An Introduction to Functional Programming Through Lambda Calculus

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- 224 -λx.(not (not (not x))) true => ... =>not (not (not true)) => ... =>((not (not true)) false true) => ... =>((not true) false true) false true => ... =>((true false true) false true) false true => ... =>(false false true) false true => ... =>true false true => ... => falseiv)a) - see 1) aboveb) λx.λy.(not (and x (not y))) false false => ... =>not (and false (not false)) => ... =>(and false (not false)) false true => ... =>(false (not false) false) false true) => ... =>false false true => ... => trueλx.λy.(not (and x (not y))) false true => ... =>not (and false (not true)) => ... =>(and false (not true)) false true => ... =>(false (not true) false) false true => ... =>false false true => ... => trueλx.λy.(not (and x (not y))) true false => ... =>not (and true (not false)) => ... =>(and true (not false)) false true => ... =>(true (not false) false) false true => ... =>(not false) false true => ... =>true false true => ... => falseλx.λy.(not (and x (not y))) true true => ... =>not (and true (not true)) => ... =>(and true (not true)) false true => ... =>(true (not true) false) false true => ... =>(not true) false true => ... =>false false true => ... => truev) a) - see 2) aboveb) λx.λy.(and (implies x y) (implies y x)) false false => ... =>and (implies false false) (implies false false) => ... =>(implies false false) (implies false false) false => ... =>(false false true) (implies false false) false => ... =>true (implies false false) false => ... =>implies false false => ... =>false false true => ... => trueλx.λy.(and (implies x y) (implies y x)) false true => ... =>and (implies false true) (implies true false)(implies false true) (implies true false) false => ... =>(false true true) (implies true false) false => ... =>true (implies true false) false => ... =>implies true false => ... =>true false false => ... => falseλx.λy.(and (implies x y) (implies y x)) true false => ... =>and (implies true false) (implies false true)(implies true false) (implies false true) false => ... =>(true false true) (implies false true) false => ... =>
- 225 -false (implies false true) false => ... => falseλx.λy.(and (implies x y) (implies y x)) true true => ... =>and (implies true true) (implies true true) => ... =>(implies true true) (implies true true) false => ... =>(true true true) (implies true true) false => ... =>true (implies true true) false => ... =>implies true true => ... =>true true true => ... => true4)λx.(succ (pred x)) λs.(s false ) =>succ (pred λs.(s false )Simplifying: pred λs.(s false ) => ... =>so: succ => ... =>λs.(s false )λx.(pred (succ x)) λs.(s false ) =>pred (succ λs.(s false number>))Simplifying: succ λs.(s false ) => ... =>λs.(s false λs.(s false ))so: pred λs.(s false λs.(s false )) => ... =>λs.(s false )λx.(succ (pred x)) zero =>(succ (pred zero))Simplifying: pred zero => ... => zeroso: succ zero ==oneλx.(pred (succ x)) zero =>(pred (succ zero))Simplifying: succ zero == oneso: pred one => ... =>zeroChapter 41)2)sum three => ... =>recursive sum1 three => ... =>sum1 (recursive sum1) three => ... =>add three ((recursive sum1) (pred three)) -> ... ->add three (sum1 (recursive sum1) two) -> ... ->add three (add two ((recursive sum1) (pred two))) -> ... ->add three (add two (sum1 (recursive sum1) one)) -> ... ->add three (add two (add one ((recursive sum1) (pred one)))) -> ... ->add three (add two (add one (sum1 (recursive sum1) zero))) -> ... ->add three (add two (add one zero)) -> ... ->six
