Enzymind Labs

Sentences can have fractal structure

Here's a brief comparison of a couple of the simplest possible L-systems, and human language sentence structure.

The elements of these systems are called letters by mathematians, words by linguists, and states by computer scientists. I'm calling them symbols. The results are called strings, sentences, or computations by these respective fields. I call them patterns because I represent them in two dimensions as images.

L-systems are recursive symbol replacement systems:

- Start with a set of symbols.
- Replace each symbol with an ordered set of symbols, a unique set for each symbol.
- Repeat these last two steps indefinitely, using the new set of symbols.

This is small class of a formal grammar (a set of rules and symbols), in particular a Type-3 grammar without a terminating symbol, which Chompsky (1956, and others) characterized with respect to more complex grammars.


Here's one of the simplest L-systems (only two symbols and two rules), with a graphical representation:
or graphically:
This system results in a one-dimensional pattern, a sentence:

or


This one-dimensional pattern is fractal, but that's not obvious on inspection -- humans don't easily recognize the self-similar and discontinuous nature of the pattern. Human language is much richer (more possible sentences, more convoluted sentence structure) than this because it has more symbols (words) and rules. But if a system like this is extended to two dimensions (by allowing replacement rules to be 2x2 instead of nx1), the fractal patterns formed are more evident. Here's one of the simplest possible 2-symbol 2x2 systems:



which evolves like this:

...
which is more obviously a fractal (a right angle Serpinski gasket).

Human language sentences have a terminating symbol, and if they are diagrammed they have a similar fractal structure. They are usually terminated very early ("Sally went to the store."), because if too many rules are applied humans can't keep in working memory all the relationships ("Sally, my sister, went, after climbing the tall hill, to the bigger store at the bottom of the tall green hill.")

Human written language can be considered one-dimensional, but spoken/signed language can have other dimensions, like prosody and intonation.

 

"Ending a sentence with a preposition is something up with which I will not put."
Winston Churchill

I havn't include any good example sentences, that demonstrate recursion in grammar, because I wasn't paying attention in 5th grade English. I don't know the names of the replacement rules of English.

What are the simplest sentences that obviously demonstrate recursion, and not just repetition like "A long, long, long, long time ago....". Here are two that demonstrate a true recurrence, where the replacement refers to the original and is not just a copy;

"I think, I think, therefore I am." [ a -> {a ", I think,}, b -> b]
and
"I think I think therefore I am." [ a -> {"I think" a} , b -> b]

These rules can be combined, or applied in succession, with this and many other English sentences. Note that it's not important that the insertion be the same symbols. "I run, I think, therefore I am." works too. Grammar is not about meaning, but about allowed structure.

I'm sure these aren't original. It seems likely that Decartes recognized the implicit recursion in "Cogito, ergo sum":

At the beginning of the second meditation, having reached what he considers to be the ultimate level of doubt — his argument from the existence of a deceiving god — Descartes examines his beliefs to see if any has survived the doubt. In his belief in his own existence he finds it: it is impossible to doubt that he exists. Even if there were a deceiving god (or an evil demon, the tool he uses to stop himself sliding back into ungrounded beliefs), his belief in his own existence would be secure, for how could he be deceived unless he existed in order to be deceived?

They may have been examples from one of Douglas Hofstadter's books; "Gödel, Escher, Bach: an Eternal Golden Braid" (GEB), which explores the meaning of recursion in math, visual art and music using stories that are recursively structured to make his point, "The Mind's I", whos title's two meanings are mutually self referential, or "I Am a Strange Loop" which he wrote in part because the point of GEB was misconstrued, and whose title is a strange loop, like Hofstadter believes himself and us to be.

Here's a of history I learned on the D. Hofstadter wiki page, demonstrating a nice cross-fertilization between cognitive science and mathematical physics:

Hofstadter, D.R. et. al., "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields", Phys. Rev. B 14 (1976) 2239.

• Written while he was at the University of Oregon, this paper was enormously influential in directing further research. Hofstadter predicted that the allowed energy level values of an electron in a crystal lattice, as a function of a magnetic field applied to the latice, formed a fractal set. That is, the distribution of energy levels for large-scale changes in the applied magnetic field repeat patterns seen in the small-scale structure. This fractal structure is generally known as "Hofstadter's butterfly",which was the first fractal ever found in physics, and it has recently been confirmed in transport measurements in two-dimensional electron systems with a superimposed nano-fabricated lattice.
• 
  

Yoshiko Yamata wrote:

I believe, in Linguistics, the replacement rules are called Phrase Structure rules.
There are different versions of these PS rules (depending on the stages of the development of formal syntactic theory), but simples ones go like this:

S -> NP VP
NP -> ART N S
VP -> NP

An example of recursive application of these rules would be:

The dog chased the cat that ate the rat that bit the baby that slept in the crib that was made by the man who lived on the street ....

The above PS rules also allow sentences such as:

[The dog [that the cat [that the rat [that the baby hated] feared] bit] slept under the tree.]

I put the brackets for the ease of parsing. The last time when I sat in a formal syntax class, sentences like this were considered grammatical. I was told that they did not sound good ( or at least were nearly impossible to comprehend) because of our processing limitation (You can see how it is grammatical from a sentence with the same basic construction but with a fewer embedded sentences, e.g., "The dog that the cat bit slept under the tree.".), and I doubt that speakers of any language produce sentences like this.

I think this sentence is comparable to:
..... (I've gotten tired of making branches...)
/ / \ \
/ / /\ \ \
/ / / \ \ \
/ / / /\ \ \ \
/ / / / \ \ \ \
A A A A B B B B

I've seen some animal studies use this rule to study non-human's ability to learn recursive rules. I can't help wondering why they use this rule. Shouldn't non-human animals have some processing limitations (therefore, may not learn recursive rules based on input like this)? I'm not familiar iwth the literature, so I'm not sure if they are actually learning this recursive rule, but it would be pretty amazing if they are...

 

Also see Chompsky's kudzu , whos caption is a joke about this topic.