An Introduction to Functional Programming Through Lambda Calculus

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- 118 -def COMP = MAPCARS λX.λY.(STRING_EQUAL X Y)def SUM2 = MAPCARS λX.λY.(X+Y)6.17. SummaryIn this chapter we have:• introduced the list type• developed a representation for the list type and typed list operations• developed elementary functions for manipulating linear lists• introduced simplified list notations• introduced strings as character lists with simplified notation• introduced list case definitions and structure matching• developed functions for constructing ordered linear lists• developed functions for indexed linear list access• developed mapping functions to generalise linear list operationsSome of these topics are summarised below.List notation:: == CONS [,] == ::[][] == ::NIL[] == NIL::(::) ==::::String notation" " == ::"""" = []List case definitionrec [] = or (::) = ==rec =IF ISNIL THEN ELSE
- 119 -‘TAIL ’>6.18. Exercises1) Define a concatenation function for linear lists whose elements are atoms of the same type.2) i) Write a fucntion which indicates whether or not a list starts with a sub-list. For example:STARTS "The" "The cat sat on the mat." => ... =>TRUESTARTS "A" "The cat sat on the mat." => ... =>FALSEii) Write a function which indicates whether or not a list contains a given sub-list. For example:CONTAINS "the" "The cat sat on the mat." => ... =>TRUECONTAINS "the" "All cats sit on all mats." => ... =>FALSEiii) Write a function which counts how often a sub-list appears in another list. For example:COUNT "at" "The cat sat on the mat." => ... =>3iv) Write a function which removes a sub-list from the start of a list, assuming that you know that the sub-liststarts the list. For example:REMOVE "The " "The cat sat on the mat." => ... =>"cat sat on the mat."v) Write a function which deletes the first occurence of a sub-list in another list. For example:DELETE "sat" "The cat sat on the mat." => ... =>"The cat on the mat."DELETE "lay" "The cat sat on the mat." => ... =>"The cat sat on the mat."vi) Write a function which inserts a sub-list after the first occurence of another sub-list in a list. For example:INSERT "sat" "cat " "The cat on the mat." => ... =>"The cat sat on the mat."INSERT "sat" "fish " "The cat on the mat." => ... =>"The cat on the mat."vii) Write a function which replaces a sub-list with another sub-list in a list. For example:REPLACE "sat" "lay" "The cat sat on the mat." => ... =>"The cat lay on the mat."REPLACE "sit" "lay" "The cat sat on the mat." => ... =>"The cat sat on the mat."
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AN INTRODUCTION TO FUNCTIONAL PROGR
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- 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
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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
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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
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- 37 -Bound variablesi) is bound i
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- 39 -i) λx.λy.(λx.y λy.x)ii)
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- 41 -(((cond false) true) x) ==(((
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- 43 -3.5. OROR is a binary operato
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- 45 -two ==(succ one) ==(λn.λs.(
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- 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
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- 53 -gobble a sweet andgobble a sw
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- 55 -then xelse f (succ x) (pred y
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- 57 -and:-> ... ->will still imply
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- 59 -If this function is now appli
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- 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
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: - 117 -"cats"::"dogs"::(λW.(APPEND
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) -> (
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- 169 -Functions have the form:fn
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- 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
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- 177 -- fun length [] = 0 |length
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- 179 -- datatype traffic_light = r
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- 181 -but this defines the new typ
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- 183 -con empty = empty : (’a tr
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- 185 -iv)Join strings s1 and s2 to
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- 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
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- 197 -(car ’(1 2 3)) ->1(cdr ’
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- 199 -(tadd 7 nil) ->(7 nil nil)(t
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- 201 -For example, suppose the ave
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- 203 -10.17. Symbols, quoting and
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- 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 ) =>
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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) (
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- 241 -(if (= m1 m2)(if (< s1 s2)tn
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An Introduction to Functional Programming Through Lambda Calculus

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