An Introduction to Functional Programming Through Lambda Calculus

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- 60 -add three ((λx.λy.if iszero ythen zeroelse add x ((λs.(mult1 (s s)) λs.(mult1 (s s)))x (pred y))) three (pred two)) -> ... ->add three if iszero (pred two)then zeroelse add three ((λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred (pred two))) -> ... ->add three (add three ((λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred (pred two)))) ->add three (add three (mult1 (λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred (pred two)))) ==add three (add three ((λx.λy.if iszero ythen zeroelse add x ((λs.(mult1 (s s)) λs.(mult1 (s s)))x (pred y))three (pred (pred two)))) -> ... ->add three (add three if iszero (pred (pred two))then zeroelse add three ((λs.(mult1 (s s)) λs.(mult1 (s s)))three (pred (pred (pred two))))) -> ... ->add three (add three zero) -> ... ->add three three => ... =>six4.7. Recursion notationThe function recursive is known as a paradoxical combinator or a fixed point finder, and is called Y in the λcalculus literature.Rather than always defining an auxiliary function with an abstraction and then using recursive to construct arecursive version, we will allow the defined name to appear in the defining expression but use a new definition form:rec = This is to indicate that the occurrence of the name in the definition should be replaced using abstraction and theparadoxical combinator should then be applied to the whole of the defining expression. For example, for addition, wewill write:instead of:rec add x y =if iszero ythen xelse add (succ x) (pred y)
- 61 -def add1 f x y =if iszero ythen xelse f (succ x) (pred y)def add = recursive add1and for multiplication we will write:rec mult x y =if iszero ythen zeroelse add x (mult x (pred y))When we expand or evaluate a recursive definition we will just leave the recursive reference in place.4.8. Arithemtic operationsWe will now use recursion to define arithmetic operations for raising to a power, subtraction, equality and inequalities,and division.4.8.1. PowerTo raise one number to the power of another number, multiply the first by the first to the power of the decrementedsecond. If the second is zero then the power is one:rec power x y =if iszero ythen oneelse mult x (power x (pred y))For example:power two three => ... =>mult two(power two (pred three)) -> ... ->mult two(mult two(power two (pred (pred three)))) -> ... ->mult two(mult two(mult two(power two (pred (pred (pred three)))))) -> ... ->mult two(mult two(mult two one)) -> ... ->mult two(mult two two) -> ... ->mult two four => ... =>
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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
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 and 24: - 23 -in the body:λarg.(func arg)g
Page 25 and 26: - 25 -We can use the self applicati
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: - 59 -If this function is now appli
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 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
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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
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- 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,
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- 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
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- 143 -and so on.def istype (t::o)=
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- 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
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- 175 -end;> val sort = fn : (int l
Page 177 and 178:
- 177 -- fun length [] = 0 |length
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- 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
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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