Come risolvere un puzzle cripto-aritmetico in Scala

Utilizzando Monad Transformer o un semplice algoritmo genetico

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1. @filippovitale Aprile 2019 FPinBO Utilizzando Monad Transformer o un semplice algoritmo genetico Come risolvere un puzzle cripto-aritmetico in Scala
6. SEND + MORE ------- MONEY @alexeyraga https://en.wikipedia.org/wiki/Verbal_arithmetic
7. SEND + MORE ------- MONEY @alexeyraga E == E == E
8. SEND + MORE ------- MONEY @alexeyraga S,M ≠ 0
10. SEND + MORE ------- MONEY M←1
11. SEND + 1ORE ------- 1ONEY M←1 O←0 S←8,9
12. M←1 Aspetta un attimo... @alexeyraga
13. https://bartoszmilewski.com/2015/05/11/using-monads-in-c-to-solve-constraints-1-the-list-monad/
14. Immutability Higher-order function Monad Referential transparency
15. Ricerca Esaustiva Implementazione naïve
16. “Brute force” 10!/(10 – 8)! ≈ 2 Milioni Quanto è esteso lo Spazio di Ricerca?
17. for (int s = 0; s < 10; ++s) for (int e = 0; e < 10; ++e) for (int n = 0; n < 10; ++n) for (int d = 0; d < 10; ++d) ... s != 0 e != s n != s && n != e d != s && d != e && d != n Approccio Imperativo: 8 loop annidati
18. for (int s = 0; s < 10; ++s) for (int e = 0; e < 10; ++e) for (int n = 0; n < 10; ++n) for (int d = 0; d < 10; ++d) ... s != 0 e != s n != s && n != e d != s && d != e && d != n Ricerca soluzioni teoricamente possibili: ➔ Generazione delle assegnazioni ➔ Verifica dei vincoli
19. for { s <- 0 to 9 if s != 0 e <- 0 to 9 if e != s n <- 0 to 9 if n != s && n != e d <- 0 to 9 if d != s && d != e && d!=n ... for (int s = 0; s < 10; ++s) for (int e = 0; e < 10; ++e) for (int n = 0; n < 10; ++n) for (int d = 0; d < 10; ++d) ... s != 0 e != s n != s && n != e d != s && d != e && d != n Ricerca soluzioni teoricamente possibili: ➔ Generazione delle assegnazioni ➔ Verifica dei vincoli
20. Ricerca soluzioni teoricamente possibili Verifica vincoli
21. val listOfDigits = (0 to 9).toList val solutions = for { s <- listOfDigits if s != 0 e <- listOfDigits if s != e n <- listOfDigits if s != n && e != n d <- listOfDigits if s != d && e != d && n != d m <- listOfDigits if m != 0 if s != m && e != m && n != m && d != m o <- listOfDigits if s != o && e != o && n != o && d != o && m != o r <- listOfDigits if s != r && e != r && n != r && d != r && m != r && o != r y <- listOfDigits if s != y && e != y && n != y && d != y && m != y && o != y && r != y } yield ??? Ricerca soluzioni teoricamente possibili Verifica vincoli
22. val listOfDigits = (0 to 9).toList val solutions = for { s <- listOfDigits if s != 0 e <- listOfDigits if s != e n <- listOfDigits if s != n && e != n d <- listOfDigits if s != d && e != d && n != d m <- listOfDigits if m != 0 if s != m && e != m && n != m && d != m o <- listOfDigits if s != o && e != o && n != o && d != o && m != o r <- listOfDigits if s != r && e != r && n != r && d != r && m != r && o != r y <- listOfDigits if s != y && e != y && n != y && d != y && m != y && o != y && r != y send = List(s, e, n, d).reduce(_ * 10 + _) more = List(m, o, r, e).reduce(_ * 10 + _) money = List(m, o, n, e, y).reduce(_ * 10 + _) if send + more == money } yield (send, more, money) Ricerca soluzioni teoricamente possibili Verifica vincoli
23. val listOfDigits = (0 to 9).toList val solutions = for { s <- listOfDigits if s != 0 e <- listOfDigits if s != e n <- listOfDigits if s != n && e != n d <- listOfDigits if s != d && e != d && n != d m <- listOfDigits if m != 0 if s != m && e != m && n != m && d != m o <- listOfDigits if s != o && e != o && n != o && d != o && m != o r <- listOfDigits if s != r && e != r && n != r && d != r && m != r && o != r y <- listOfDigits if s != y && e != y && n != y && d != y && m != y && o != y && r != y send = List(s, e, n, d).reduce(_ * 10 + _) more = List(m, o, r, e).reduce(_ * 10 + _) money = List(m, o, n, e, y).reduce(_ * 10 + _) if send + more == money } yield (send, more, money)
24. “Jimmy” e il Multiverso
26. 0 1 2 3 4 5 6 7 8 9
27. 1 2 3 4 5 6 7 8 9 0
28. ? ? ? ? ? ? ? ? ? ? S = …
29. 1 2 3 4 5 6 7 8 9 0 0 S = 0!
