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Diffstat (limited to 'lambda-calcul/haskell/challenges.json')
| -rw-r--r-- | lambda-calcul/haskell/challenges.json | 124 |
1 files changed, 92 insertions, 32 deletions
diff --git a/lambda-calcul/haskell/challenges.json b/lambda-calcul/haskell/challenges.json index 544bf0a..119115b 100644 --- a/lambda-calcul/haskell/challenges.json +++ b/lambda-calcul/haskell/challenges.json @@ -109,198 +109,258 @@ }, { "level": 18, + "description": "Integer literals are native values in this interpreter — they are already fully reduced and need no evaluation. What does the integer 42 reduce to?", + "expr": "42", + "answer": "42" + }, + { + "level": 19, + "description": "Zero is also a valid integer literal that self-evaluates. What does 0 reduce to?", + "expr": "0", + "answer": "0" + }, + { + "level": 20, + "description": "Negative integers are written with a minus sign immediately preceding the digits, with no space: -7 is a single integer literal, not the identifier - applied to 7. What does -7 reduce to?", + "expr": "-7", + "answer": "-7" + }, + { + "level": 21, + "description": "Apply the identity function to an integer. ((lam (x) x) 99) beta-reduces by substituting 99 for x. What is the result?", + "expr": "((lam (x) x) 99)", + "answer": "99" + }, + { + "level": 22, + "description": "A two-argument constant function applied to an integer and a free variable. (lam (x y) y) ignores its first argument and returns the second. What does ((lam (x y) y) 42 z) reduce to?", + "expr": "((lam (x y) y) 42 z)", + "answer": "z" + }, + { + "level": 23, + "description": "A let-binding wraps an integer in a local name. (let ((n 5)) n) desugars to ((lam (n) n) 5) and reduces to 5. What is the result?", + "expr": "(let ((n 5)) n)", + "answer": "5" + }, + { + "level": 24, + "description": "An integer passed as argument to a free variable f produces a stuck neutral term — f has no definition, so (f 42) cannot reduce further. What does (f 42) reduce to?", + "expr": "(f 42)", + "answer": "(f 42)" + }, + { + "level": 25, + "description": "An integer in function position applied to an argument cannot reduce — 42 is not a lambda. The result is a stuck neutral application. What does (42 x) reduce to?", + "expr": "(42 x)", + "answer": "(42 x)" + }, + { + "level": 26, + "description": "Chained neutral applications. (42 a) is already stuck; applying that to b gives a left-associative neutral term. Remember that (f a b) pretty-prints as (f a b). What does ((42 a) b) reduce to?", + "expr": "((42 a) b)", + "answer": "(42 a b)" + }, + { + "level": 27, + "description": "A def declaration binds an integer to a name globally. Evaluating n afterwards should return the integer. What does the following program produce?", + "expr": "(def n -3)\nn", + "answer": "-3" + }, + { + "level": 28, "description": "Introduce And. And a b = a False b: if a is True it selects its second argument (b); if a is False it selects its first argument (False). Evaluate (And True True).", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Or (lam (a b) (a b True)))\n(def And (lam (a b) (a False b)))\n(And True True)", "answer": "(lam (f t) t)" }, { - "level": 19, + "level": 29, "description": "Combine Or, And and Not. Evaluate (And (Or False True) (Not False)). Both sub-expressions must reduce to True for And to return True.", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def Or (lam (a b) (a b True)))\n(def And (lam (a b) (a False b)))\n(And (Or False True) (Not False))", "answer": "(lam (f t) t)" }, { - "level": 20, + "level": 30, "description": "Xor (exclusive or) is True when exactly one argument is True. It can be defined as And (Or a b) (Not (And a b)). Evaluate (Xor True False).", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def Xor (lam (a b) (And (Or a b) (Not (And a b)))))\n(Xor True False)", "answer": "(lam (f t) t)" }, { - "level": 21, + "level": 31, "description": "Evaluate (Xor False False). Two identical inputs to Xor should always yield False.", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def Xor (lam (a b) (And (Or a b) (Not (And a b)))))\n(Xor False False)", "answer": "(lam (f t) f)" }, { - "level": 22, + "level": 32, "description": "Church-encoded if-then-else. Because our True selects its second argument and False selects its first, ite must swap the branches: ite c t e = c e t. Evaluate (ite True a b) — which branch is returned?", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def ite (lam (c t e) (c e t)))\n(ite True a b)", "answer": "a" }, { - "level": 23, + "level": 33, "description": "Evaluate (ite False a b). When the condition is False, the else-branch should be returned.", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def Not (lam (b) (b True False)))\n(def And (lam (a b) (a False b)))\n(def Or (lam (a b) (a b True)))\n(def ite (lam (c t e) (c e t)))\n(ite False a b)", "answer": "b" }, { - "level": 24, + "level": 34, "description": "Church-encoded pairs. pair x y = lam f -> f x y stores two values by passing them to a selector function. fst p = p (lam x y -> x) retrieves the first. Evaluate (fst (pair a b)).", "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(fst (pair a b))", "answer": "a" }, { - "level": 25, + "level": 35, "description": "snd p = p (lam x y -> y) retrieves the second element of a Church pair. Evaluate (snd (pair a b)).", "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(snd (pair a b))", "answer": "b" }, { - "level": 26, + "level": 36, "description": "Pairs can be nested. (pair a (pair b c)) is a pair whose second element is itself a pair. Evaluate (snd (snd (pair a (pair b c)))).", "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(snd (snd (pair a (pair b c))))", "answer": "c" }, { - "level": 27, + "level": 37, "description": "swap exchanges the two elements of a pair: swap p = pair (snd p) (fst p). Evaluate (fst (swap (pair a b))). After swapping, fst should return what was previously snd.", "expr": "(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def swap (lam (p) (pair (snd p) (fst p))))\n(fst (swap (pair a b)))", "answer": "b" }, { - "level": 28, + "level": 38, "description": "Church numerals encode natural numbers as higher-order functions: n applies its first argument (the successor step) to its second argument (the base) exactly n times. zero = lam s z -> z applies s zero times. Evaluate (zero f x).", "expr": "(def zero (lam (s z) z))\n(zero f x)", "answer": "x" }, { - "level": 29, + "level": 39, "description": "two = lam s z -> s (s z) applies s twice. Evaluate (two f x) — you should see f applied to x twice.", "expr": "(def two (lam (s z) (s (s z))))\n(two f x)", "answer": "(f (f x))" }, { - "level": 30, + "level": 40, "description": "three = lam s z -> s (s (s z)) applies s three times. Evaluate (three f x).", "expr": "(def three (lam (s z) (s (s (s z)))))\n(three f x)", "answer": "(f (f (f x)))" }, { - "level": 31, + "level": 41, "description": "isZero tests whether a Church numeral is zero. isZero n = n (lam x -> False) True: if n > 0 the step function (lam x -> False) is applied at least once, discarding its argument and returning False; if n = 0 the base True is returned. Evaluate (isZero zero).", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def zero (lam (s z) z))\n(def isZero (lam (n) (n (lam (x) False) True)))\n(isZero zero)", "answer": "(lam (f t) t)" }, { - "level": 32, + "level": 42, "description": "Evaluate (isZero three). Three is not zero, so the step function fires and the result is False.", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def three (lam (s z) (s (s (s z)))))\n(def isZero (lam (n) (n (lam (x) False) True)))\n(isZero three)", "answer": "(lam (f t) f)" }, { - "level": 33, + "level": 43, "description": "The I combinator (identity): I x = x. It is the simplest combinator and the building block of the SKI calculus. Evaluate (I a).", "expr": "(def I (lam (x) x))\n(I a)", "answer": "a" }, { - "level": 34, + "level": 44, "description": "The K combinator (constant): K x y = x. It ignores its second argument and always returns its first. Evaluate (K a b).", "expr": "(def K (lam (x y) x))\n(K a b)", "answer": "a" }, { - "level": 35, + "level": 45, "description": "The S combinator (substitution): S f g x = f x (g x). It distributes its third argument over two functions and applies the results. Evaluate (S f g x) — f and g are free variables here.", "expr": "(def S (lam (f g x) ((f x) (g x))))\n(S f g x)", "answer": "(f x (g x))" }, { - "level": 36, + "level": 46, "description": "SKI completeness: S K K encodes the identity function. S K K x = K x (K x) = x. Evaluate (S K K a) and confirm it equals a.", "expr": "(def S (lam (f g x) ((f x) (g x))))\n(def K (lam (x y) x))\n(S K K a)", "answer": "a" }, { - "level": 37, + "level": 47, "description": "The B combinator (composition): B f g x = f (g x). It is function composition. Evaluate (B f g x) — you should see f applied to the result of g applied to x.", "expr": "(def B (lam (f g x) (f (g x))))\n(B f g x)", "answer": "(f (g x))" }, { - "level": 38, + "level": 48, "description": "The C combinator (flip): C f x y = f y x. It flips the order of the last two arguments of a function. Evaluate (C f a b).", "expr": "(def C (lam (f x y) (f y x)))\n(C f a b)", "answer": "(f b a)" }, { - "level": 39, + "level": 49, "description": "The W combinator (duplication): W f x = f x x. It applies f to x twice, duplicating the argument. Evaluate (W f a).", "expr": "(def W (lam (f x) (f x