An Introduction to Functional Programming Through Lambda Calculus

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- 104 -STRING_EQUAL "og" "og" ==STRING_EQUAL (’o’::"g") (’o’::"g") -> ... ->STRING_EQUAL "g" "g" ==STRING_EQUAL (’g’::"") (’g’::"") -> ... ->STRING_EQUAL "" "" -> ... ->TRUEIn our notation:rec STRING_EQUAL S1 S2 =IF (ISNIL S1) AND (ISNIL S2)THEN TRUEELSEIF (ISNIL S1) OR (ISNIL S2)THEN FALSEELSEIF CHAR_EQUAL (HEAD S1) (HEAD S2)THEN STRING_EQUAL (TAIL S1) (TAIL S2)ELSE FALSESimilarly, one string comes before another if its empty and the other is not:STRING_LESS "" (C::S) = TRUESTRING_LESS (C::S) "" = FALSEor if the character in its head comes before the character in the others head:STRING_LESS (C1::S1) (C2::S2) = TRUEif CHAR_LESS C1 C2or the head characters are the same and the first string’s tail comes before the second string’s tail:STRING_LESS (C1::S1) (C2::S2) = (CHAR_EQUAL C1 C2) AND(STRING_LESS S1 S2)if NOT (CHAR_LESS C1 C2)For example:STRING_LESS "porridge" "potato" ==STRING_LESS (’p’::"orridge") (’p’::"otato") -> ... ->(CHAR_EQUAL ’p’ ’p’) AND(STRING_LESS "orridge" "otato") ==(CHAR_EQUAL ’p’ ’p’) AND(STRING_LESS (’o’::"rridge") (’o’::"tato")) -> ... ->(CHAR EQUAL ’p’ ’p’) AND((CHAR_EQUAL ’o’ ’o’) AND(STRING_LESS "rridge" "tato")) ==(CHAR EQUAL ’p’ ’p’) AND((CHAR_EQUAL ’o’ ’o’) AND
- 105 -(STRING_LESS (’r’::"ridge") (’t’::"ato")) -> ... ->(CHAR EQUAL ’p’ ’p’) AND((CHAR_EQUAL ’o’ ’o’) ANDTRUE)) -> ... ->TRUEIn our notation this is:rec STRING_LESS S1 S2 =IF ISNIL S1THEN NOT (ISNIL S2)ELSEIF ISNIL L2THEN FALSEELSEIF CHAR_LESS (HEAD S1) (HEAD S2)THEN TRUEELSE (CHAR_EQUAL (HEAD S1) (HEAD S2)) AND(STRING_LESS1 (TAIL S1) (TAIL S2))6.12. Numeric string to number conversionGiven a string of digits, we might wish to find the equivalent number. Note first of all that the number equivalent to adigit is found by taking ’0’s value away from its value. For example:so:value ’0’ => ... =>forty_eightsub (value ’0’) (value ’0’) -> ... ->sub forty_eight forty_eight => ... =>zeroSimilarly:so:and:value ’1’ => ... =>forty_ninesub (value ’1’) (value ’0’) -> ... ->sub forty_nine forty_eight => ... =>one
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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
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)
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- 41 -(((cond false) true) x) ==(((
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- 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
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: - 103 -In general, the string:" "is
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
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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
Page 167 and 168:
- 167 -- tl;> fn : (’a list) -> (
Page 169 and 170:
- 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
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
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- 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
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- 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 ) =>
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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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