### Lexical Problems

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];
otherwise A[3n+1].

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?

Update:

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.

1. Paraplegia is the paralysis of the lower half of the body. Since paralysis of this type is usually caused by a spinal cord injury, paralyzing everything below the injured point, I would imagine that there's no word for upper body paralysis simply because it doesn't occur often enough to need naming.

Paralysis of the arms and legs is, of course, quadriplegia. On a hunch, I looked up "triplegia", which is apparently:

1. Paralysis of an upper and a lower extremity and of the face.
2. Paralysis of both extremities on one side and one extremity on the opposite side.

Then, there is "diplegia" which may be what you're looking for:

Paralysis of corresponding parts on both sides of the body.

It may be a little broad, as it would technically include paralysis of both ears, or both pinkies.

A reverse dictionary is, of course, a fairly difficult problem to encode, yet is a task we perform all the time ourselves. "What's the word for..."

2. In the case of non-words, are you referring to neologisms and portmanteaus (portmanteux?) that you come across while reading? I would say that if you're a reasonably intelligent native speaker of the language, and the meaning of a neologism is not obvious to you, the author didn't do his job.

If you're not, well, that sucks.

Good (meaning clear, not necessarily clever) neologisms:
eco-terrorism
vinophilia
demopublican/republicrat
reasonectomy
communerd

moible: weakness for using a cell phone
brunk: a boring drunk
arbrop: to fall out of a tree

A bad neologism can occasionally be saved by context clues.

3. Anonymous11:33 AM

Why is moible a "bad" neologism - I thought it was rather clever.

4. Because absent context, it is undecipherable. Mayber it's a marble with a moire pattern.

5. "Mayber" is a typonym for "maybe," not a neologism. "Typonym" is a (hopefully clever) neologism.

Anyhow, I just proved that the Collatz Problem contains no cycles. Is that worth anything?

6. Apropos "demopublican/republicrat"-- Al Capp used "demican" (sp.?) and "republicrat" in the 40s or 50s (but these terms were not carried over into "The Country's in the Very Best of Hands" in the musical version of Li'l Abner).
Apropos Collatz cycles--if the proof is short, please post it.

7. I sent my proof to a mathematician I know. That way I won't needlessly embarrass myself.

8. Okay, all I proved is that if a sequence goes to zero, it can't loop on the way there.

And also that there are an infinite number of numbers whose sequences go to zero.

9. Anonymous7:13 AM

If a sequence loops at all, won't it necessarily loop forever?

10. Exactly. It can approach a loop from one direction, but once it enters can never leave.

Great article! Thanks.

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