Page 1 and 2:
AN INTRODUCTION TO FUNCTIONAL PROGR
Page 3 and 4:
- 3 -LISP was one of the earliest l
Page 5 and 6:
- 5 -1.4. Execution order in impera
Page 7 and 8:
- 7 -B[1] + B[2] + ... + B[M] + SUM
Page 9 and 10:
- 9 -quantifiers to describe proper
Page 11 and 12:
- 11 -program specification all use
Page 13 and 14:
- 13 -To begin with, we will briefl
Page 15 and 16:
- 15 -Now, compare this with the fo
Page 17 and 18:
- 17 -wherefor example: ::= λx.x
Page 19 and 20:
- 19 -giving:λx.xAn identity opera
Page 21 and 22:
- 21 -λx.xThis is now evaluated as
Page 23 and 24:
- 23 -in the body:λarg.(func arg)g
Page 25 and 26:
- 25 -We can use the self applicati
Page 27 and 28:
- 27 -and body:λsecond.firstWhen a
Page 29 and 30:
- 29 -2.12.3. Making pairs from two
Page 31 and 32:
- 31 -(λx.x λx.x) =>λx.xIt is po
Page 33 and 34:
- 33 -i) the expression is an appli
Page 35 and 36:
- 35 -((λfunc.λarg.(func arg) arg
Page 37 and 38:
- 37 -Bound variablesi) is bound i
Page 39 and 40:
- 39 -i) λx.λy.(λx.y λy.x)ii)
Page 41 and 42:
- 41 -(((cond false) true) x) ==(((
Page 43 and 44:
- 43 -3.5. OROR is a binary operato
Page 45 and 46:
- 45 -two ==(succ one) ==(λn.λs.(
Page 47 and 48:
- 47 -So:def pred = λn.(((cond zer
Page 49 and 50:
- 49 -For example, we could re-writ
Page 51 and 52:
- 51 -4) Show that:λx.(succ (pred
Page 53 and 54:
- 53 -gobble a sweet andgobble a sw
Page 55 and 56:
- 55 -then xelse f (succ x) (pred y
Page 57 and 58:
- 57 -and:-> ... ->will still imply
Page 59 and 60:
- 59 -If this function is now appli
Page 61 and 62:
- 61 -def add1 f x y =if iszero yth
Page 63 and 64:
- 63 -the second minus the first wi
Page 65 and 66:
- 65 -succ (succ (div1 one four)) -
Page 67 and 68:
- 67 -n * n-1 * n-2 * ... * 1in nor
Page 69 and 70:
- 69 -within our intended interpret
Page 71 and 72:
- 71 -Sometimes it may be necessary
Page 73 and 74:
- 73 -λvalue.λs.(s error_type val
Page 75 and 76:
- 75 -def AND X Y =if and (isbool X
Page 77 and 78:
- 77 -def ISBOOL B = MAKE_BOOL (isb
Page 79 and 80:
- 79 -Note that we return NUMB_ERRO
Page 81 and 82:
- 81 -def ’1’ = MAKE_CHAR (succ
Page 83 and 84:
- 83 -will do so. However, for a no
Page 85 and 86:
- 85 -To simplify the presentation
Page 87 and 88:
- 87 -rec 0 = or (SUCC ) = Thus,
Page 89 and 90:
- 89 -def signed_type = ...def SIGN
Page 91 and 92:
- 91 -If the head of a list and the
Page 93 and 94:
- 93 -def MAKE_LIST = make_obj list
Page 95 and 96:
- 95 -LIST_ERRORThe test for an emp
Page 97 and 98:
- 97 -CONS 1 (APPEND (TAIL (CONS 1
Page 99 and 100:
- 99 -For example:1::(2::(3::(4::[]
Page 101 and 102:
- 101 -LIST_EQUAL [] [] = TRUELIST_
Page 103 and 104:
- 103 -In general, the string:" "is
Page 105 and 106:
- 105 -(STRING_LESS (’r’::"ridg
Page 107 and 108:
- 107 -gives value of "" with 10*(1
Page 109 and 110:
- 109 -In general, for:rec =IF IS
Page 111 and 112:
- 111 -Note that this description t
Page 113 and 114:
- 113 -Similarly, to remove a speci
Page 115 and 116:
- 115 -rec DOUBLE [] = []or DOUBLE
Page 117 and 118:
- 117 -"cats"::"dogs"::(λW.(APPEND
Page 119 and 120:
- 119 -‘TAIL ’>6.18. Exercises1
Page 121 and 122:
