1 Here are a bunch of pre-tested operations for the untyped lambda calculus. In some cases multiple versions are offered.
3 Some of these are drawn from:
5 * [[!wikipedia Lambda calculus]]
6 * [[!wikipedia Church encoding]]
7 * Oleg's [Basic Lambda Calculus Terms](http://okmij.org/ftp/Computation/lambda-calc.html#basic)
9 and all sorts of other places. Others of them are our own handiwork.
12 **Spoilers!** Below you'll find implementations of map and filter for v3 lists, and several implementations of leq for Church numerals. Those were all requested in Assignment 2; so if you haven't done that yet, you should try to figure them out on your own. (You can find implementations of these all over the internet, if you look for them, so these are no great secret. In fact, we'll be delighted if you're interested enough in the problem to try to think through alternative implementations.)
16 let true = \y n. y in ; aka K
17 let false = \y n. n in ; aka K I
18 let and = \p q. p q false in ; or
19 let and = \p q. p q p in ; aka S C I
20 let or = \p q. p true q in ; or
21 let or = \p q. p p q in ; aka M
22 let not = \p. p false true in ; or
23 let not = \p y n. p n y in ; aka C
24 let xor = \p q. p (not q) q in
25 let iff = \p q. not (xor p q) in ; or
26 let iff = \p q. p q (not q) in
29 let make_pair = \x y f. f x y in
30 let get_fst = \x y. x in ; aka true
31 let get_snd = \x y. y in ; aka false
34 let make_triple = \x y z f. f x y z in
37 ; Church numerals: basic operations
39 let zero = \s z. z in ; aka false
40 let one = \s z. s z in ; aka I
41 let succ = \n s z. s (n s z) in
42 ; for any Church numeral n > zero : n (K y) z ~~> y
43 let iszero = \n. n (\x. false) true in
48 let empty = \f z. z in
49 let make_list = \h t f z. f h (t f z) in
50 let isempty = \lst. lst (\h sofar. false) true in
51 let head = \lst. lst (\h sofar. h) junk in
52 let tail_empty = empty in
53 let tail = \lst. (\shift. lst shift (make_pair empty tail_empty) get_snd)
55 (\h p. p (\t y. make_pair (make_list h t) t)) in
56 let length = \lst. lst (\h sofar. succ sofar) 0 in
57 let map = \f lst. lst (\h sofar. make_list (f h) sofar) empty in
58 let filter = \f lst. lst (\h sofar. f h (make_list h sofar) sofar) empty in ; or
59 let filter = \f lst. lst (\h. f h (make_list h) I) empty in
60 let singleton = \x f z. f x z in
61 ; append [a;b;c] [x;y;z] ~~> [a;b;c;x;y;z]
62 let append = \left right. left make_list right in
63 ; very inefficient but correct reverse
64 let reverse = \lst. lst (\h sofar. append sofar (singleton h)) empty in ; or
65 ; more efficient reverse builds a left-fold instead
66 ; (make_left_list a (make_left_list b (make_left_list c empty)) ~~> \f z. f c (f b (f a z))
67 let reverse = (\make_left_list lst. lst make_left_list empty) (\h t f z. t f (f h z)) in
68 ; zip [a;b;c] [x;y;z] ~~> [(a,x);(b,y);(c,z)]
69 let zip = \left right. (\base build. reverse left build base (\x y. reverse x))
71 (make_pair empty (map (\h u. u h) right))
73 (\h sofar. sofar (\x y. isempty y
75 (make_pair (make_list (\u. head y (u h)) x) (tail y))
77 let all = \f lst. lst (\h sofar. and sofar (f h)) true in
78 let any = \f lst. lst (\h sofar. or sofar (f h)) false in
83 let empty = make_pair true junk in
84 let make_list = \h t. make_pair false (make_pair h t) in
85 let isempty = \lst. lst get_fst in
86 let head = \lst. isempty lst err (lst get_snd get_fst) in
87 let tail_empty = empty in
88 let tail = \lst. isempty lst tail_empty (lst get_snd get_snd) in
91 ; more math with Church numerals
93 let add = \m n. m succ n in ; or
94 let add = \m n s z. m s (n s z) in
95 let mul = \m n. m (\z. add n z) zero in ; or
96 let mul = \m n s. m (n s) in
97 let pow = \b exp. exp (mul b) one in ; or
99 ; b (b succ) ; adds b b times, ie adds b^2
100 ; b (b (b succ)) ; adds b^2 b times, ie adds b^3
101 ; exp b succ ; adds b^exp
102 let pow = \b exp s z. exp b s z in
105 ; three strategies for predecessor
106 let pred_zero = zero in
107 let pred = (\shift n. n shift (make_pair zero pred_zero) get_snd)
109 (\p. p (\x y. make_pair (succ x) x)) in ; or
110 ; from Oleg; observe that for any Church numeral n: n I ~~> I
111 let pred = \n. iszero n zero
113 (n (\x. x I ; when x is the base term, this will be K zero
114 ; when x is a Church numeral, it will be I
119 ; from Bunder/Urbanek
120 let pred = \n s z. n (\u v. v (u s)) (K z) I in ; or
123 ; inefficient but simple comparisons
124 let leq = \m n. iszero (n pred m) in
125 let lt = \m n. not (leq n m) in
126 let eq = \m n. and (leq m n) (leq n m) in ; or
129 ; more efficient comparisons, Oleg's gt provided some simplifications
130 let leq = (\base build consume. \m n. n consume (m build base) get_fst)
132 (make_pair true junk)
134 (\p. make_pair false p)
136 (\p. p get_fst p (p get_snd)) in
137 let lt = \m n. not (leq n m) in
