An Introduction to Functional Programming Through Lambda Calculus

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- 140 -7.15. Curried and uncurried functionsIn imperative languages like Pascal and C we use procedures and functions declared with several formal parametersand we cannot separate a procedure or function from the name it is declared with. Here, however, all our functions arebuilt from nested λ functions with single bound variables: names and definitions of name/function associations are justa convenient simplification. Our notation for function definitions and applications has led us to treat a name associatedwith a nested function as if it were a function with several bound variables. Now we have introduced another form ofmultiple bound variables through bound variable lists.In fact, nested functions of single bound variables and functions with multiple bound variables are equivalent. Thetechnique of defining multi-parameter functions as nested single parameter functions was popularised by the Americanmathematician Haskell Curry and nested single parameter functions are called curried functions.We can construct functions to transform a curried function into an uncurried function and vice versa. For a function fwith a bound variable list containing two bound variables:def curry f x y = f [x,y]will convert from uncurried form to curried form.For example, consider:Then for:def SUM_SQ1 [X,Y] = (X*X)+(Y*Y)def curry_SUM_SQ = curry SUM_SQ1the right hand side expands as:so:λf.λx.λy.(f [x,y]) SUM_SQ1 =>λx.λy.(SUM_SQ1 [x,y])def curry_SUM_SQ x y = SUM_SQ1 [x,y]Now, the use of curry_SUM_SQ with nested arguments is the same as the use of SUM_SQ1 with an argument list.Similarly, for a function g with a single bound variable a, which returns a function with single bound variable bdef uncurry g [a,b] = g a bwill convert from curried form to uncurried form.For example, withthen for:def SUM_SQ2 X Y = (X*X)+(Y*Y)def uncurry_SUM_SQ = uncurry SUM_SQ2the right hand side expands as:λg.λ[a,b].(g a b) SUM_SQ2 =>
- 141 -λ[a,b].(SUM_SQ2 a b)so:def uncurry_SUM_SQ [a,b] = SUM_SQ2 a bNow, the use of uncurry_SUM_SQ with an argument list is equivalent to the use of SUM_SQ2 with nestedarguments.The functions curry and uncurry are inverses. For an arbitrary function:consider:uncurry (curry ) ==λg.λ[a,b].(g a b) (λf.λx.λy.(f [x,y]) ) ->λg.λ[a,b].(g a b) λx.λy.( [x,y]) =>λ[a,b].(λx.λy.( [x,y]) a b)which simplifies to:λ[a,b].( [a,b]) ==Here we have used a form of η reduction to simplify:to:Similarly:λ[,]).( [,])curry (uncurry ) ==λf.λx.λy.(f [x,y]) (λg.λ[a,b].(g a b) ) ->λf.λx.λy.(f [x,y]) (λ[a,b].(λx.λy.(λ[a,b].( a b) [x,y])which simplifies to:λx.λy.( x y) ==Again we have used a form of η reduction to simplify:λ.λ.( )
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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)) -
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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
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- 73 -λvalue.λs.(s error_type val
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- 75 -def AND X Y =if and (isbool X
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- 77 -def ISBOOL B = MAKE_BOOL (isb
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- 79 -Note that we return NUMB_ERRO
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- 81 -def ’1’ = MAKE_CHAR (succ
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- 83 -will do so. However, for a no
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- 85 -To simplify the presentation
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- 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
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: - 139 -7.14. Binary tree efficiency
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
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
Page 185 and 186: - 185 -iv)Join strings s1 and s2 to
Page 187 and 188: - 187 -( + )( - )( * )( / ) == numb
Page 189 and 190: - 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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