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# OUDL Y Combinator

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A brief intro to the Y Combinator, using Clojure.

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### OUDL Y Combinator

1. 1. We are OUDL. Organization for the Understanding of Dynamic Languages http://meetup.com/dynamic/Wednesday, August 22, 12We are programming language enthusiastsCheck us out on Meetup.comAll events have been by members, for membersEach event has a theme, a selected pastry or baked good, and a horrible logo
2. 2. What makes Objective C dynamic?Wednesday, August 22, 12
3. 3. What makes Objective C dynamic? Kamehameha Bakery donuts Otto Cake cheesecake Cake Couture cupcakes Fendu Bakery croissants & cookies Saint Germain Bakery palmiersWednesday, August 22, 12
4. 4. Mahalo.Wednesday, August 22, 12
5. 5. Y combinator Examples in Clojure. Also includes: blenders and kittens. Caveat emptor: I make no effort to teach you Clojure. Kyle Oba @mudphone Pas de ChocolatWednesday, August 22, 12
6. 6. Not this one.Wednesday, August 22, 12Paul Graham did name his company after the *REAL* Y combinator.But, why?
7. 7. This one. (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y))))))Wednesday, August 22, 12Which is this thing, in Clojure.
8. 8. Let’s get started. Here’s a non-recursive deﬁnition of factorial, using the Y combinator.Wednesday, August 22, 12Here it is. Thank you, good night and good luck.
9. 9. (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y)))))) (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))) (defn factorial [n] ((Y almost-factorial) n))Wednesday, August 22, 12Here it is. Thank you, good night and good luck.
10. 10. Questions?Wednesday, August 22, 12
11. 11. An example: 5! (factorial 5) = 5 * 4 * 3 * 2 * 1 = 120 (defn factorial [n] (if (= n 0) 1 (* n (factorial (dec n)))))Wednesday, August 22, 12Let’s back up. This is how a sane person would deﬁne factorial... recursively.
12. 12. here to here? (defn factorial (defn almost-factorial [n] [f] (if (= n 0) (fn [n] 1 (if (= n 0) (* n (factorial (dec n))))) 1 (* n (f (dec n)))))) (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y)))))) (defn factorial [n] ((Y almost-factorial) n))Wednesday, August 22, 12Recursive deﬁnition to non-recursive
13. 13. 2 Things 1) recursion 2) functionsWednesday, August 22, 12
14. 14. 2 Things 1) recursion 2) functionsWednesday, August 22, 12
15. 15. “The Y combinator allows recursion recursion... as a set of rewrite rules, without requiring native recursion support in the language.” -- Someone on WikipediaWednesday, August 22, 12
16. 16. replace “native recursion” with manual recursionWednesday, August 22, 12
17. 17. (defn factorial [n] (defn fact (if (= n 0) [n] 1 (if (= n 0) (* n (factorial (dec n))))) 1 (* n (ERROR (dec n)))))Wednesday, August 22, 12
18. 18. (defn factorial [n] (defn fact (if (= n 0) [n] 1 (if (= n 0) (* n (factorial (dec n))))) 1 (* n (ERROR (dec n))))) n = 0 OK n = 1 BOOM!Wednesday, August 22, 12
19. 19. (defn factorial [n] (defn fact (defn fact (if (= n 0) [n] [n] 1 (if (= n 0) (if (= n 0) (* n (factorial (dec n))))) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n) (defn fact (defn fact [n] [n] (if (= n 0) (if (= n 0) 1 1 n = 0 OK (* n (ERROR (dec n))))) (* n (ERROR (dec n)) (defn fact (defn fact [n] [n] n = 1 BOOM! (if (= n 0) (if (= n 0) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n) (defn fact (defn fact [n] [n] (if (= n 0) (if (= n 0) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n)Wednesday, August 22, 12
20. 20. (defn fact (fn [n] (fn [n] [n] (if (= n 0) (if (= n (if (= n 0) 1 1 1 (* n (ERROR (dec n))))) (* n ( (* n (ERROR (dec n))))) (fn [n] (fn [n] (if (= n 0) (if (= n 0) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n)))))Wednesday, August 22, 12
21. 21. (defn fact (fn [n] (fn [n] [n] (if (= n 0) (if (= n (if (= n 0) 1 1 1 (* n (ERROR (dec n))))) (* n ( (* n (ERROR (dec n))))) (fn [n] (fn [n] (if (= n 0) (if (= n 0) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n))))) n=0 n=1 n=2 n=3 n=4Wednesday, August 22, 12
22. 22. replace “native recursion” with manual recursion “rewrite rules”Wednesday, August 22, 12
23. 23. 2 Things 1) recursion 2) functionsWednesday, August 22, 12
24. 24. 2 Things 1) recursion 2) functionsWednesday, August 22, 12
25. 25. Functions are machines. Functions are relationships, between inputs and outputs.Wednesday, August 22, 12
26. 26. A function is a blender.Wednesday, August 22, 12
27. 27. FIRST ORDER BLENDER A normal blender that consumes single input and creates output.Wednesday, August 22, 12
28. 28. FIRST ORDER BLENDER A normal blender that consumes single input and creates output. HIGHER ORDER BLENDER A special blender that consumes a blender and outputs another blender.Wednesday, August 22, 12
29. 29. FIRST ORDER BLENDER A normal blender that consumes single input and creates output. HIGHER ORDER BLENDER A special blender that consumes a blender and outputs another blender. FIXPOINT (BLENDER) COMBINATOR Y Consumes a blender and produces a new blender that can consume any number of inputs.Wednesday, August 22, 12
30. 30. ONEWednesday, August 22, 12
31. 31. ONE (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12
32. 32. ONE ANY (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12
33. 33. ONE ANY Y (defn almost-factorial [f] factorial (fn [n] (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12
34. 34. if you squint (defn factorial (defn almost-factorial [n] [f] (if (= n 0) (fn [n] 1 (if (= n 0) (* n (factorial (dec n))))) 1 (* n (f (dec n)))))) (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y)))))) (defn factorial [n] ((Y almost-factorial) n))Wednesday, August 22, 12Derivation not possible in 3 minutes.