- 121 -["Betty","Baker"].For exampl
Page 123 and 124:
- 123 -For example:IF STRING_EQUAL
Page 125 and 126:
- 125 -["VDU",25,12]::["modem",20,1
Page 127 and 128:
- 127 -matches the argument:so:[,,]
Page 129 and 130:
- 129 -For example, suppose we have
Page 131 and 132:
- 131 -to place the Nth item, skip
Page 133 and 134:
- 133 -[7,EMPTY,EMPTY]To add 4 to t
Page 135 and 136:
- 135 -[3,EMPTY,EMPTY],[5,EMPTY,[6,
Page 137 and 138:
- 137 -TFIND 5 [4,[3,EMPTY,EMPTY],[
Page 139 and 140:
- 139 -7.14. Binary tree efficiency
Page 141 and 142:
- 141 -λ[a,b].(SUM_SQ2 a b)so:def
Page 143 and 144:
- 143 -and so on.def istype (t::o)=
Page 145 and 146:
- 145 - ::= ( + ) |( + ) |( + ) |(
Page 147 and 148:
- 147 -IF ISZERO (ADD 1 2)THEN 1ELS
Page 149 and 150:
- 149 -cond one (add (succ one) (pr
Page 151 and 152:
- 151 -λdummy.e2Instead, we have t
Page 153 and 154:
- 153 -The first Church-Rosser theo
Page 155 and 156:
- 155 -(λx.x λy.y) 2=>λy.yand th
Page 157 and 158:
- 157 -For example, consider:SQUARE
Page 159 and 160:
- 159 -SQUARE 1 => ... =>IFIND 1 SQ
Page 161 and 162:
- 161 - : For objects which do not
Page 163 and 164:
- 163 -is a tuple consisting of two
Page 165 and 166:
- 165 -to indicate both integer and
Page 167 and 168:
- 167 -- tl;> fn : (’a list) -> (
Page 169 and 170:
- 169 -Functions have the form:fn
Page 171 and 172:
- 171 -For example, to find the lar
Page 173 and 174: - 173 -9.19. Pattern matchingSML fu
Page 175 and 176: - 175 -end;> val sort = fn : (int l
Page 177 and 178: - 177 -- fun length [] = 0 |length
Page 179 and 180: - 179 -- datatype traffic_light = r
Page 181 and 182: - 181 -but this defines the new typ
Page 183 and 184: - 183 -con empty = empty : (’a tr
Page 185 and 186: - 185 -iv)Join strings s1 and s2 to
Page 187 and 188: - 187 -( + )( - )( * )( / ) == numb
Page 189 and 190: - 189 -10.4. Logictis the primitive
Page 191 and 192: - 191 -(> 9 8 7 6 5) ->tThe primiti
Page 193 and 194: - 193 -It is important to note that
Page 195 and 196: - 195 -nilwhich may be also written
Page 197 and 198: - 197 -(car ’(1 2 3)) ->1(cdr ’
Page 199 and 200: - 199 -(tadd 7 nil) ->(7 nil nil)(t
Page 201 and 202: - 201 -For example, suppose the ave
Page 203 and 204: - 203 -10.17. Symbols, quoting and
Page 205 and 206: - 205 -Note once again that there m
Page 207 and 208: - 207 -iv) find the sum of applying
Page 209 and 210: - 209 -ii) Write a function which i
Page 211 and 212: - 211 -Chapter 3 - Conditions, bool
Page 213 and 214: - 213 -11. R. M. Burstall, J. S. Co
Page 215 and 216: - 215 -55. P. Wegner, Programming L
Page 217 and 218: - 217 - - (g h) - - g - - h - f
Page 219 and 220: - 219 -λb.b3)i) a) (identity ) =>
Page 221 and 222: - 221 -λb.(λa.{a} b))λa.{a}a fre
Page 223: - 223 -not (or false false) => ...
Page 227 and 228: - 227 -then fun nelse add (fun n) (
Page 229 and 230: - 229 -i) ISBOOL 3 => ... =>MAKE_BO
Page 231 and 232: - 231 -iii)def SIGN_+ X Y =if and (
Page 233 and 234: - 233 -THEN [H,M,S]ELSElet M = M +
Page 235 and 236: - 235 -λy.y5 reductionsλf.λa.(f
Page 237 and 238: - 237 -gconstr (v:int) ((h::t):int
Page 239 and 240: - 239 -0(+ n (nsum (- n 1)))))ii) (
Page 241: - 241 -(if (= m1 m2)(if (< s1 s2)tn
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An Introduction to Functional Programming Through Lambda Calculus

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