An Introduction to Functional Programming Through Lambda Calculus

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- 102 -LIST_EQUAL (TAIL [2,3]) (TAIL [2,4]) -> ... ->LIST_EQUAL [3] [4] -> ... ->{ EQUAL (HEAD [3]) (HEAD [4])) -> ... ->EQUAL 3 4 -> ... ->FALSE}FALSE6.10. StringsStrings are the basis of text processing in many languages. Some provide strings as a distinct type, for exampleBASIC and ML. Others base strings on arrays of characters, for example Pascal and C. Here we will introduce stringsas linear lists of characters.We can define a test for stringness:ISSTRING [] = TRUEISSTRING (H::T) = (ISCHAR H) AND (ISSTRING T)For example:ISSTRING [’a’,’p’,’e’] -> ... ->(ISCHAR ’a’) AND(ISSTRING [’p’,’e’]) -> ... ->(ISCHAR ’a’) AND((ISCHAR ’p’) AND(ISSTRING [’e’])) -> ... ->(ISCHAR ’a’) AND((ISCHAR ’p’) AND((ISCHAR ’e’) AND(ISSTRING []))) -> ... ->(ISCHAR ’a’) AND((ISCHAR ’p’) AND((ISCHAR ’e’) ANDTRUE) -> ... ->TRUEIn our notation, this function is:rec ISSTRING S =IF ISNIL STHEN TRUEELSE AND (ISCHAR (HEAD S)) (ISSTRING (TAIL S))We will represent a string as a sequence of characters within"Here is a string!"
- 103 -In general, the string:" "is equivalent <strong>to</strong> the list:’’::""For example:"cat" ==’c’::"at" ==’c’::’a’::"t" ==’c’::’a’::’t’::""If we represent the empty string:""as:[]then the string notation mirrors the compact list notation. For example:"cat" ==’c’::’a’::’t’::"" ==’c’::’a’::’t’::[] ==[’c’,’a’,’t’]6.11. String comparisonString comparison is a type specific version of list comparison. Two strings are the same if both are empty:STRING_EQUAL "" "" = TRUESTRING_EQUAL "" (C::S) = FALSESTRING_EQUAL (C::S) "" = FALSEor if the characters in the heads are the same and the strings in the tails are the same:STRING_EQUAL (C1::S1) (C2::S2) = STRING_EQUAL S1 S2if CHAR_EQUAL C1 C2STRING_EQUAL (C1::S1) (C2::S2) = FALSEif NOT (CHAR_EQUAL C1 C2)For example:STRING_EQUAL "dog" "dog" ==STRING_EQUAL (’d’::"og") (’d’::"og") -> ... ->
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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
Page 45 and 46:
- 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
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 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: - 101 -LIST_EQUAL [] [] = TRUELIST_
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
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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
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
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- 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
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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
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 ) =>
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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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