely versatile in the domains that they cope with, unlike current artificial vision systems.
  10. The possibility of mathematical understanding of knots (and many other things) that extends far beyond the limits of what humans can perceive (or learn to perceive) is one of several objections to statistics-based models of human learning and intelligence (e.g. Bayesian brains...), since those can only provide statistical generalisations, not mathematical proofs, as in Euclidean geometry, topology, etc. [To be expanded.]

Background: Possibility transducers
(Read this after you have tried the string problems, above.)

Sloman (1996) suggested that an important aspect of human and, more generally, animal intelligence is the ability to perceive, think about and make use of possible variants of a perceived or known structure, process or situation.

Moreover we can discover systematic relationships between the set of possible changes that can be made to some entity (inputs) and the set of possible outcomes of those changes (outputs). In that sense particular objects, processes or situations can be thought of as possibility transducers.

The ability to think about possibilities and to recognise properties of possibility transducers is part of the ability to perceive and make use of what J.J. Gibson referred to as "affordances" in the environment, a concept that can be extended far beyond the sorts of cases Gibson considered. (REF) This is also related to philosophical discussions regarding the nature of causation and causal powers. (Many philosophers base their analyses on the notion of "sets of possible worlds" some including this world. I don't believe that any such semantic competence was available to our ancestors and other animals that make intelligent use of possibilities and constraints in the environment. Instead we and they need to be able to think about possible variations in an object, a process or a state of affairs that is a bounded portion of this universe.)

The examples above present some of the ways of thinking of a flexible string resting on a flat surface as a transducer of sets of possibilities in which the string is preserved, its total length does not change, but its shape, the geometrical and topological relationships of parts of the string and portions of space that they occupy, change.

Learning about such things seems to be important for some nest-building animals (most obviously weaver birds) and for young children growing up in certain environments, e.g. with shoe-laces, slings, bows and arrows, tents with tent ropes, and others. As far as I know very little is understood about how biological brains acquire and use such competences, and I don't know of any work on intelligent robotics addressing this problem, though the problem is recognized in Cabalar and Santos (2011)

There are now many online videos of weaver birds. Here's a short BBC video: http://www.youtube.com/watch?v=6svAIgEnFvw&feature=kp

REFERENCES

(To be expanded) 

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