An Introduction to Functional Programming Through Lambda Calculus

More documents

Recommendations

Info

- 44 -true3.6. Natural numbersWe tend to take numbers for granted in programming but we now have to consider how to represent them explicitly.Our approach will be based on the ability to define natural numbers - non-negative integers - as successors of zero:1 = successor of 02 = successor of 1= successor of successor of 03 = successor of 2= successor of successor of 1= successor of successor of successor of 0etc.Thus, the definition of an arbitrary integer will be that number of successors of zero. We need to find a function zeroto represent zero and a successor function succ so that we can define:and so on.Note that:and so on.def one = (succ zero)def two = (succ one)def three = (succ two)two == (succ (succ zero))three == (succ (succ one)) == (succ (succ (succ zero)))There are a variety of ways of representing zero and succ. We will use:def zero = identitydef succ = λn.λs.((s false) n)so each time succ is applied to a number n it builds a pair function with false first and the original number second.For example:Similarly:one ==(succ zero) ==(λn.λs.((s false) n) zero) =>λs.((s false) zero)
- 45 -two ==(succ one) ==(λn.λs.((s false) n) one) =>λs.((s false) one) ==λs.((s false) λs.((s false) zero))and:three ==(succ two) ==(λn.λs.((s false) n) two) =>λs.((s false) two) ==λs.((s false) λs.((s false) one) ==λs.((s false) λs.((s false) λs.((s false) zero)))This representation enables the definition of a unary function iszero which returns true if its argument is zeroand false otherwise. Remember that a number is a function with an argument which may be used as a selector. Foran arbitrary number:λs.((s false) )if the argument is set to select_first then false will be selected:(λs.((s false) ) select_first) =>((select_first false) ) ==((λfirst.λsecond.first false) ) =>(λsecond.false ) =>falseIf zero, which is the identity function, is applied to select_first then select_first, which is the same astrue by definition, will be returned:(zero select_first) ==(λx.x select_first) =>select_first ==trueThis suggests using:def iszero = λn.(n select_first)
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: - 43 -3.5. OROR is a binary operato
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 and 224:
- 223 -not (or false false) => ...
Page 225 and 226:
- 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 +
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
show all

An Introduction to Functional Programming Through Lambda Calculus

Create successful ePaper yourself

Delete template?

Save as template?