Next: About this document ...
Intelligent Patterning
or
Why I've been doing computer science
Brief overview of where I'm headed:
- General problem solving
- Pattern recognition
- Symbols and signs
- Intelligent patterning
- Some history
- What's wrong in computing today
- The intelligent mathematical assistant
General problem solving
- Understanding the problem
- 1.
- Problem context and statement of
the problem
- 2.
- Solving the right problem (ill-posed and
ill-conditioned problems)
- 3.
- Preconceptions
- 4.
- Language and restating the problem
- The role of experience
- 1.
- Similar problems and analogy
- 2.
- Appropriate tools
- 3.
- Specific experience
- Three basic methods
- 1.
- Plug and grind
- 2.
- Guess and prove
- 3.
- Look it up
- Hypothesis generation and testing
- 1.
- Flexibility and freedom -- willingness to
try and fail
- 2.
- Recognizing blind alleys, and the value of
exploring
- 3.
- Appropriate hypotheses
- 4.
- Lateral thinking
- Recognizing solutions
- 1.
- ``A'' solution vs. ``the'' solution
- 2.
- Useful solutions
- 3.
- When a ``solution'' solves an un-posed, but more
significant problem
Pattern recognition
- Images (``visual patterns'') vs.
``syntactic'' patterns
- Symbols as patterns, and symbols as
pattern labels
- Patterns of symbols
- Hierarchies of patterns, and symbols as
tools for recognizing patterns
- Pattern manipulation
- Learning to recognize patterns, and pattern
recognition as learning
Pattern recognition examples
- What number comes next in the sequence?
1, 1, 2, 3, 5, 8, 13, ...
- What number comes next in the sequence?
8, 5, 4, 9, 1, 7, 6, 3, ...
- What letter comes next in the sequence?
E, T, A, O, I, N, S, H, ...
- In which row does Z go?
A, E, F, H, I, K, L, M, N, T, V, W, X, Y
B, C, D, G, J, O, P, Q, R, S, U
- What letter comes next in the sequence?
W, L, C, N, I, T, ...
Symbols and signs
- The utility and power of symbols
- Choosing symbols, naming and pointing
- Symbols as ``chunking'' tools
- When to use symbols
- 1.
- The importance of anonymity (e.g., the
lambda calculus)
- 2.
- Place holders (variables)
- 3.
- Temporary and tentative symbols
- Signs, symbols, content and meaning
Intelligent patterning
- Creativity and Art
- 1.
- Knowing when to pattern
- 2.
- Symbol attachment and creation;
patterns/symbols as revealers and
concealers
- 3.
- Levels of patterning
- Multiple patterns and selection
(x - 1)(x - 2)(x - 3) - 6
x3 - 6x2 + 11x - 12
(x - 4)(x2 - 2x + 3)
- Adaptive pattern recognition
- Are the patterns really there?
Some history
- Physics
- Philosophy (theory of knowledge)
- Mathematics
- 1.
- Matrix manipulation
- 2.
- Topology
- 3.
- Algebra
- 4.
- Lie groups
- 5.
- Manifolds and relativity theory
- 6.
- Algebraic topology
We have the map
given by
where i(g) is the inclusion,
and
vn(r,s) =
where
We have the map
$ b_n: \Sigma^2U(n) \rightarrow SU(n+1) $ \newline
given by
\[ b_n(g, r, s) = \left[ i(g), v_n(r, s) \right] \]
where $i(g)$ is the inclusion,
$\left[g, h\right] = ghg^{-1}h^{-1}$ \newline
and
$ v_n(r,s) = $
\[
\left[ \begin{array}{cccccc}
\alpha & 0 & 0 & \cdots & 0 & \beta (-\overline{\alpha})^0 \\
\beta (-\overline{\alpha})^0\overline{\beta} &
\alpha & 0 & \cdots & 0 &
\beta (-\overline{\alpha})^1 \\
\beta (-\overline{\alpha})^1\overline{\beta} &
\beta (-\overline{\alpha})^0\overline{\beta} &
\alpha & \cdots & 0 & \beta (-\overline{\alpha})^2 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
\beta (-\overline{\alpha})^{n-1}\overline{\beta} &
\beta (-\overline{\alpha})^{n-2}\overline{\beta} &
\cdots & \cdots & \alpha &
\beta (-\overline{\alpha})^n \\
-(-\overline{\alpha})^n\overline{\beta} &
-(-\overline{\alpha})^{n-1}\overline{\beta} &
\cdots & \cdots & -(-\overline{\alpha})^0
\overline{\beta} & -(-\overline{\alpha})^n \\
\end{array} \right]
\]
where
\[ \alpha = \alpha(r,s) =
\cos(\pi r) + i \sin(\pi r)\cos(\pi s) \]
\[ \beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s) \]
What's wrong in computing today
- Not enough resolution on displays
- Not enough processing power and memory
- Not enough parallelism
- Software tools are ``flat'' and sequential
rather than hierarchical
The intelligent mathematical assistant
- Adaptive symbolic input and output
- Strong basic skills (all of arithmetic
through college calculus and
elementary discrete structures)
- First order logic capabilities
- Adaptive ``patterning'' and ``symboling''
- Elementary hypothesis generation
and testing
Next: About this document ...
Tom Carter
1999-04-21