An Introduction to Functional Programming Through Lambda Calculus

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- 124 -def ESURNAME E = SURNAME (NAME E)def ADDRESS E = HEAD (TAIL E)def PHONE E = HEAD (TAIL (TAIL E))These selector functions disguises the underlying representation and makes it easier to understand the functions thatuse them.For example, given a circulation list we might want to delete a name, knowing the surname. If the list is empty thenreturn the empty list. If the list is not empty then if the surname is that of the first name then return the rest of the list.Otherwise, add the first name to the result of deleting the required name from the rest of the list:rec NDELETE S [] = []or NDELETE S (H::T) =IF STRING_EQUAL S (SURNAME H)THEN TELSE H::(NDELETE S T)For example:NDELETE "Charlie" [["Anna","Able"],["Betty","Baker"],["Clarice","Charlie"]] -> ... ->["Anna","Able"]::(NDELETE "Charlie" [["Betty","Baker"],["Clarice","Charlie"]]) -> ... ->["Anna","Able"]::["Betty","Baker"]::(NDELETE "Charlie" [["Clarice","Charlie"]]) -> ... ->["Anna","Able"]::["Betty","Baker"]::[] ==[["Anna","Able"],["Betty","Baker"]]For example, given a stock control list, we might want to increment the stock level, knowing the item name. If the listis empty then return the empty list. If the list is not empty then if the first item is the required one then increment itsstock level and add the changed item to the rest of the list. Otherwise, add the first item to the result of searching therest of the list:rec SINCREMENT I V [] = []or SINCREMENT I V (H::T) =IF STRING_EQUAL I (ITEM H)THEN [(ITEM H),(STOCK H)+V,(REORDER H)]::TELSE H::(SINCREMENT I V T)For example:SINCREMENT "modem" 10 [["VDU",25,12],["modem",10,12],["printer",125,10]] -> ... ->["VDU",25,12]::(SINCREMENT "modem" 10 [["modem",10,12],
- 125 -["VDU",25,12]::["modem",20,12]::[["printer",125,10]] ==[["VDU",25,12],["modem",20,12]["printer",125,10]]["printer",125,10]]) -> ... ->For example, given a telephone directory, we might want to add a new entry in alphabetic surname order. If thedirectory is empty then make a new directory from the new entry. If the directory is not empty then if the new entrycomes before the first entry then add it to the front of the directory. Otherwise, add the first entry to the result ofadding the new entry to the rest of the directory:rec DINSERT E [] = [E]or DINSERT E (H::T) =IF STRING_LESS (ESURNAME E) (ESURNAME H)THEN E::H::TELSE H::(DINSERT E T)For example:DINSERT[["Chris","Catnip"],"Credit",3333][[["Anna","Able"],"Accounts",1212],[["Betty","Baker"],"Boiler room",4242],[["Clarice","Charlie"],"Customer orders",2424]] -> ... ->[["Anna","Able"],"Accounts",1212]::(DINSERT[["Chris","Catnip"],"Credit",3333][[["Betty","Baker"],"Boiler room",4242],[["Clarice","Charlie"],"Customer orders",2424]]) -> ... ->[["Anna","Able"],"Accounts",1212]::[["Betty","Baker"],"Boiler room",4242]::(DINSERT[["Chris","Catnip"],"Credit",3333][[["Clarice","Charlie"],"Customer orders",2424]]) -> ... ->[["Anna","Able"],"Accounts",1212]::[["Betty","Baker"],"Boiler room",4242]::[["Chris","Catnip"],"Credit",3333]::[[["Clarice","Charlie"],"Customer orders",2424]] ==[[["Anna","Able"],"Accounts",1212],[["Betty","Baker"],"Boiler room",4242],[["Chris","Catnip"],"Credit",3333],[["Clarice","Charlie"],"Customer orders",2424]]7.5. Generalised structure matchingIn chapter 6 we introduced structure matching into function definitions. Objects are defined in terms of constant basecases and structured recursion cases. Thus, function definitions have base cases with constants instead of boundvariables for matching against constant arguments, and recursion cases with structured bound variable for matchingagainst structured arguments. In particular, for list processing we have used bound variable lists of the form:
Page 1 and 2:
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.(
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
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- 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
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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)) -
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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
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 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: - 123 -For example:IF STRING_EQUAL
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) -> (
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
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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
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
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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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