;; Packing and Unpacking Numbers ;; This post is kind of pointless, since while writing it I discovered a better way to do the same thing. ;; But I also think it's kind of cool, so I'm putting it up in case anyone else finds a use for it. ;; Suppose we've got a function which gives random results (I mean, deliberately...) (defn randomfunction [] (list (case (randint 2) 0 '+ 1 ') (randint 100) (list (case (randint 2) 0 '+ 1 ') (randint 100) (randint 100)))) (randomfunction) ;> (+ 43 ( 71 75)) (randomfunction) ;> ( 18 ( 13 22)) (randomfunction) ;> ( 36 (+ 60 3)) (randomfunction) ;> (+ 56 (+ 34 4)) (randomfunction) ;> ( 87 ( 61 76)) ;; And it comes to testing it: (= (randomfunction) '(+ 26 ( 69 0))) ;> false ;; We'd prefer it to be deterministic. ;; So we can separate the concerns thus: (defn deterministicfunction [a b c d e] (list (case a 0 '+ 1 ') b (list (case c 0 '+ 1 ') d e))) (defn randomfunction [] (deterministicfunction (randint 2) (randint 100) (randint 2) (randint 100) (randint 100))) ;; tadaa! (= (deterministicfunction 0 26 1 69 0) '(+ 26 ( 69 0))) ;> true (randomfunction) ;> (+ 30 (+ 52 95)) (randomfunction) ;> ( 97 ( 57 61)) (randomfunction) ;> (+ 88 (+ 26 15)) ;; Although useful, this is a right royal pain in the neck, and horribly errorprone ;; It occurred to me that it might be made less painful by setting the ;; seed of the random number generator before running tests, but I ;; also had a need to make 'deterministic random' values in the code ;; itself, and having everything hanging off a global variable that ;; any old function occasionally resets struck me as a recipe for ;; disaster. ;; So I thought, well why not pack all the random parameters into one big number: (def << bitshiftleft) ;; If all the random ranges are powers of two then this is easy: (let [[a b c d] [(randint 8) (randint 4) (randint 8) (randint 2)]] [[a b c d] (+ (<< a (+ 1 3 2)) (<< b (+ 1 3)) (<< c 1) (<< d 0))]) ; [[2 2 4 0] 168] ; [[3 1 4 0] 216] ; [[5 3 6 0] 380] ;; Looking at the last one ;; 5/3/6/0 is ;; 101/11/110/0 in binary ;; and binary 101111100 is: 2r101111100 ; 380 ;; in decimal ;; To extract the numbers, we can do: (mod 380 2) ; 0 (/ ( 380 (mod 380 2)) 2) ; 190 (mod 190 8) ; 6 (/ ( 190 (mod 190 8)) 8) ; 23 (mod 23 4) ; 3 (/ ( 23 (mod 23 4)) 4) ; 5 (mod 5 8) ; 5 ;; And this motivates the following recursion: (defn ^:dynamic unpack [rl n] (if (empty? rl) '() (let [[rh & rr] rl] (cons (mod n rh) (unpack rr (/ ( n (mod n rh)) rh)))))) ;; Which behaves as I was hoping (unpack '() 0) ;> () (unpack '() 1) ;> () (unpack '(8) 1) ;> (1) (unpack '(8) 9) ;> (1) (unpack '(8 8) 9) ;> (1 1) (unpack '(2 8 4 8) 380) ;> (0 6 3 5) (unpack '(2 8 4 8) 216) ;> (0 4 1 3) (unpack '(2 8 4 8) 168) ;> (0 4 2 2) ;; And that then motivates the following packing function (defn ^:dynamic pack [rlist vlist] (if (empty? rlist) 0 (let [[rh & rr] rlist [vh & vr] vlist] (+ (* rh (pack rr vr)) vh)))) ;; Which also works (pack '()()) ;> 0 (pack '(8) '(1)) ;> 1 (pack '(8 8) '(1 1)) ;> 9 (pack '(2 8 4 8) '(0 6 3 5)) ;> 380 (pack '(2 8 4 8) '(0 4 1 3)) ;> 216 (pack '(2 8 4 8) '(0 4 2 2)) ;> 168 ;; But the cool thing is that it works whether or not the ranges are powers of two: (use 