051014 Fri 1830 Novato, CA rev. 051015 Sat 1750 (c) 2005 by Walter Bloch
THREE RELATED LEXICAL PROBLEMS IN RETRIEVING DATA
1) HOW CAN YOU GOOGLE WHEN YOU DON’T HAVE A SEARCH KEY?
Consider the following (famous) algorithm A applying to positive integers n:
A[n]: if n = 1 then terminate;
If n is even, then A[n/2];
For any n, there are three possibilities:
The sequence terminates, e.g.:
9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The sequence repeats a number and loops forever.
The sequence is nonterminating and nonrepeating, and so is unbounded, since under a ceiling a repeat is inevitable.
For any n, in fact, the sequence terminates; last time I checked, this was unprovable.
How can I google to check for progress, when I have forgotten the name of the sequence?
This example is from life; it inspired me to think about this and the related questions. It illustrates the point well enough, but it is not a perfect example, as it turns out.
Carol von Haden immediately found the Collatz Problem (1937), also known as Kakutani's Problem (I knew him at Yale in the '50s) (see mathworld.wolfram.com/CollatzProblem.html).
The search key she used after reading the first version of this communique?
"9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1"
2) HOW CAN YOU ASK SAM WHEN YOU CAN’T FRAME THE QUESTION?
“Hemiplegia” is paralysis of one side of the body, e.g., including left arm and left leg.
What is the word W for paralysis of top or bottom half, e.g., left arm and right arm? This is not hemiplegia. “Hemiplegia” and W are not synonyms or antonyms. Theirs is a related relation, we could say “transposes” using matrix terminology,
or think of a 180-degree rotation about a 45-degree axis. Thus we wish to find the transpose of “hemiplegia.” But lexicography has no word that Ask Sam would understand, so how do we frame the question?
3) HOW CAN YOU LOOK UP A WORD IN THE DICTIONARY WHEN IT DOESN’T EXIST?
Mangor, n. (obviously) the taste, aroma, or quality of being mango.
Mangid, adj. (obviously) having the taste, aroma, or quality of being mango.
But if it isn’t obvious to me, how can I look up nonwords in the dictionary?
These problems are clearly related (somehow), and indeed are aspects of a single problem, although I despair of describing them as such.
"If your approach to mathematics is mechanical not mystical, you're not going to go anywhere." -- Nassim Nicholas Taleb
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