138 let eq = (\base build consume. \m n. n consume (m build base) get_fst)
139 ; 2nd element of a pair will now be of the form (K sthg) or I
140 ; we supply the pair being consumed itself as an argument
141 ; getting back either sthg or the pair we just consumed
143 (make_pair true (K (make_pair false I)))
145 (\p. make_pair false (K p))
150 ; -n is a fixedpoint of \x. add (add n x) x
151 ; but unfortunately Y that_function doesn't normalize
153 let sub = \m n. n pred m in ; or
154 ; how many times we can succ n until m <= result
155 let sub = \m n. (\base build. m build base (\cur fin sofar. sofar))
157 (make_triple n false zero)
159 (\t. t (\cur fin sofar. or fin (leq m cur)
160 (make_triple cur true sofar) ; enough
161 (make_triple (succ cur) false (succ sofar)) ; continue
164 let sub = (\base build consume. \m n. n consume (m build base) get_fst)
166 (make_pair zero I) ; see second defn of eq for explanation of 2nd element
168 (\p. p (\x y. make_pair (succ x) (K p)))
173 let min = \m n. sub m (sub m n) in
174 let max = \m n. add n (sub m n) in
177 ; (m/n) is a fixedpoint of \x. add (sub (mul n x) m) x
178 ; but unfortunately Y that_function doesn't normalize
180 ; how many times we can sub n from m while n <= result
181 let div = \m n. (\base build. m build base (\cur go sofar. sofar))
183 (make_triple m true zero)
185 (\t. t (\cur go sofar. and go (leq n cur)
186 (make_triple (sub cur n) true (succ sofar)) ; continue
187 (make_triple cur false sofar) ; enough
189 ; what's left after sub n from m while n <= result
190 let mod = \m n. (\base build. m build base (\cur go. cur))
194 (\p. p (\cur go. and go (leq n cur)
195 (make_pair (sub cur n) true) ; continue
196 (make_pair cur false) ; enough
200 let divmod = (\base build mtail. \m n.
201 (\dhead. m (mtail dhead) (\sel. dhead (sel 0 0)))
202 (n build base (\x y z. z junk))
203 (\t u x y z. make_pair t u) )
205 (make_triple succ (K 0) I) ; see second defn of eq for explanation of 3rd element
207 (\t. make_triple I succ (K t))
209 (\dhead d. d (\dz mz df mf drest sel. drest dhead (sel (df dz) (mf mz))))
211 let div = \n d. divmod n d get_fst in
212 let mod = \n d. divmod n d get_snd in
215 ; sqrt n is a fixedpoint of \x. div (div (add n (mul x x)) 2) x
216 ; but unfortunately Y that_function doesn't normalize
219 ; (log base b of m) is a fixedpoint of \x. add (sub (pow b x) m) x
220 ; but unfortunately Y that_function doesn't normalize
222 ; how many times we can mul b by b while result <= m
223 let log = \m b. (\base build. m build base (\cur go sofar. sofar))
225 (make_triple b true 0)
227 (\t. t (\cur go sofar. and go (leq cur m)
228 (make_triple (mul cur b) true (succ sofar)) ; continue
229 (make_triple cur false sofar) ; enough
233 ; Rosenbloom's fixed point combinator
234 let Y = \f. (\h. f (h h)) (\h. f (h h)) in
235 ; Turing's fixed point combinator
236 let Theta = (\u f. f (u u f)) (\u f. f (u u f)) in
239 ; length for version 1 lists
240 let length = Y (\self lst. isempty lst 0 (succ (self (tail lst)))) in
247 ; numhelper 0 f z ~~> z
248 ; when n > 0: numhelper n f z ~~> f (pred n)
249 ; compare Bunder/Urbanek pred
250 let numhelper = \n. n (\u v. v (u succ)) (K 0) (\p f z. f p) in
252 ; accepts fixed point combinator as a parameter, so you can use different ones
253 let fact = \y. y (\self n. numhelper n (\p. mul n (self p)) 1) in
257 fact Theta 3 ; returns 6
261 ; my original efficient comparisons
262 let leq = (\base build consume. \m n. n consume (m build base) get_fst (\x. false) true)
264 (make_pair zero I) ; supplying this pair as an arg to its 2nd term returns the pair
266 (\p. p (\x y. make_pair (succ x) (K p))) ; supplying the made pair as an arg to its 2nd term returns p (the previous pair)
269 let lt = \m n. not (leq n m) in
270 let eq = (\base build consume. \m n. n consume (m build base) true (\x. false) true)
272 (make_pair zero (K (make_pair one I)))
274 (\p. p (\x y. make_pair (succ x) (K p)))
276 (\p. p get_snd p) in ; or
284 show Oleg's definition of integers:
285 church_to_int = \n sign. n
286 church_to_negint = \n sign s z. sign (n s z)
291 sign_case = \int ifpos ifzero ifneg. int (K ifneg) (K ifpos) ifzero
293 negate_int = \int. sign_case int (church_to_negint (abs int)) zero (church_to_int (abs int))
295 for more, see http://okmij.org/ftp/Computation/lambda-calc.html#neg
305 ; deconstruct our sofar-pair
306 sofar (\might_be_equal right_tail.
309 (and (and might_be_equal (not (isempty right_tail))) (eq? hd (head right_tail)))
313 ; we pass along the fold a pair
314 ; (might_for_all_i_know_still_be_equal?, tail_of_reversed_right)
315 ; when left is empty, the lists are equal if right is empty
317 (not (isempty right))
320 ; when fold is finished, check sofar-pair
321 (\might_be_equal right_tail. and might_be_equal (isempty right_tail))