Enzymind Labs

Sentences can (and do) 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 any formal grammar (a set of rules and symbols), in particular a Type-3 grammar without a terminting 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.

 

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