35. 35. I’m so sorry. No kittens were blended during the creation of this presentation.Wednesday, August 22, 12
36. 36. No really, done now. No kittens were blended during the creation of this presentation.Wednesday, August 22, 12
37. 37. A Clojure Primer PARENTHESIS (+ 1 2 3) ;; => 6 PREFIX NOTATION (operator arg1 arg2 arg3) FUNCTIONS (defn multby2 (fn [n] (* n 2)) [n] (* n 2)) ;; (multby2 4) => 8Wednesday, August 22, 121) First, a primer on LISP & Clojure- parens for function call- preﬁx notation, followed by arguments2) And, function deﬁnition and anonymous functions
38. 38. (defn simple-factorial [n] (if (= n 0) 1 (* n (simple-factorial (dec n)))))Wednesday, August 22, 12Here, we remove the recursive deﬁnition. Kind of delaying it, for now.
39. 39. (defn simple-factorial [n] (if (= n 0) 1 (* n (simple-factorial (dec n))))) (defn part [self n] (if (= n 0) 1 (* n (self self (dec n))))) ;; (part part 5) => 120Wednesday, August 22, 12Here, we remove the recursive deﬁnition. Kind of delaying it, for now.
40. 40. (defn simple-factorial [n] (if (= n 0) 1 (* n (simple-factorial (dec n))))) (defn part [self n] (if (= n 0) 1 (* n (self self (dec n))))) ;; (part part 5) => 120Wednesday, August 22, 12Change part to take a single arg, returning a function that takes n.
41. 41. (defn part [self n] (if (= n 0) 1 (* n (self self (dec n))))) ;; (part part 5) => 120 (defn part2 [self] (fn [n] (if (= n 0) 1 (* n ((self self) (dec n)))))) ;; ((part2 part2) 5) => 120Wednesday, August 22, 12Change part to take a single arg, returning a function that takes n.
42. 42. (defn part [self n] (if (= n 0) 1 (* n (self self (dec n))))) ;; (part part 5) => 120 (defn part2 [self] (fn [n] (if (= n 0) 1 (* n ((self self) (dec n)))))) ;; ((part2 part2) 5) => 120Wednesday, August 22, 12Replace (self self) with f, which blows up in a Stack Overﬂow... but, we press on.
43. 43. (defn part2 [self] (fn [n] (if (= n 0) 1 (* n ((self self) (dec n)))))) ;; ((part2 part2) 5) => 120 (defn part3 [self] (let [f (self self)] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))))Wednesday, August 22, 12Replace (self self) with f, which blows up in a Stack Overﬂow... but, we press on.
44. 44. (defn part2 [self] (fn [n] (if (= n 0) 1 (* n ((self self) (dec n)))))) ;; ((part2 part2) 5) => 120 (defn part3 [self] (let [f (self self)] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))))Wednesday, August 22, 12Bury, the (self self) call in a lambda.
45. 45. (defn part3 [self] (let [f (self self)] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))) (defn part4 [self] (let [f (fn [y] ((self self) y))] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))))Wednesday, August 22, 12Bury, the (self self) call in a lambda.
46. 46. (defn part3 [self] (let [f (self self)] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))) (defn part4 [self] (let [f (fn [y] ((self self) y))] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))))Wednesday, August 22, 12Rip out the function that looks almost like the factorial function. This is what we’regeneralizing. The Y combinator computes the ﬁxpoint of this function.
47. 47. (defn part4 [self] (let [f (fn [y] ((self self) y))] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))) (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12Rip out the function that looks almost like the factorial function. This is what we’regeneralizing. The Y combinator computes the ﬁxpoint of this function.
48. 48. (defn part4 [self] (let [f (fn [y] ((self self) y))] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))) (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12Insert almost-factorial into the part function.
