An Introduction to Functional Programming Through Lambda Calculus

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- 24 -theories for λ calculus without developing them further.Furthermore, we are going to use λ calculus as an time order independent language to investigate time ordering. Thus,our substitution rules should also be time order independent. Otherwise, different substitution orders might lead todifferent substitutions being made and these might result in expressions with different meanings. The simplest way toensure time order independence is to insist that all substitutions be made statically or be capable of being madestatically. We can then apply them to produce pure λ calculus before evaluation starts.We won’t always actually make all substitutions before evaluation as this would lead to pages and pages ofincomprehensible λ expressions for large higher level expressions. We will insist, however, that making allsubstitutions before evaluation always remains a possibility.2.10. Notation for naming functions and application reductionIt is a bit tedious writing functions out over and over again. We will now name functions using:def = to define a name/function association.For example, we could name the functions we looked at in the previous sections:def identity = λx.xdef self_apply = λs.(s s)def apply = λfunc.λarg.(func arg)Now we can just use the in expressions to stand for the .Strictly speaking, all defined names in an expression should be replaced by their definitions before the expression isevaluated. However, for now we will only replace a name by its associated function when the name is the functionexpression of an application. We will use the notation:( ) == ( )to indicate the replacement of a by its associated .Formally, the replacement of a bound variable with an argument in a function body is called β reduction (betareduction). In future, instead of spelling out each β reduction blow by blow we will introduce the notation:( ) => to mean that the results from the application of the to the .When we have seen a sequence of reductions before or we are familiar with the functions involved we will omit thereductions and write:=> ... =>to show where they should be.2.11. Functions from functions
- 25 -We can use the self application function to build versions of other functions. For example, let us define a functionwith the same effect as the identity function:def identity2 = λx.((apply identity) x)Let us apply this to the identity function:(identity2 identity) ==(λx.((apply identity) x) identity) =>((apply identity) identity) ==((λfunc.λarg.(func arg) identity) identity) =>(λarg.(identity arg) identity) =>(identity identity) => ... =>identityLet us show that identity and identity2 are equivalent. Suppose:stands for any expression. Then:(identity2 ) ==(λx.((apply identity) x) ) =>((apply identity) ) => ... =>(identity ) => ... =>so identity and identity2 have the same effect.We can use the function application function to define a function with the same effect as the function applicationfunction itself. Suppose:is any function. Then:(apply ) ==(λf.λa.(f a) ) =>λa.( a)Applying this to any argument:we get:
Page 1 and 2: AN INTRODUCTION TO FUNCTIONAL PROGR
Page 3 and 4: - 3 -LISP was one of the earliest l
Page 5 and 6: - 5 -1.4. Execution order in impera
Page 7 and 8: - 7 -B[1] + B[2] + ... + B[M] + SUM
Page 9 and 10: - 9 -quantifiers to describe proper
Page 11 and 12: - 11 -program specification all use
Page 13 and 14: - 13 -To begin with, we will briefl
Page 15 and 16: - 15 -Now, compare this with the fo
Page 17 and 18: - 17 -wherefor example: ::= λx.x
Page 19 and 20: - 19 -giving:λx.xAn identity opera
Page 21 and 22: - 21 -λx.xThis is now evaluated as
Page 23: - 23 -in the body:λarg.(func arg)g
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 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
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- 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
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- 91 -If the head of a list and the
Page 93 and 94:
- 93 -def MAKE_LIST = make_obj list
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- 95 -LIST_ERRORThe test for an emp
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- 97 -CONS 1 (APPEND (TAIL (CONS 1
Page 99 and 100:
- 99 -For example:1::(2::(3::(4::[]
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- 101 -LIST_EQUAL [] [] = TRUELIST_
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- 103 -In general, the string:" "is
Page 105 and 106:
- 105 -(STRING_LESS (’r’::"ridg
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- 107 -gives value of "" with 10*(1
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- 109 -In general, for:rec =IF IS
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- 111 -Note that this description t
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- 113 -Similarly, to remove a speci
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- 115 -rec DOUBLE [] = []or DOUBLE
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- 117 -"cats"::"dogs"::(λW.(APPEND
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- 119 -‘TAIL ’>6.18. Exercises1
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- 121 -["Betty","Baker"].For exampl
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- 123 -For example:IF STRING_EQUAL
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- 125 -["VDU",25,12]::["modem",20,1
Page 127 and 128:
- 127 -matches the argument:so:[,,]
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- 129 -For example, suppose we have
Page 131 and 132:
- 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,
Page 137 and 138:
- 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
Page 143 and 144:
- 143 -and so on.def istype (t::o)=
Page 145 and 146:
- 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
Page 167 and 168:
- 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
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
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- 189 -10.4. Logictis the primitive
Page 191 and 192:
- 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
Page 219 and 220:
- 219 -λb.b3)i) a) (identity ) =>
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- 221 -λb.(λa.{a} b))λa.{a}a fre
Page 223 and 224:
- 223 -not (or false false) => ...
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- 225 -false (implies false true) f
Page 227 and 228:
- 227 -then fun nelse add (fun n) (
Page 229 and 230:
- 229 -i) ISBOOL 3 => ... =>MAKE_BO
Page 231 and 232:
- 231 -iii)def SIGN_+ X Y =if and (
Page 233 and 234:
- 233 -THEN [H,M,S]ELSElet M = M +
Page 235 and 236:
- 235 -λy.y5 reductionsλf.λa.(f
Page 237 and 238:
- 237 -gconstr (v:int) ((h::t):int
Page 239 and 240:
- 239 -0(+ n (nsum (- n 1)))))ii) (
Page 241:
- 241 -(if (= m1 m2)(if (< s1 s2)tn
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An Introduction to Functional Programming Through Lambda Calculus

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