x)))\n(W f a)", "answer": "(f a a)" }, { - "level": 40, + "level": 50, "description": "Scott-encoded lists. nil = lam c n -> n (the empty list). cons h t = lam c n -> c h t stores head and tail directly. head l = l (lam h t -> h) nil extracts the first element. Evaluate (head (cons a nil)).", "expr": "(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def head (lam (l) (l (lam (h t) h) nil)))\n(head (cons a nil))", "answer": "a" }, { - "level": 41, + "level": 51, "description": "tail l = l (lam h t -> t) nil retrieves the rest of a Scott-encoded list. Evaluate (head (tail (cons a (cons b nil)))) — the head of the tail of a two-element list is its second element.", "expr": "(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def head (lam (l) (l (lam (h t) h) nil)))\n(def tail (lam (l) (l (lam (h t) t) nil)))\n(head (tail (cons a (cons b nil))))", "answer": "b" }, { - "level": 42, + "level": 52, "description": "isNil tests whether a Scott-encoded list is empty. isNil l = l (lam h t -> False) True: a non-empty list immediately calls its cons branch returning False; nil returns its base True. Evaluate (isNil nil).", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def isNil (lam (l) (l (lam (h t) False) True)))\n(isNil nil)", "answer": "(lam (f t) t)" }, { - "level": 43, + "level": 53, "description": "Evaluate (isNil (cons a nil)). A non-empty list is not nil, so the result should be False.", "expr": "(def False (lam (f t) f))\n(def True (lam (f t) t))\n(def nil (lam (c n) n))\n(def cons (lam (h t c n) (c h t)))\n(def isNil (lam (l) (l (lam (h t) False) True)))\n(isNil (cons a nil))", "answer": "(lam (f t) f)" }, { - "level": 44, + "level": 54, "description": "Church-encoded sum types (Either). Left x = lam l r -> l x tags a value as Left; Right x = lam l r -> r x tags it as Right. either f g e = e f g eliminates the sum by applying the matching handler. Evaluate (either f g (Left a)).", "expr": "(def Left (lam (x l r) (l x)))\n(def Right (lam (x l r) (r x)))\n(def either (lam (f g e) (e f g)))\n(either f g (Left a))", "answer": "(f a)" }, { - "level": 45, + "level": 55, "description": "Evaluate (either f g (Right a)). The Right handler g should be applied to a.", "expr": "(def Left (lam (x l r) (l x)))\n(def Right (lam (x l r) (r x)))\n(def either (lam (f g e) (e f g)))\n(either f g (Right a))", "answer": "(g a)" }, { - "level": 46, + "level": 56, "description": "Church-encoded option (Maybe). None = lam n s -> n is the absent case. Some x = lam n s -> s x wraps a value. fromOption default opt = opt default (lam x -> x) returns the wrapped value or the default. Evaluate (fromOption a None).", "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def fromOption (lam (default opt) (opt default (lam (x) x))))\n(fromOption a None)", "answer": "a" }, { - "level": 47, + "level": 57, "description": "Evaluate (fromOption a (Some b)). The option contains b, so the default a is ignored and b is returned.", "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def fromOption (lam (default opt) (opt default (lam (x) x))))\n(fromOption a (Some b))", "answer": "b" }, { - "level": 48, + "level": 58, "description": "mapOption applies a function inside an Option without unwrapping it. mapOption f opt = opt None (lam x -> Some (f x)): None maps to None; Some x maps to Some (f x). Evaluate (mapOption f (Some a)) and give its fully reduced form.", "expr": "(def None (lam (n s) n))\n(def Some (lam (x n s) (s x)))\n(def mapOption (lam (f opt) (opt None (lam (x) (Some (f x))))))\n(mapOption f (Some a))", "answer": "(lam (n s) (s (f a)))" }, { - "level": 49, + "level": 59, "description": "The Kleene predecessor uses a pair-stepping trick to compute pred n = n - 1 (with pred zero = zero). Starting from (pair zero zero), each step shifts: (pair (snd p)) (succ (snd p)). After n steps fst of the pair is n-1. Evaluate (pred zero) — it should return zero.", "expr": "(def zero (lam (s z) z))\n(def one (lam (s z) (s z)))\n(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def succ (lam (n s z) (s ((n s) z))))\n(def pred (lam (n) (fst (n (lam (p) (pair (snd p) (succ (snd p)))) (pair zero zero)))))\n(pred zero)", "answer": "(lam (s z) z)" }, { - "level": 50, + "level": 60, "description": "Evaluate (pred one) using the same Kleene predecessor. After one pair-step starting from (pair zero zero) the fst is still zero. Confirm that the predecessor of one is zero.", "expr": "(def zero (lam (s z) z))\n(def one (lam (s z) (s z)))\n(def pair (lam (x y f) (f x y)))\n(def fst (lam (p) (p (lam (x y) x))))\n(def snd (lam (p) (p (lam (x y) y))))\n(def succ (lam (n s z) (s ((n s) z))))\n(def pred (lam (n) (fst (n (lam (p) (pair (snd p) (succ (snd p)))) (pair zero zero)))))\n(pred one)", "answer": "(lam (s z) z)" |