31. ? ? ? ? ? ? ? ? ? ? S = …
32. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5 1 2 3 4 0 6 7 8 9 0 6 2 3 4 5 1 7 8 9 0 1 7 3 4 5 6 2 8 9 0 1 3 2 4 5 6 7 8 9 0 1 8 3 4 5 6 7 8 9 0 1 4 3 2 5 6 7 8 9 0 1 9 3 4 5 6 7 8 2
33. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5 1 2 3 4 0 6 7 8 9 0 6 2 3 4 5 1 7 8 9 0 1 7 3 4 5 6 2 8 9 0 1 3 2 4 5 6 7 8 9 0 1 8 3 4 5 6 7 8 9 0 1 4 3 2 5 6 7 8 9 0 1 9 3 4 5 6 7 8 2 S = 1
34. 0 1 2 3 4 5 6 7 8 9 S = 1
35. 0 1 2 3 4 5 6 7 8 9 S = 1 E = …
36. 0 1 2 3 4 5 6 7 8 9 S = 1 E = … 0 2 3 4 5 6 7 8 9
37. Cosa Farebbe Lambda Man in questa situazione? http://homepages.inf.ed.ac.uk/wadler/
38. Implementazione delle Dimensioni Parallele utilizzando il List Monad
39. val l = for { a <- List(1, 2) b <- List(1, 2) } yield Map('a' -> a, 'b' -> b) val l = ???
40. val l = for { a <- List(1, 2) b <- List(1, 2) } yield Map('a' -> a, 'b' -> b) val l: List[A] = ???
41. val l = for { a <- List(1, 2) b <- List(1, 2) } yield Map('a' -> a, 'b' -> b) val l: List[Map[Char, Int]] = ???
42. val l = for { a <- List(1, 2) b <- List(1, 2) } yield Map('a' -> a, 'b' -> b) List(Map('a' -> 1, 'b' -> 1), Map('a' -> 1, 'b' -> 2), Map('a' -> 2, 'b' -> 1), Map('a' -> 2, 'b' -> 2))
43. val l = for { a <- List(1, 2) b <- List(1, 2) if a != b } yield Map('a' -> a, 'b' -> b) List(Map('a' -> 1, 'b' -> 2), Map('a' -> 2, 'b' -> 1))
44. Implementazione delle Dimensioni Parallele utilizzando lo State Monad https://www.manning.com/books/functional-programming-in-scala#chapter-6
45. Implementazione delle Dimensioni Parallele utilizzando lo State Monad https://www.manning.com/books/functional-programming-in-scala#chapter-6 L’analogia del burrito comincia a scricchiolare
46. “The future is a function of the past.” – A. P. Robertson https://goo.gl/jt6bvz
47. // stateful computation type State[S, +A] = S => (S, A) https://bartoszmilewski.com/2016/11/30/monads-and-effects/#State
48. // stateful computation type State[S, +A] = S => (S, A) https://stackoverflow.com/a/10231786/81444
49. object State { def apply[S, A](f: S => (S, A)): State[S, A] }
50. // stateful computation f: S => (S, A) def select(xs: List[Int]): (List[Int], Int) = (xs.tail, xs.head)
51. // stateful computation f: S => (S, A) def select(xs: List[Int]): (List[Int], Int) = (xs.tail, xs.head) 1 2 3 4 5 6 7 8 9 0
52. // stateful computation f: S => (S, A) def select(xs: List[Int]): (List[Int], Int) = (xs.tail, xs.head)
53. // stateful computation f: S => (S, A) def select(xs: List[Int]): (List[Int], Int) = (xs.tail, xs.head)
54. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s = ???
55. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ S, A ] = ???
56. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ S, A ] = ??? def select(xs: List[Int]): (List[Int], Int)
57. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], A ] = ???
58. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], Map[Char, Int] ] = ???
59. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], Map[Char, Int] ] = ???
60. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], Map[Char, Int] ] = ???
61. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], Map[Char, Int] ] = ??? s.eval(List(1, 2)) ==== Map('a' -> 1, 'b' -> 2)
62. import scalaz.State val s = for { a <- State(select) b <- State(select) } yield Map('a' -> a, 'b' -> b) val s: State[ List[Int], Map[Char, Int] ] = ??? s.eval(List(8, 3)) ==== Map('a' -> 8, 'b' -> 3)
63. val s = for { a <- State(select) b <- State(select) c <- State(select) } yield Map('a' -> a, 'b' -> b, 'c' -> c) s.eval(1 |-> 3) ==== Map('a' -> 1, 'b' -> 2, 'c' -> 3)
64. val s = for { a <- State(select) b <- State(select) c <- State(select) d <- State(select) } yield Map('a' -> a, 'b' -> b, 'c' -> c, 'd' -> d) s.eval(1 |-> 4) ==== Map('a' -> 1, 'b' -> 2, 'c' -> 3, 'd' -> 4) a b c d
66. Implementazione delle Dimensioni Parallele utilizzando StateT[List, S, A]
67. Implementazione delle Dimensioni Parallele utilizzando StateT[List, S, A] https://speakerdeck.com/gabro/monad-transformers-what-and-why
70. def select[A](xs: List[A]): (List[A], A) State S => ( S , A)
71. def select[A](xs: List[A]): (List[A], A) State S => ( S , A) StateT[F, S, A] S => F[( S , A)]
72. def select[A](xs: List[A]): (List[A], A) State S => ( S , A) def select[A](xs: List[A]): List[(List[A], A)] StateT[F, S, A] S => F[( S , A)]
73. def select[A](xs: List[A]): List[(List[A], A)] = xs map { x => ( ??? ) }
74. def select[A](xs: List[A]): List[(List[A], A)] = xs map { x => (xs.filterNot(_ == x), x) }
75. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5 1 2 3 4 0 6 7 8 9 0 6 2 3 4 5 1 7 8 9 0 1 7 3 4 5 6 2 8 9 0 1 3 2 4 5 6 7 8 9 0 1 8 3 4 5 6 7 8 9 0 1 4 3 2 5 6 7 8 9 0 1 9 3 4 5 6 7 8 2
76. sl.eval(1 |-> 2) == ??? import scalaz.StateT import scalaz.std.list._ val sl = for { a <- StateT(select) b <- StateT(select) } yield Map('a' -> a, 'b' -> b)
77. import scalaz.StateT import scalaz.std.list._ val sl = for { a <- StateT(select) b <- StateT(select) } yield Map('a' -> a, 'b' -> b) sl.eval(1 |-> 2) == List(Map('a' -> 1, 'b' -> 2), Map('a' -> 2, 'b' -> 1)) https://gist.github.com/filippovitale/6cc45396ed917a2a8411
78. val sel = StateT(select[Int]) for { s <- sel e <- sel n <- sel d <- sel m <- sel o <- sel r <- sel y <- sel … }
79. val sel = StateT(select[Int]) for { s <- sel e <- sel n <- sel d <- sel m <- sel o <- sel r <- sel y <- sel … } Per ogni chiamata `.flatMap`, `sel` restituisce un numero diverso! 0 1 2 3 4 5 6 7 8 9
80. Risolvere il puzzle con StateT[List, S, A]
81. val sel = StateT(select[Int]) for { s <- sel if s != 0 e <- sel n <- sel d <- sel m <- sel if m != 0 o <- sel r <- sel y <- sel send = List(s, e, n, d).reduce(_ * 10 + _) more = List(m, o, r, e).reduce(_ * 10 + _) money = List(m, o, n, e, y).reduce(_ * 10 + _) if send + more == money } yield (s, e, n, d, m, o, r, y) MonadPlus MonadPlus https://speakerdeck.com/filippovitale/send-plus-more-equals-money-scalasyd-july-2015
82. Risolvere il puzzle con le API di Scala
