An Introduction to Functional Programming Through Lambda Calculus

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- 138 -[["Mark","Monkey"],[["Graham","Goat"],[["David","Duck"],EMPTY,EMPTY],[["James","Jaguar"],EMPTY,EMPTY]],[["Quentin","Quail"],EMPTY,EMPTY]]The tree traversal function above may be applied to binary trees with arbitrary node values as it only inspects branchesduring traversal. For example:TRAVERSE [["Mark","Monkey"],[["Graham","Goat"],[["David","Duck"],EMPTY,EMPTY],[["James","Jaguar"],EMPTY,EMPTY]],[["Quentin","Quail"],EMPTY,EMPTY]] -> ... ->[["David","Duck"],["Graham","Goat"],["James","Jaguar"],["Mark","Monkey"],["Quentin","Quail"]]Finally, the tree search function above may be modified to return some or all of a required composite value. Forexample, we might find the forename corresponding to a surname, using the surname to identify the required node:rec CTFIND S EMPTY = ""or CTFIND S [[NF,NS],L,R] =IF STRING_EQUAL S NSTHEN NFELSEIF STRING_LESS S NSTHEN CTFIND S LELSE CTFIND S RFor example:CTFIND "Duck" [["Mark","Monkey"],[["Graham","Goat"],[["David","Duck"],EMPTY,EMPTY],[["James","Jaguar"],EMPTY,EMPTY]],[["Quentin","Quail"],EMPTY,EMPTY]] -> ... ->CTFIND "Duck" [["Graham","Goat"],[["David","Duck"],EMPTY,EMPTY],[["James","Jaguar"],EMPTY,EMPTY]] -> ... ->CTFIND "Duck" [["David","Duck"],EMPTY,EMPTY]"David"
- 139 -7.14. Binary tree efficiencyWhen an ordered binary tree is formed from a value sequence, each node holds an ordered sub-sequence. Every subnodeon the left branch of a node contains values which are less than the node’s value and every sub-node on the rightbranch contains values which are greater than the node’s value. Thus, when searching a tree for a node given a value,the selection of one branch discounts all the sub-nodes, and hence all the values, on the other branch. The number ofcomparisons required to find a node depends on how many layers of sub-nodes there are between the root and thatnode.A binary tree is said to be balanced if for any node, the number of values in both branches is the same. For abalanced binary tree, if a node holds N values then there are (N-1)/2 values in its left branch and (N-1)/2 in theits right branch. Thus, the total number of branch layers depends on how often the number of values can be halved.This suggests that in general, if:2 L ... ->[1,EMPTY,[2,EMPTY,[3,EMPTY,[4,EMPTY,EMPTY]]]]We will not consider the construction of balanced trees here.
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
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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
Page 17 and 18:
- 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
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- 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
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
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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
Page 63 and 64:
- 63 -the second minus the first wi
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- 65 -succ (succ (div1 one four)) -
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- 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
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: - 137 -TFIND 5 [4,[3,EMPTY,EMPTY],[
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
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- 189 -10.4. Logictis the primitive
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- 191 -(> 9 8 7 6 5) ->tThe primiti
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- 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
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- 203 -10.17. Symbols, quoting and
Page 205 and 206:
- 205 -Note once again that there m
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- 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 ) =>
Page 221 and 222:
- 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
Page 227 and 228:
- 227 -then fun nelse add (fun n) (
Page 229 and 230:
- 229 -i) ISBOOL 3 => ... =>MAKE_BO
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- 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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