[FOM] Mandelbrot set, Catalan numbers, Context-free grammars

[Andrius Kulikauskas]

I wish to share a curious link that I noticed between the Mandelbrot set 
and the Catalan numbers which may be useful for studying the kinds of 
logical expressions that automata can generate.  I appreciate any 
related references or ideas that you might offer.

I am interested to understand the big picture in math.  That is, I am 
looking for a perspective from which I could understand which ideas are 
cognitively most basic and how more sophisticated mathematics unfolds.  
This wish supposes that math is not merely "explicit", not merely what 
gets written down, but that there are also "implicit" notions at play 
which may yet be distinguished as "basic" or "sophisticated".  I am 
interested to discuss with mathematicians how we might investigate math 
as a whole.

I was reading mathematical physicist Roger Penrose's "Road to Reality", 
which is an impressive 1100+ page survey of all he knows about math, 
available for free online.  I was curious why he mentions the Mandelbrot 
set, which I think of as an example of what is most accidental in math, 
and thus least interesting from my point of view.  The Mandelbrot set is 
generated by considering any complex number c and what happens in the 
limit to infinity upon iterating the substitution  z -> z^2 + c and then 
discarding the z (setting z=0) at the end.  If the iterations Pn stay 
bounded (<=2), then we include c in the Mandelbrot set, and otherwise 
they diverge to infinity and we don't include them.  Roger Penrose 
writes this out much more elegantly as the sequence:
... c^2 + c)^2  + c)^2  + c)^2 + c

Well, as a combinatorialist, I realized that the leading terms settle 
down and might actually be a nice sequence.  It turns out that they are 
in fact the Catalan numbers, as can be shown by induction to yield their 
recurrence relation.
https://en.wikipedia.org/wiki/Catalan_number
These are some of the most basic numbers in combinatorics.  They count 
the number of correct expressions for n pairs of parentheses. In other 
words, they count the semantical possibilities that our mind has in 
associating operators in a string of operators. Furthermore, this is 
related to the types of strings that can be generated by context-free 
grammars and context-sensitive grammars, which are important in the 
Chomsky hierarchy. For example, we could code the words that such a 
grammar generates and if we plug in 1 for all of the symbols then we 
should typically get the Catalan numbers or something related.

The generating function G(c) is given by G(c) = c*G(c)^2 + 1.  Thus 0 = 
c*G(c)^2 - G(c) + 1.   Solving the quadratic equation yields G(c) = 2/(1 
+ (1-4c)^(1/2)).

One question then is whether the behavior of Pn as n goes to infinity is 
given by G(c).  The coefficients are all positive and the coefficients 
of G(c) are always greater than Pn.  The first half or so of the 
coefficients of Pn match those of G(c).  So if Pn diverges then it seems 
that G(c) should diverge, too.  And if Pn does not diverge, then neither 
should G(c).  But I'm not an expert and I'm not sure.

So it appears that the exceedingly bizarre Mandelbrot set is simply 
defined by the behavior (convergence, divergence or otherwise) of the 
generating function of the Catalan numbers G(c).  Which apparently is 
not trivial as I gather from this post:
http://math.stackexchange.com/questions/1097097/generating-series-of-catalan-numbers

Also, surprisingly, the relation between the Mandelbrot set and the 
Catalan numbers may not be well known.  There is a note at the Online 
Encyclopedia of Integer Sequences https://oeis.org/A000108 thanks to an 
observation in 2009 by Donald D. Cross 
http://cosinekitty.com/mandel_poly.html

But now imagine this as a tool for Foundations of Math.  We can 
interpret the Catalan generating function at each complex number (or 
pair of real numbers, or map from one real number to another real 
number).  This generating function can be thought of as enumerating all 
of the possibilities for a grammar.  In particular, consider a hierarchy 
of grammars where a matrix A describes the rules (and the symbols 
accepted) and a matrix X is the nonterminal generator of the possibilities:
X -> AX finite automata, regular languages
X -> AXA push-down automata, context free languages
XA -> AX linear bounded automata, context sensitive languages
XA -> X Turing machines, recursively enumerable languages

A rule like X -> AXA is involved in making sure that each open 
parenthesis is ultimately paired with a closed parenthesis.  This is 
what I suppose the Catalan numbers are coding when they converge.  A 
rule like XA->AX allows the cursor to glide across many symbols. This 
perhaps could be the case when the Catalan generating function doesn't 
settle down.  And a rule like XA->X means that the automata is able to 
destroy information and introduce irreversibility like a full fledged 
Turing machine, and then perhaps in that case the Catalan generating 
function diverges.

So that is some poetic thinking which might perhaps yet spark ideas even 
for the P/NP complete problem.

Also, I note that the Catalan numbers count the number of semiorders on 
n unlabeled items.
https://en.wikipedia.org/wiki/Semiorder
This makes me think of cardinality and the Continuum hypothesis. For the 
semiorders seem to be "almost" orders.  And in fact it may be possible 
to play with that "almost" in the limit, making it more or less 
favorable.  Thus they seem relevant if one is to try to construct a set 
that is bigger than the integers but less than the continuum.

The Mandelbrot set could be an amazing object for tackling nasty 
problems.  Apparently, the Mandelbrot set may be locally connected. I 
think that would mean that you could always find a small enough 
neighborhood where the set is "nonchaotic" but if you move farther away 
then you will get chaotic results.

I appreciate any critique or feedback.  I also want to point out that 
what I thought was a most idiosyncratic object in math is in fact 
related to what is cognitively most fundamental (the possibilities for 
associativity) and this link may be quite practical and powerful.  So 
this suggests that math is not evolving haphazardly but rather if we 
survey all of math so far then we might get a sense of some overarching 
organizational principles.

I just want to add that I am very grateful for this forum.  I am glad to 
witness Harvey Friedman's thinking out loud even though I can only 
understand just a bit.  Also, my first talk in graduate school was about 
Hilbert's Tenth Problem which Martin Davis helped solve and which is 
inspiring in terms of what math can encode.

Andrius

Andrius Kulikauskas
ms at ms.lt
+370 607 27 665
Vilnius, Lithuania