49. 49. (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))) (defn part5 [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))Wednesday, August 22, 12Insert almost-factorial into the part function.
50. 50. (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))) (defn part5 [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))Wednesday, August 22, 12fact5 is a working factorial function, but we can generalize it
51. 51. (defn part5 [self] (let [f (fn [y] ((self self) y))] (almost-factorial f))) (defn fact5 [n] ((part5 part5) n))Wednesday, August 22, 12fact5 is a working factorial function, but we can generalize it
52. 52. (defn part5 [self] (let [f (fn [y] ((self self) y))] (almost-factorial f))) (defn fact5 [n] ((part5 part5) n))Wednesday, August 22, 12here we embed the deﬁnition the “part” function
53. 53. (defn fact5 [n] ((part5 part5) n)) (def fact6 (let [part (fn [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))] (part part)))Wednesday, August 22, 12here we embed the deﬁnition the “part” function
54. 54. (defn fact5 [n] ((part5 part5) n)) (def fact6 (let [part (fn [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))] (part part)))Wednesday, August 22, 12rename part => xand self => xfor kicks really
55. 55. (def fact6 (let [part (fn [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))] (part part))) (def fact7 (let [x (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))] (x x)))Wednesday, August 22, 12rename part => xand self => xfor kicks really
56. 56. (def fact6 (let [part (fn [self] (let [f (fn [y] ((self self) y))] (almost-factorial f)))] (part part))) (def fact7 (let [x (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))] (x x)))Wednesday, August 22, 12replace the (x x) invocation with a lambda of the same
57. 57. (def fact7 (let [x (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))] (x x))) (def fact8 ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))))Wednesday, August 22, 12replace the (x x) invocation with a lambda of the same
58. 58. (def fact7 (let [x (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))] (x x))) (def fact8 ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f)))))Wednesday, August 22, 12Rename to Yand generalize, by accepting a function g and using this to replace almost-factorial
59. 59. (def fact8 ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f))))) (defn nearly-Y [g] ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (g f)))))Wednesday, August 22, 12Rename to Yand generalize, by accepting a function g and using this to replace almost-factorial
60. 60. (def fact8 ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (almost-factorial f))))) (defn nearly-Y [g] ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (g f)))))Wednesday, August 22, 12Replace f with the anonymous function bound to it
61. 61. (defn nearly-Y [g] ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (g f))))) (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y))))))Wednesday, August 22, 12Replace f with the anonymous function bound to it
62. 62. (defn nearly-Y [g] ((fn [x] (x x)) (fn [x] (let [f (fn [y] ((x x) y))] (g f))))) (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y))))))Wednesday, August 22, 12
63. 63. (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y)))))) (defn factorial [n] ((Y almost-factorial) n))Wednesday, August 22, 12
64. 64. (defn Y [g] ((fn [x] (x x)) (fn [x] (g (fn [y] ((x x) y)))))) I’m so sorry. (defn factorial [n] ((Y almost-factorial) n))Wednesday, August 22, 12
65. 65. I’m so sorry. No kittens were blended during the creation of this presentation.Wednesday, August 22, 12
66. 66. (defn almost-factorial (Y almost-factorial) [f] (fn [n] ;; ((Y almost-factorial) 5) => 120 (if (= n 0) 1 (* n (f (dec n))))))Wednesday, August 22, 12
67. 67. (defn almost-factorial (Y almost-factorial) [f] (fn [n] ;; ((Y almost-factorial) 5) => 120 (if (= n 0) 1 (* n (f (dec n)))))) YWednesday, August 22, 12
68. 68. (defn almost-factorial (Y almost-factorial) [f] (fn [n] ;; ((Y almost-factorial) 5) => 120 (if (= n 0) 1 (* n (f (dec n)))))) Y FACTORIALWednesday, August 22, 12
69. 69. (defn almost-factorial (Y almost-factorial) [f] (fn [n] ;; ((Y almost-factorial) 5) => 120 (if (= n 0) 1 (* n (f (dec n)))))) (defn fact (defn fact [n] [n] (if (= n 0) (if (= n 0) Y 1 1 (* n (ERROR (dec n))))) (* n (ERROR (dec n))))) (defn fact (defn fact [n] [n] (if (= n 0) (if (= n 0) 1 1 (* n (ERROR (dec n))))) (* n (ERROR (d FACTORIALWednesday, August 22, 12
70. 70. (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))) Y factorialWednesday, August 22, 12
71. 71. (defn almost-factorial [f] (fn [n] (if (= n 0) 1 (* n (f (dec n)))))) Y factorialWednesday, August 22, 12
72. 72. (defn almost-factorial [f] (fn [n] (if (= n 0) 1 ONE (* n (f (dec n)))))) Y factorialWednesday, August 22, 12
73. 73. (defn almost-factorial [f] (fn [n] (if (= n 0) ANY 1 ONE (* n (f (dec n)))))) Y factorialWednesday, August 22, 12