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 x^{3} − 6x^{2} + 11x − 12 (x − 4)(x^{2} − 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 b_{n} : Σ^{2}U(n) → SU(n + 1) given by

b_{n}(g, r, s) = [i(g), v_{n}(r, s)]

where i(g) is the inclusion, [g, h] = ghg^{−1}h^{−1} and

v_{n}(r, s) =

[ \begin{matrix}α & 0 & 0 & ⋯ & 0 & β(−α)^{0} \\ β(−α)^{0}β & α & 0 & ⋯ & 0 & β(−α)^{1} \\ β(−α)^{1}β & β(−α)^{0}β & α & ⋯ & 0 & β(−α)^{2} \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ β(−α)^{n−1}β & β(−α)^{n−2}β & ⋯ & ⋯ & α & β(−α)^{n} \\ −(−α)^{n}β & −(−α)^{n−1}β & ⋯ & ⋯ & −(−α)^{0}β & −(−α)^{n} \end{matrix} ]

where

α = α(r, s) = cos (πr) + i sin(πr) cos (πs)
β = β(r, s) = i sin(πr) sin(πs)

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