An Introduction to Functional Programming Through Lambda Calculus

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- 158 -(SQLIST (SUCC (SUCC 0) 1) 3) 4=> ... =>(SQ (SUCC (SUCC 0) 1) 3) 6::(SQLIST (SUCC (SUCC (SUCC 0) 1) 3) 7) 8Occurrences of bound pair 4 are replaced by the new bound pair 8 so SQUARES is now:(SQ 0)::(SQ (SUCC 0) 1)::(SQ (SUCC (SUCC 0) 1) 3) 6::(SQLIST (SUCC (SUCC (SUCC 0) 1) 3) 7) 8Thus, evaluation of:HEAD (TAIL (TAIL SQUARES))requires the selection of:(SQ (SUCC (SUCC 0) 1) 3) 6This requires the evaluation of:(SUCC (SUCC 0) 1) 3which in turn requires the evaluation of:(SUCC 0) 1=> ... => 1which replaces all occurrences of bound pair 1. Now, SQUARES is:(SQ 0)::(SQ 1)::(SQ (SUCC 1) 3) 6::(SQLIST (SUCC (SUCC 1) 3) 7) 8Thus evaluation of:(SUCC (SUCC 0) 1) 3=> ... =>(SUCC 1) 3gives:2which replaces all occurrences of bound pair 3 so SQUARES is now:(SQ 0)::(SQ 1)::(SQ 2) 6::(SQLIST (SUCC 2) 7) 8Finally, evaluation of:(SQ 2) 6gives:4which replaces all occurrences of bound pair 6 so SQUARES is now:(SQ 0)::(SQ 1)::4::(SQLIST (SUCC 2))If we now try:
- 159 -SQUARE 1 => ... =>IFIND 1 SQUARES => ... =>IFIND 0 (TAIL SQUARES) => ... =>HEAD (TAIL SQUARES) ==HEAD (TAIL ((SQ 0)::(SQ 1) 9::4::(SQLIST (SUCC 2)))) -> ... ->HEAD ((SQ 1) 9::4::(SQLIST (SUCC 2))) => ... =>(SQ 1) 9=> ... => 1which replaces all occurrences of bound pair 9 so SQUARES is now:(SQ 0)::1::4::(SQLIST (SUCC 2))Thus, repeated list access evaluates more and more of the infinite list but avoids repetitive evaluation.Lazy evaluation is used for Miranda lists.8.10. SummaryIn this chapter we have:• compared normal and applicative order β reduction and seen that normal order reduction may be less efficient• seen that consitent applicative order β reduction with our conditional expression representation leads to nontermination• considered ways of delaying applicative order evaluation• seen that the halting problem is unsolvable• met the Church-Rosser theorems which suggest that normal order β reduction is most likely to terminate• seen that normal order β reduction enables the construction of infinite objects• met lazy evaluation as a way of combining the best aspects of normal and applicative order β reductionLazy evaluation is summarised below.Lazy evaluationi) number every bound pairii)To lazy evaluate ( ) ia) lazy evaluate to b) if is λ.then replace all free occurences of in with and renumber consistently all surrounding bound pairsand replace all occurences of ( ) iwith the new and lazy evaluate the new
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
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- 7 -B[1] + B[2] + ... + B[M] + SUM
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- 9 -quantifiers to describe proper
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- 11 -program specification all use
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- 13 -To begin with, we will briefl
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- 15 -Now, compare this with the fo
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- 17 -wherefor example: ::= λx.x
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- 19 -giving:λx.xAn identity opera
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- 21 -λx.xThis is now evaluated as
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- 23 -in the body:λarg.(func arg)g
Page 25 and 26:
- 25 -We can use the self applicati
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- 27 -and body:λsecond.firstWhen a
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- 29 -2.12.3. Making pairs from two
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- 31 -(λx.x λx.x) =>λx.xIt is po
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- 33 -i) the expression is an appli
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- 35 -((λfunc.λarg.(func arg) arg
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- 37 -Bound variablesi) is bound i
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- 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
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- 49 -For example, we could re-writ
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- 51 -4) Show that:λx.(succ (pred
Page 53 and 54:
- 53 -gobble a sweet andgobble a sw
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- 55 -then xelse f (succ x) (pred y
Page 57 and 58:
- 57 -and:-> ... ->will still imply
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- 59 -If this function is now appli
Page 61 and 62:
- 61 -def add1 f x y =if iszero yth
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- 63 -the second minus the first wi
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- 65 -succ (succ (div1 one four)) -
Page 67 and 68:
- 67 -n * n-1 * n-2 * ... * 1in nor
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- 69 -within our intended interpret
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- 71 -Sometimes it may be necessary
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- 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
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- 91 -If the head of a list and the
Page 93 and 94:
- 93 -def MAKE_LIST = make_obj list
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- 95 -LIST_ERRORThe test for an emp
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- 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: - 157 -For example, consider:SQUARE
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
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- 209 -ii) Write a function which i
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- 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
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- 219 -λb.b3)i) a) (identity ) =>
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- 221 -λb.(λa.{a} b))λa.{a}a fre
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- 223 -not (or false false) => ...
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- 225 -false (implies false true) f
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- 227 -then fun nelse add (fun n) (
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- 229 -i) ISBOOL 3 => ... =>MAKE_BO
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- 231 -iii)def SIGN_+ X Y =if and (
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- 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
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- 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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