'clojure.tools.trace) (let [a (for [i (range (inc (randint 7)))] (randint 30)) b (for [i a] (randint i))] (clojure.tools.trace/dotrace [unpack pack] (unpack a (pack a b)))) ;; TRACE t3191: (pack (12 23 5 5 8 3) (4 9 3 4 0 0)) ;; TRACE t3192:  (pack (23 5 5 8 3) (9 3 4 0 0)) ;; TRACE t3193:   (pack (5 5 8 3) (3 4 0 0)) ;; TRACE t3194:    (pack (5 8 3) (4 0 0)) ;; TRACE t3195:     (pack (8 3) (0 0)) ;; TRACE t3196:      (pack (3) (0)) ;; TRACE t3197:       (pack nil nil) ;; TRACE t3197:       => 0 ;; TRACE t3196:      => 0 ;; TRACE t3195:     => 0 ;; TRACE t3194:    => 4 ;; TRACE t3193:   => 23 ;; TRACE t3192:  => 538 ;; TRACE t3191: => 6460 ;; TRACE t3198: (unpack (12 23 5 5 8 3) 6460) ;; TRACE t3199:  (unpack (23 5 5 8 3) 538) ;; TRACE t3200:   (unpack (5 5 8 3) 23) ;; TRACE t3201:    (unpack (5 8 3) 4) ;; TRACE t3202:     (unpack (8 3) 0) ;; TRACE t3203:      (unpack (3) 0) ;; TRACE t3204:       (unpack nil 0) ;; TRACE t3204:       => () ;; TRACE t3203:      => (0) ;; TRACE t3202:     => (0 0) ;; TRACE t3201:    => (4 0 0) ;; TRACE t3200:   => (3 4 0 0) ;; TRACE t3199:  => (9 3 4 0 0) ;; TRACE t3198: => (4 9 3 4 0 0) ;; (4 9 3 4 0 0) ;; Which I found slightly smugmaking ;; So now we can say: (defn deterministicfunction [bigrand] (let [[a b c d e] (unpack [2 100 1 100 100] bigrand)] (list (case a 0 '+ 1 ') b (list (case c 0 '+ 1 ') d e)))) (defn randomfunction [] (deterministicfunction (randint (* 2 100 2 100 100)))) ;; Behold the many benefits of this scheme: ;; We can enumerate all possible varieties (deterministicfunction 1) ;> ( 0 (+ 0 0)) (deterministicfunction 2) ;> (+ 1 (+ 0 0)) (deterministicfunction 3) ;> ( 1 (+ 0 0)) ;; We can generate examples using a single number (deterministicfunction 10345778979) ;> ( 89 (+ 94 88)) ;; Or we can get random examples (randomfunction) ;> (+ 99 (+ 72 8)) ;; If we are not desperately fussed about the statistical properties of things ;; we do not need to worry about the exact range of the random number at all: ;; I think that this will produce only tiny biases: (defn randomfunction [] (deterministicfunction (randint (<< 1 30)))) ;; And it still looks pretty random to me (randomfunction) ;> (+ 76 (+ 10 87)) (randomfunction) ;> (+ 76 (+ 81 3)) (randomfunction) ;> ( 27 (+ 18 63))
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Wednesday, February 20, 2013
Packing and Unpacking Numbers : Deterministic Randomness
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Maybe it is pointless for you, but I have got tons of emotions from first part (separation of concerns), found about `clojure.tools.trace/dotrace`
ReplyDeleteThanks
Great, thanks for leaving the note!
ReplyDeleteActually I was thinking about this more only this morning, and I suspect it might provide a motivating scheme for the informationtheoretic concept of Entropy, and in particular the KullbackLiebler divergence and Mutual Information. These are really crucial basic ideas in science, but very obscure and difficult, to the point where they aren't even widely known amongst Mathematicians.
I can feel a book called 'Quantative Epistemology' coming on. Written in a mixture of Clojure and English. I feel it might reach a nontrivial potential audience.