Mandelbrot: SETL & Lisp

SETL is a general-purpose programming language developed by Jack Schwartz back in the late 1960s. The language is multiparadigm – both Procedural (containing subroutines and control structures like if statements) and Set Theoretical. The latter is not often mentioned when talking about language paradigms – and is not to be confused with Logical Programming. The concept is outlined in the 1974 paper An Introduction to the Set Theoretical Language SETL. Importantly though data in SETL is of three main types: Atoms, Sets, and Sequences (also referred to as tuples or ordered sets).

A good overview of SETL can be found in the GNU SETL Om.

setl-mandelbrot

Pixels inside the Mandelbrot Set are marked in black, the code used to generate them (mandelbrot.setl) can be found here.

Generation of the Mandelbrot Set
On the Wikipedia page for the Mandelbrot Set you will find its formal definition, which looks like so: M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}.

The most fantastic thing about SETL is how powerful its Set Builder Notation is. Almost exactly as you would in the formal case (removing ‘there exists some real s’ for the practical application):

M := {c : c in image | forall num in {0..iterations} | cabs(pofc(c, num)) <= 2};

For some reason SETL doesn’t support complex numbers, but they are easily handled by writing the necessary procedures we need like cabs, ctimes, and cplus dealing with tuples in the form [x, y]. The variable ‘image’ is a sequence of these tuples. Another procedure is written, pofc, which is a more practical version of P_c^n(0).

Interaction with the Lisp graphical display
The goal of the SETL program is to produce a set of points that lie inside the Mandelbrot set. To actually display the image I used Common Lisp and displayed a widget with the image using CommonQt. In a very rudimentary way I had mandelbrot.setl take arguments about the image then print the set of pixel co-ordinates. All lisp had to do was swap and ‘{}’ or ‘[]’ for ‘()’ and read it as lisp code then update the image.

(setf in-set (read-from-string (parse-line (uiop:run-program "setl mandelbrot.setl -- -250 -250 250 8" :output :string))))
(draw-mandel instance in-set)

An after-thought was to make a Mandelbrot searcher where you could zoom and move using the mouse but the SETL code is such an inefficient way of doing it that it’s not worth it. As an attempt to mimic the formal definition it was highly successful and fun. Though much quicker SETL code could be written for generating the Mandelbrot Set.

Source file for mandelbrot.setl and the Lisp/CommonQt front end can be found here.

Accuracy of Generated Fractals

Note: I refer to the Mandelbrot set in general as the M-set for short.

When I was writing the post on Rough Mandelbrot Sets I tried out some variations on the rough set. One variation was to measure the generated M-set against a previously calculated master M-set of high precision (100000 iterations of z = z^2 + C). In the image below the master M-set is in white and the generated M-sets are in green (increasing in accuracy):

50 Against MasterHere instead of approximating with tiles I measured the accuracy of the generated sets against the master set by pixel count. Where P = \{ \text{set of all pixels} \} the ratio of P_{master} / P_{generated} produced something that threw me, the generated sets made sudden but periodic jumps in accuracy:

Graph OneLooking at the data I saw the jumps were, very roughly, at multiples of 256. The size of the image being generated was 256 by 256 pixels so I changed it to N by N for N = {120, 360, 680} and the increment was still every ~256. So I’m not really sure why, it might be obvious, if you know tell me in the comments!

I am reminded of the images generated from Fractal Binary and other Complex Bases where large geometric entities can be represented on a plane by iteration through a number system. I’d really like to know what the Mandelbrot Number System is…

Below is a table of the jumps and their iteration index:

Iterations Accuracy measure
255
256
0.241929
0.397073
510
511
0.395135
0.510806
765
766
0.510157
0.579283
1020
1021
0.578861
0.644919
1275
1276
0.644919
0.679819
1530
1531
0.679696
0.718911

Rough Mandelbrot Sets

I’ve been reading up on Zdzisław Pawlak’s Rough Set Theory recently and wanted to play with them. They are used to address vagueness in data so fractals seem like a good subject.

Super Quick Intro to Rough Sets:
A rough set is a tuple (ordered pair) of sets R(S) = \langle R_*, R^* \rangle which is used to model some target set S. The set R_* has every element definitely in set S and set R^* has every element that is possibly in set S . It’s roughness can be measured by the accuracy function \alpha(S) = \frac{|R_*|}{|R^*|} . So when |R_*| = |R^*| then the set is known as crisp (not vague) with an accuracy of 1.

A more formal example can be found on the wiki page but we’ll move on to the Mandelbrot example because it is visually intuitive:

The tiles are 36x36 pixels, the Mandelbrot set is marked in yellow. The green and white tiles are possibly i the Mandelbrot set, but the white tiles are also definitely in the Mandelbrot set.

The tiles are 36×36 pixels, the Mandelbrot set is marked in yellow. The green and white tiles are possibly in the Mandelbrot set, but the white tiles are also definitely in it.

Here the target set S contains all the pixels inside the Mandelbrot set, but we are going to construct this set in terms of tiles. Let T_1, T_2, T_3,\dots , T_n be the tile sets that contain the pixels. R^* is the set of all tiles T_x where the set T_x contains at least one pixel that is inside the Mandelbrot set, R_* is the set of all tiles T_x that contain only Mandelbrot pixels. So in the above example there are 28 tiles possibly in the set including the 7 tiles definitely in the set. Giving R(S) an accuracy of 0.25.

Tile sizes: 90, 72, 60, 45, 40, 36, 30, 24, 20, 18, 15, 12, 10, 9, 8, 6, 5, 4.

Tile width: 90, 72, 60, 45, 40, 36, 30, 24, 20, 18, 15, 12, 10, 9, 8, 6, 5, 4. There seems to be a lack of symmetry but it’s probably from computational precision loss.

Obviously the smaller the tiles the better the approximation of the set. Here the largest tiles (90×90 pixels) are so big that there are no tiles definitely inside the target set and 10 tiles possibly in the set, making the accuracy 0. On the other hand, the 4×4 tiles give us |R_*| = 1211 and |R^*| = 1506 making a much nicer:

\alpha(S) = 0.8 \overline{04116865869853917662682602921646746347941567065073}

For much more useful applications of Rough Sets see this extensive paper by Pawlak covering the short history of Rough Sets, comparing them to Fuzzy Sets and showing uses in data analysis and Artificial Intelligence.