Registered: Aug 2003
Posts: 712

At this year's summer program at Brown, we considered ways of furthering NKS in education. One idea I had was to smuggle in NKS considerations through standardized tests, given the manner in which teachers and students these days focus on such tests. NKS seems to lend itself to such questions, which should not require previous familiarity with the subject, and should be readily answered, accurately by any decent student, in less than a minute. I thought others here might have some ideas for simple questions along these lines.

Here are some sample subjects -

Following finite state machine transitions

Composing transformation steps

Noticing cycles or necessary end states in FSMs

"Recoding" a FSM

e.g. states 1,2,3 are called A, states 4, 5 are called B,
give the A-B sequence this FSM follows starting at 2.

FSM rules given in English.

There are some very simple exercises like this early in Ashby.

Also sSimple substitution system questions, followed for just 1, 2, 3 steps.

Slightly harder ones, e.g. can you reach string xxxx from yyy using rules a b c, where there is a simple trick.

Filling in a pair of blank squares in a CA given the entire rule (pair used to get 4 multiple choice outcomes).

Filling in a pair of blank squares in a CA given only the pattern (where one must notice the rule - given its form, though). Simple rule cases.

Sequence questions, like how many steps does 3n+1 take to get to 1 starting from this number or that number - relatively easy cases.

Matthew Szudzik suggested the following substitution system questions -

(1) Using the substitution rules

0 -> 00
0 -> 11
1100 -> 01

can 00 be transformed into 111?

Solution: Yes, 00 -> 000 -> 1100 -> 01 -> 111.

This example comes from page 776 of the NKS book.

(2) Using the substitution rules

1 -> 010
001 -> 1
11 -> 0

can 101 be transformed into 01110?

Solution: No, the rules can never increase the number of 1's in a
sequence.

(3) Using the substitution rules

0 -> 001
10 -> 011
0011 -> 101

can 0100 be transformed into 101?

Solution: No. Starting from 101 and following the rules backwards we see that there are only three possible sequences that can be transformed into 101, namely 0011, 010, and 01.

Note that I chose to use 0's and 1's instead of A's and B's to avoid confusion, since standardized tests like the SAT usually assign the letters A, B, C, etc. to potential answers.

Matthew Szudzik

Here are some Ashby like questions about simple transformation rules -

A transformation is a list of rules that say what each starting state, marked as some letter, goes to next, written as e.g. -

A: a -> c, b -> c, c -> a

A composition or
product of two transformations is simply the result of applying each individual transformation in sequence.

If A is as given above, what is the transformation A A = A ^ 2?

(1) A^2 : a -> a, b -> a, c-> c
(2) A^2: a->c, b -> c, c->c
(3) A^2: a->e, b->d, c->a
(4) A^2: a->c, b->a, c->c

Which of the following is A^4? (...)

If A is as above and B is B: a->b, b->c, c->a, then which of the following is the transformation B A B? (...)

It seems to me there is a fine opportunity here to smuggle NKS style thinking into the curriculum. Anyone can answer such simple questions without ever having seen them before, but students will prefer to have seen them before, in the interest of testing speed among other things.

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Registered: Nov 2004
Posts: 23

As schools start to replace the books with computers - iBook from Apple - jump to me from the New York Times, it would be good if a modern implementation of NKS Explorer in Java or Cocoa could be placed on those iBooks and let the kids discover them.

If a videogame could be created where all the visionary effect are the consequences of CA evolutions and the kids could find "cheat" codes /automata rules/ to put in via a command line interface /like Mathematica has/, I think you just could not tell the kids to STOP discovering :)

__________________
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"..because Annushka has already bought
sunflower oil, and not only bought it, but
spilled it too."
Bulgakov: Master and Margarita

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Registered: Aug 2003
Posts: 712

NKS games will happen. In fact, I wrote a simple one last week. It uses some routines that aren't out yet, so I can't post it here. But the idea is perfectly feasible.

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