## October 30, 2006

### Puzzle #4

#### Posted by John Baez

This is the 100th blog entry at the n-Category Café! David, Urs and I thank all the people who have come together to make this a fascinating place to talk about philosophy, physics and math.

So, let’s celebrate with a puzzle.

What is the following sentence about?

“Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.”

Posted at October 30, 2006 1:01 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1008

### Re: Puzzle #4

Christine

Posted by: Christine Dantas on October 30, 2006 1:57 AM | Permalink | Reply to this

### Re: Puzzle #4

From JC Brown 1955, ‘Lojban’, an artificial language designed as “phonetic, logical and culturally neutral”.

Posted by: Doug on October 30, 2006 2:04 AM | Permalink | Reply to this

### Re: Puzzle #4

That was quick!

Q: What is the following sentence about?

“Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.”

A: The grammar of an artificial language called Lojban, deliberately designed to be phonetic, logical, and culturally neutral. Work on this language was begun by James Cooke Brown in 1955. He originally called the language “Loglan”, for “logical language”.

A “brivla” is a predicate in Lojban. A “cmavo” is a structure word, the equivalent of an article, conjunction, preposition, number, or punctuation mark. A “gismu” is a primitive brivla; gismu and cmavo these can be put together to form “lujvo”, or compound brivla. When a gismu or cmavo appears as part of a lujvo it can appear in various forms which are called “rafsi”; for example, its final vowel can be deleted.

Source:

As a little extra followup: which expert on Lie groups and geometry invented his own language, and what was it called?

Posted by: John Baez on October 30, 2006 3:50 AM | Permalink | Reply to this

### Re: Puzzle #4

One possibility is John Horton Conway with ‘Fractran’.

Posted by: Doug on November 1, 2006 1:03 AM | Permalink | Reply to this

### Re: Puzzle #4

I think he meant a human language rather than a computer language.

But Fractran is really cool; in fact, we were (implicitly) talking about it at the CS/CT seminar yesterday morning.

It was mentioned that one can move all the branching to the top of a program and have a single switch statement in a big loop; Fractran is a particularly clever language in which this is the only way to write programs.

In case anyone hasn’t heard of it before, a Fractran program is a list of fractions and the program state is an integer. You update the state by multiplying the current state by the first fraction in the list such that the result is an integer.

Here’s a simple program that given 2n produces the Hamming weight of n, i.e. the number of 1’s in the binary expansion of n:

(1)$\frac{33}{20},\frac{5}{11},\frac{13}{10},\frac{1}{5},\frac{2}{3},\frac{10}{7},\frac{7}{2}$
Posted by: Mike Stay on November 1, 2006 9:26 PM | Permalink | Reply to this

### Re: Puzzle #4

Mike wrote:

It was mentioned that one can move all the branching to the top of a program and have a single switch statement in a big loop; Fractran is a particularly clever language in which this is the only way to write programs.

Hmm!

Looking around for stuff about Fractran, I ran into this:

Devienne, Philippe; Lebegue, Patrick; Routier, Jean-Christophe; Wurtz, Jorg.

One binary Horn clause is enough.

STACS 94 (Caen, 1994), 21–32, Lecture Notes in Comput. Sci., 775, Springer, Berlin, 1994.

Summary: This paper proposes an equivalent form of the famous Bohm-Jacopini theorem for declarative languages. C. Bohm and G. Jacopini [Comm. ACM 9 (1966), 366–371; Zbl 145, 242] proved that all programming can be done with at most one single while-do. That result is cited as a mathematical justification for structured programming. A similar result can be shown for declarative programming. Indeed, the simplest class of recursive programs in Horn clause languages can be defined by the following scheme:

[gobbledytex deleted]

where the […] are positive first-order literals. This class is shown here to be as expressive as Turing machines and all simpler classes would be trivial. The proof is based on a remarkable and insufficiently known codification of any computable function by unpredictable iterations proposed by J. H. Conway [in Proceedings of the Number Theory Conference (Boulder, CO, 1972), 49–52, Univ. Colorado, Boulder, CO, 1972; MR 52 #13717]. Then, we prove effectively by logical transformations that all conjunctive formulas of Horn clauses can be translated into an equivalent conjunctive 4-formula (as above). Some consequences are presented in several contexts (mathematical logic, unification modulo a set of axioms, compilation techniques and other program patterns).

Posted by: John Baez on November 3, 2006 5:20 AM | Permalink | Reply to this

### Re: Puzzle #4

Yes, I meant a language that you could talk in - though as you’ll see, the word human is a bit misleading here!