83. val listOfDigits = (0 to 9).toList val solutions = for { s <- listOfDigits if s != 0 e <- listOfDigits if s != e n <- listOfDigits if s != n && e != n d <- listOfDigits if s != d && e != d && n != d m <- listOfDigits if m != 0 if s != m && e != m && n != m && d != m o <- listOfDigits if s != o && e != o && n != o && d != o && m != o r <- listOfDigits if s != r && e != r && n != r && d != r && m != r && o != r y <- listOfDigits if s != y && e != y && n != y && d != y && m != y && o != y && r != y send = List(s, e, n, d).reduce(_ * 10 + _) more = List(m, o, r, e).reduce(_ * 10 + _) money = List(m, o, n, e, y).reduce(_ * 10 + _) if send + more == money } yield (send, more, money) Ricerca soluzioni teoricamente possibili Verifica vincoli
84. "sendmoremoney".distinct.size // 8 distinct `char` val maps = for { ??? } yield ??? Ricerca soluzioni teoricamente possibili Verifica vincoli
85. val maps = for { ??? } yield ("sendmoremoney".distinct zip ???).toMap Ricerca soluzioni teoricamente possibili Verifica vincoli
86. val maps = for { combination <- (0 to 9).combinations(8) // 45 combinazioni // Vector(0, 1, 2, 3, 4, 5, 6, 7) // (8, 9) // Vector(0, 1, 2, 3, 4, 5, 6, 8) // (7, 9) // Vector(0, 1, 2, 3, 4, 5, 6, 9) // (7, 8) // Vector(0, 1, 2, 3, 4, 5, 7, 8) // (6, 9) // Vector(0, 1, 2, 3, 4, 5, 7, 9) // (6, 8) // Vector(0, 1, 2, 3, 4, 5, 8, 9) // (6, 7) // Vector(0, 1, 2, 3, 4, 6, 7, 8) // (5, 9) // Vector(0, 1, 2, 3, 4, 6, 7, 9) // (5, 8) // Vector(0, 1, 2, 3, 4, 6, 8, 9) // (5, 7) // ... } yield ("sendmoremoney".distinct zip ???).toMap Ricerca soluzioni teoricamente possibili Verifica vincoli
87. val maps = for { combination <- (0 to 9).combinations(8) // 45 combinazioni permutation <- combination.permutations // 40K permutazioni } yield ("sendmoremoney".distinct zip permutation).toMap Ricerca soluzioni teoricamente possibili Verifica vincoli
88. val maps = for { combination <- (0 to 9).combinations(8) // 45 combinazioni permutation <- combination.permutations // 40K permutazioni } yield ("sendmoremoney".distinct zip permutation).toMap // ~2M Ricerca soluzioni teoricamente possibili Verifica vincoli
89. val maps = for { combination <- (0 to 9).combinations(8) // 45 combinazioni permutation <- combination.permutations // 40K permutazioni } yield ("sendmoremoney".distinct zip permutation).toMap // ~2M val solutions = for { m <- maps if m('s') != 0 && m('m') != 0 send = "send" map m reduce (_ * 10 + _) more = "more" map m reduce (_ * 10 + _) money = "money" map m reduce (_ * 10 + _) if send + more == money } yield (send, more, money) Ricerca soluzioni teoricamente possibili Verifica vincoli
90. val maps = for { combination <- (0 to 9).combinations(8) // 45 combinazioni permutation <- combination.permutations // 40K permutazioni } yield ("sendmoremoney".distinct zip permutation).toMap // ~2M val solutions = for { m <- maps if m('s') != 0 && m('m') != 0 send = "send" map m reduce (_ * 10 + _) more = "more" map m reduce (_ * 10 + _) money = "money" map m reduce (_ * 10 + _) if send + more == money } yield (send, more, money) solutions.next() // (9567,1085,10652) Ricerca soluzioni teoricamente possibili Verifica vincoli
91. Risolvere il puzzle a compile-time
92. Risolvere il puzzle a compile-time Peano's arithmetic Shapeless: HList, Sum, … http://jto.github.io/articles/typelevel_quicksort/
94. Finora abbiamo visto solo ricerche depth-first
96. Risolvere il puzzle con un Algoritmo Genetico http://dl.acm.org/citation.cfm?id=1725467 Solving cryptarithmetic problems using a Genetic Algorithm
