An Introduction to Functional Programming Through Lambda Calculus

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- 74 -5.6. BooleansWe will represent the boolean type as one:def bool_type = oneConstructing a boolean type involves preceding a boolean value with bool_type:def MAKE_BOOL = make_obj bool_typewhich expands as:λvalue.λs.(s bool_type value)We can now construct the typed booleans TRUE and FALSE from the untyped versions by:def TRUE = MAKE_BOOL truewhich expands as:and:λs.(s bool_type true)def FALSE = MAKE_BOOL falsewhich expands as:λs.(s bool_type false)As with the error type, the test for a boolean type involves checking for bool_type:def isbool = istype bool_typeThis definition expands as:λobj.(equal (type obj) bool_type)A boolean error object will be an error object with type bool_type:def BOOL_ERROR = MAKE_ERROR bool_typewhich expands as:λs.(s error_type bool_type)The typed function NOT should either return an error if the argument is not a boolean or extract the value from theargument, use untyped not <strong>to</strong> complement it and make a new boolean from the result:def NOT X =if isbool Xthen MAKE_BOOL (not (value X))else BOOL_ERRORSimilarly, the typed function AND should either return an error if either argument is non boolean or make a newboolean from ‘and’ing the values of the arguments:
- 75 -def AND X Y =if and (isbool X) (isbool Y)then MAKE_BOOL (and (value X) (value Y))else BOOL_ERRORWe will now consider how these definitions bolt <strong>to</strong>gether by looking at:AND TRUE FALSEAfter definition replacement and initial bound variable substitution we have:First of all:if and (isbool TRUE) (isbool FALSE)then MAKE_BOOL (and (value TRUE) (value FALSE))else BOOL_ERRORisbool TRUE ==λobj.(equal (type obj) bool_type) TRUE =>equal (type TRUE) bool_type ==equal (λobj.(obj select_first) TRUE) bool_type ->equal (TRUE select_first) bool_type ==equal (λs.(s bool_type true) select_first) bool_type -> ... ->equal bool_type bool_type => ... =>trueSimilarly:Thus:isbool FALSE => ... =>trueand (isbool TRUE) (isbool FALSE) -> ... ->and true (isbool FALSE) -> ... ->and true true => ... =>trueWe now evaluate:MAKE_BOOL (and (value TRUE) (value FALSE))For the and:value TRUE ==λobj.(obj select_second) TRUE =>
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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
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 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: - 73 -λvalue.λs.(s error_type val
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
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- 125 -["VDU",25,12]::["modem",20,1
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- 127 -matches the argument:so:[,,]
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- 129 -For example, suppose we have
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- 131 -to place the Nth item, skip
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- 133 -[7,EMPTY,EMPTY]To add 4 to t
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- 135 -[3,EMPTY,EMPTY],[5,EMPTY,[6,
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- 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
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- 143 -and so on.def istype (t::o)=
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- 145 - ::= ( + ) |( + ) |( + ) |(
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- 147 -IF ISZERO (ADD 1 2)THEN 1ELS
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- 149 -cond one (add (succ one) (pr
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- 151 -λdummy.e2Instead, we have t
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- 153 -The first Church-Rosser theo
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- 155 -(λx.x λy.y) 2=>λy.yand th
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- 157 -For example, consider:SQUARE
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- 159 -SQUARE 1 => ... =>IFIND 1 SQ
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- 161 - : For objects which do not
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- 163 -is a tuple consisting of two
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- 165 -to indicate both integer and
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- 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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