(That’s a hint.)

Also, Conway is not mainly known as an expert on Lie groups and geometry - he’s mainly known for his work on finite groups, lattices, sphere packings, and codes, games, surreal numbers, the Game of Life, and being a wild and crazy guy.

Posted by: John Baez on November 1, 2006 9:50 PM | Permalink | Reply to this

### Re: Puzzle #4

Another possibility may be Hermann Weyl with ‘mathematizing’ as “… a creative activity of man, like language or music …”

Posted by: Doug on November 1, 2006 10:52 PM | Permalink | Reply to this

### Re: Puzzle #4

No, I didn’t mean Weyl. I’m talking about someone famous for their work on Lie groups and geometry, who invented an artificial language, a bit like Lojban or Ithkuil or Toki Pona, but with different design criteria.

Posted by: John Baez on November 2, 2006 9:43 PM | Permalink | Reply to this

### Re: Puzzle #4

Hans Freudenthal? Lincos?

Posted by: David Corfield on November 2, 2006 10:05 PM | Permalink | Reply to this

### Re: Puzzle #4

Yes, David Corfield got it! Hans Freudenthal is famous for his work on Lie groups and geometry, especially the Freudenthal character formula and his work on octonionic geometry and the magic square. In topology he’s famous for the Freudenthal suspension theorem, which underlies stable homotopy theory. But, he also created an artificial language is called LINCOS. It’s pretty interesting.

In fact, Freudenthal wrote a book called LINCOS: Design of a Languge for Cosmic Intercourse. I must have a dirty mind or something, because I can’t help smirking when I see that title. Is LINCOS some sort of pidgin for people who want sex with space aliens?

No: it’s designed to facilitate radio communication with extraterrestrial civilizations! It’s heavily mathematical in flavor. I hope we get to try it out someday.

Here’s part of a critical review by Bruno Bassi, which shouldl give you a bit of the flavor:

2.1. Spoken and written Lincos

Lincos proper (‘spoken Lincos’) will be broadcasted in space, its ‘phonemes’ being radio signals of different duration and wavelength. For us humans though it is obviously convenient to work on a written version of it. This written Lincos will use symbols already familiar to us, corresponding to Lincos words in an arbitrary fashion. Lincos ‘phonetics’ is not discussed in Dr. Freudenthal’s book, though it is claimed that it should be as systematic as possible, in the sense that syntactic and semantic categories should be marked phonetically.

2.2. Mathematics

After a presentation of some Lincos vocabulary, in the form of loose words transmitted out of context, the Lincos program starts with simple comparisons and operations on natural numbers. A good reason for this choice is that we need to start talking about something that is presumably known to the receiver, and that we have some means to show.

* * * * * > * * *

In this instance the sequences of dots are transmitted as sequences of peeps and should work as ‘ideophonetic words’, naming numbers by ostension. The ‘greater than’ sign is a Lincos word initially incomprehensible for the receiver, who is assumed to infer its meaning after a large number of utterances in which it is applied to different numbers.

* * * < * * * * *

[and so on]

* * * * = * * * *

[and so on]

* * * * + * * = * * * * * *

[and so on]

* * * * * * - * * = * * * *

[and so on]

Once ‘equals’ is known, algorithmic numerals are substituted for ostensive ones:

* = 1

* * = 10

* * * = 11

* * * * = 100

* * * * * = 101

* * * * * * = 110

* * * * * * * * * * * * * = 1101

[and so on] (p. 46)

Posted by: John Baez on November 2, 2006 10:38 PM | Permalink | Reply to this

### Re: Puzzle #4

What happened to Cvitanovic’s Magic Triangle which includes Freudenthal’s Magic Square (e.g., p. 21 of this)? The next two pages of that reference show that Deligne got involved.

Posted by: David Corfield on November 3, 2006 8:52 AM | Permalink | Reply to this

### Re: Puzzle #4

Try this:

I haven’t had time to understand this stuff yet… too much fun stuff to think about!

Posted by: John Baez on November 3, 2006 4:59 PM | Permalink | Reply to this

### Re: Puzzle #4

Posted by: David Roberts on January 11, 2007 1:45 AM | Permalink | Reply to this

### Re: Puzzle #4

A recent post explaining Fractran describes a prime number generator:

“While researching this post, I discovered (via mathworld) that Conway figured out a way of writing an astonishing prime number generator in Fractran. If you take the following sequence as a fractran program, in the numbers that it generates, the exponent on 2 in every number that it generates will always be prime.

17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55/1”