An Introduction to Functional Programming Through Lambda Calculus

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- 58 -mult1 ==(λf.λx.λy.if iszero ythen zeroelse add x (f x (pred y))) =>λx.λy.if iszero ythen zeroelse add x ( x (pred y))In the body we have: x (pred y)but we require:so that:mult1 x (pred y)gets passed on again through mult1’s bound variable f to the next level of recursion. Thus, the copy mechanismmust be such that: => ... => mult1 In general, from function f passed to recursive, we need: => ... => f so the copy mechanism must be an application and that application must be self-replicating.We know that the self-application function:λs.(s s)will self-replicate when applied to itself but the replication never ends.Self-application may be delayed through abstraction with the construction of a new function:λf.λs.(f (s s))Here, the self-application:(s s)becomes an argument for f. This might, for example, be a function with a conditional expression in its body whichwill only lead to the evaluation of its argument when some condition is met.When this new function is applied to an arbitrary function, we getλf.λs.(f (s s)) =>λs.( (s s))
- 59 -If this function is now applied to itself:λs.( (s s)) λs.( (s s)) => (λs.( (s s)) λs.( (s s)))then we have a copy mechanism which matches our requirement.Thus, we can define recursive as:def recursive f = λs.(f (s s)) λs.(f (s s))For example, in:def mult = recursive mult1the definition evaluates to:λf.(λs.(f (s s)) λs.(f (s s))) mult1 =>λs.(mult1 (s s)) λs.(mult1 (s s)) =>mult1 (λs.(mult1 (s s)) λs.(mult1 (s s))) ==(λf.λx.λy.if iszero ythen zeroelse add x (f x (pred y))) (λs.(mult1 (s s)) λs.(mult1 (s s))) =>λx.λy.if iszero ythen zeroelse add x ((λs.(mult1 (s s)) λs.(mult1 (s s)))x (pred y))Again, note that we don’t strictly need to replace other occurrences of mult1 as its definition contains no referencesto itself.For example, we will try:mult three two => ... =>(λx.λy.if iszero ythen zeroelse add x ((λs.(mult1 (s s)) λs.(mult1 (s s)))x (pred y))) three two => ... =>if iszero twothen zeroelse add three ((λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred two)) => ... =>add three ((λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred two)) ->add three (mult1 (λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred two)) ==
Page 1 and 2:
AN INTRODUCTION TO FUNCTIONAL PROGR
Page 3 and 4:
- 3 -LISP was one of the earliest l
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- 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: - 57 -and:-> ... ->will still imply
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
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- 119 -‘TAIL ’>6.18. Exercises1
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- 121 -["Betty","Baker"].For exampl
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- 123 -For example:IF STRING_EQUAL
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- 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
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- 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],[
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- 139 -7.14. Binary tree efficiency
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- 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
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- 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
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- 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
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- 173 -9.19. Pattern matchingSML fu
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- 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
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- 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
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- 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
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- 213 -11. R. M. Burstall, J. S. Co
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- 215 -55. P. Wegner, Programming L
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- 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 and 224:
- 223 -not (or false false) => ...
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- 225 -false (implies false true) f
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 +
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- 235 -λy.y5 reductionsλf.λa.(f
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- 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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