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(ns clj-automata.core
"This module visualizes elementary cellular automata. It's primarily intended
to show off the fun aspects of functional programming and lazy-sequences in clojure to those who are coming from an OO background.
If you're interested in how elementary cellular automata work, see: and
Also, I highly recommend reading this source file bottom to top in order to understand it best.
In particular, the code here makes heavy use of lazy-sequences, both directly through
`lazy-seq` and also through functions like map, for, partition, and other functions that
in clojure return lazy sequences. In some cases, we force lazy-evaluation for side-effects with
Since the visualization is of a stream of states for the automata, lazy-sequences are a very apt model.
We can define a sequence of all future results, and simply iterate our way toward where we'd like to be.
At a high level this library revolves around a single lazy seq that represents all future states.
The rendering model here is quite simple, we use pure clojure to fill a 2D array of 1s and 0s that contains
the current on-screen state. This 2D buffer is, itself, a partition (e.g. slice) of the lazy sequence of all
future states. See the run-rule function for more detail.
The core functions behind the laziness are (run-rule) which sets up the initial state and UI,
(simulation), which returns a lazy sequence of results,
and (simulate) which does the legwork of applying the automaton's rules.
There are other functional aspects here at play. The (rule) function for instance, is a higher-order function,
meaning that it returns a brand new function. The function it returns implements a given rule, matching the patterns
of the rule to outputs"
(:require [quil.core :as qc])
(:use clojure.pprint)
(:import java.lang.Math))
;; Colors for each cell
(def live-color [242 233 99])
(def dead-color [64 37 27])
;; Elementary automata use a clever trick to allow us to describe an entire rule with only a few numbers.
;; Since the automata are essentially pattern matchers, describing what a trio of living / dead cells evaluate to
;; we can effeciently encode their behaviour as a sequence of 1s and 0s. A good visualization of this can be found
;; here, at wolfram math world:
(defn int->bdigits
"Gets the binary digits that comprise an integer as a seq of ints."
(for [c (Integer/toBinaryString number)] (Integer/valueOf (str c))))
(defn zero-pad
"Forward pads a seq with 0s to match a given length. Used for making sure int->bdigits hits byte boundaries"
[x len]
(let [shortage (- len (count x))]
(if (< shortage 1)
(concat (repeat shortage 0) x))))
(def input-patterns
"The list of possible input sequences for elementary cellular automata, which are easily generated
by counting down from 8 in binary, and making sure we have at least three digits.
This should produce a list like: ((111 110 ...))"
(map #(zero-pad (int->bdigits %1) 3) (range 8)))
(defn rule-mappings
"Returns a mapping of patterns to new states. Returns a structure like:
{(0 1 1) 1
;; Zipmap combines two sequences into a map, much like a real-life zipper!
;; The key here is that the magic rule numbers are not numbers at all
;; but a sequence rather (their individual binary digits) that get mapped
;; onto the list of possible inputs described in input-patterns.
;; The heart of the generic solution here is really just checking
;; equality of a sequence, which we can do via a lookup in the hashmap
;; this generates
(zipmap input-patterns
(reverse (zero-pad (int->bdigits number) 8))))
(defn rule
"Returns a function that will process a triad of input values according to a given rule #.
Since rules are simple lookup tables, this maps to nothing more than a get really.
We use a function here only to be able to close over the rule-mappings and only evaluate those once."
;; Applying a rule is really simple, since we've reduced the problem to pattern matching, and
;; clojrue can match lists well (e.g. (= [1 2 3] [1 2 3]) => true even though they're separate
;; objects), can simply see which pattern the 3 given values match with a map lookup via get.
(let [mappings (rule-mappings number)]
(fn [triad] (get mappings triad))))
;; Since we assume cells that are off the grid are zeroes, and the far left and far
;; right calculatoins both require these cells, we make our calculation a bit easier by
;; simply pretending the previous row has two extra 0s on either side
(defn bookend
"Pads a seq with a given value on both sides."
[x v]
(concat [v] x [v]))
(defn simulate
"Runs a single iteration of a given rule-fn on a given-state"
[rule-fn state]
(let [rule (rule-mappings 110)]
;; We bookend the value below to add a 0 on both sides of the previous state
;; as it makes calculations simpler
(for [triad (partition 3 1 (bookend state 0))]
(rule-fn triad))))
(defn simulation
"Returns a lazy-seq of future states for a given rule-fn and state"
[rule-fn state]
(let [new-state (simulate rule-fn state)]
;; This is an infinitely recursive lazy sequence! Notice how we start
;; by considing (prepending) to a new-state onto the head of a not-yet extant
;; lazy sequence.
;; You'll notice that the lazy sequence is declared with a
;; body that will recurse from the present state, passing the current state into
;; itself. Lazy sequences such as this are inherently tail-recursive, so they won't
;; blow the stack.
(cons new-state (lazy-seq (simulation rule-fn new-state)))))
(defn draw-buffer
"Redraw what's on screen given a buffer of cell data at a given scale"
[buffer scale]
;; We use letfn here because we want both of these functions to
;; have access to the variables `buffer` and `scale`. Closing over them
;; here rather than defining them separately is simply a stylistic choice.
;; We use two nested `map-index` calls to iterated over the canvas row by row,
;; cell by cell, rendering each to the UI
(letfn [(draw-row [y row]
(dorun (map-indexed (fn [x col] (draw-cell x y col)) row)))
(draw-cell [x y col]
(apply qc/fill (if (= 1 col) live-color dead-color))
(qc/rect (* scale x) (* scale y) scale scale))]
(dorun (map-indexed draw-row buffer))))
(defn setup
"Setup the UI"
(qc/smooth) ;; Enable AA
(qc/frame-rate 24))
(defn run-rule [rule-num {:keys [width height scale]}]
(let [width (or width 100)
height (or height 100)
scale (or scale 5) ; Scale factor for rendering
;; Our initial state is a single row of random 0s and 1s
initial (repeatedly height #(rand-int 2))
sim (simulation (rule rule-num) initial)
;; We use partition as a sliding wintdow here. Since sim is
;; an infinite lazy sequence of future rows we use the 3-arity
;; version of partition here to create a 2D view of the visible range
;; of results. Since this is the 3-arity version of partition, with 1
;; specified as the second parameter, each time we grab the next item from
;; the sim sequence we wind up with the same 2D view as before, but shifted ahead
;; one row. This is how the simulation scrolls down!
time-slices (atom (partition height 1 sim))]
(println "Rule " rule-num " mappings:")
(pprint (rule-mappings rule-num))
;; Initialize the graphics
(qc/defsketch automata
:title (str "Rule " rule-num)
:setup setup
:draw (fn drawfn [] ;; Named anonymous functions are easier to stacktrace
;; Lazy sequences can also re-integrate into a more imperitive style
;; What follows below is convenient, but not quite functional.
;; Since the :draw callback is inherently not recursive I went with a
;; more imperitive approach here. We use an atom to maintain the current position
;; within our infinite sequence. This could also have been archieved with a
;; tail recursive loop and not-utilizing the draw callback, but that's left
;; as an excercise for the reader.
(draw-buffer (first @time-slices) scale)
(swap! time-slices (fn [_] (rest @time-slices))))
:size [(* scale width) (* scale height)])))
(defn -main [rule-num & args]
(run-rule (Integer/valueOf rule-num)
{:width 100 :height 100 :scale 4}))
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