97. Algoritmo genetico type Digit = Int type Individual = Vector[Digit] // “sendmory” - Individual Vector di lunghezza 8 2 Digit non utilizzate
98. Algoritmo genetico type Digit = Int type Individual = Vector[Digit] // “sendmory” type Population = List[Individual] - Individual - Population
99. Algoritmo genetico type Digit = Int type Individual = Vector[Digit] // “sendmory” type Population = List[Individual] def fitness(individual: Individual): Int // 0 - Individual - Population - Fitness function
100. Algoritmo genetico - Individual - Population - Fitness function - Mutation type Digit = Int type Individual = Vector[Digit] // “sendmory” type Population = List[Individual] def fitness(individual: Individual): Int def mutation(individual: Individual): Individual // asexual genetic algorithm
101. Semplice implementazione di un Algoritmo Genetico per risolvere il puzzle
102. /** side-effect */ val r = util.Random val popolationSize = 512 val generationRetainment = 32 // top 6% // 16 of the 28 possible mutations val individualProlificness: Int = generationSize / generationRetainment
103. def fitness(individual: Individual): Int = { math.abs(send + more - money) }
104. def fitness(individual: Individual): Int = { val cs = "sendmoremoney".distinct val m = cs.zip(individual).toMap math.abs(send + more - money) }
105. def fitness(individual: Individual): Int = { val cs = "sendmoremoney".distinct val m = cs.zip(individual).toMap val send = "send" map m reduce (_ * 10 + _) val more = "more" map m reduce (_ * 10 + _) val money = "money" map m reduce (_ * 10 + _) math.abs(send + more - money) }
106. val mutationIndices = List(0 |-> 7).combinations(2) def everyMutation(individual: Individual): Iterator[Individual] = r.shuffle(mutationIndices) .map { case Seq(i, j) => individual .updated(i, individual(j)) .updated(j, individual(i)) } // +-+-+-+-+-+-+-+-+ // |S|E|N|D|M|O|R|Y| // +-+-+-+-+-+-+-+-+ // |0|1|2|3|4|5|6|7| // +-+-+-+-+-+-+-+-+ // ^ ^ // | | // +-----+ // [i] [j]
107. val solution = ???
108. val solution = Iterator .iterate(generatePopulation(generationSize)) (createNextGeneration)
109. def createNextGeneration(currentGeneration: Population) : Population = ???
110. def createNextGeneration(currentGeneration: Population) : Population = currentGeneration .take(generationRetainment) // top 6%
111. def createNextGeneration(currentGeneration: Population) : Population = currentGeneration .take(generationRetainment) .flatMap(everyMutation(_).take(individualProlificness))
112. def createNextGeneration(currentGeneration: Population) : Population = currentGeneration .take(generationRetainment) .flatMap(everyMutation(_).take(individualProlificness)) .sortBy(fitness)
113. def createNextGeneration(currentGeneration: Population) : Population = if (fitness(currentGeneration.head) == 0) { currentGeneration.take(1) } else { ??? }
114. def createNextGeneration(currentGeneration: Population) : Population = if (fitness(currentGeneration.head) == 0) { currentGeneration.take(1) } else { currentGeneration.take(generationRetainment) .flatMap(everyMutation(_).take(individualProlificness)) .sortBy(fitness) }
115. val solution = Iterator .iterate(generatePopulation(generationSize)) (createNextGeneration) .dropWhile(_.size > 1)
116. val solution = Iterator .iterate(generatePopulation(generationSize)) (createNextGeneration) .dropWhile(_.size > 1) .map(_.head).next()
117. Benchmarking
119. @filippovitale Aprile 2019 FPinBO $ tail -f questions Bologna, September 20th-21st, 2019