An Introduction to Functional Programming Through Lambda Calculus

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- 186 -4) Write functions to:i) Extract the leftmost n letters from string s:- left 4 "goodbye";> "good" : stringii) Extract the rightmost n letters from string s:- right 3 "goodbye";> "bye" : stringiii) Extract n letters starting with the l’th letter from string s:- middle 2 5 "goodbye";> "by" : stringiv) Find the position of the first occurence of string s1 in string s2:- find "by" "goodbye";> 5 : int5) The train travelling east from Glasgow Queen’s Street to Edinburgh Waverly passes through Bishopbriggs,Lenzie, Croy, Polmont, Falkirk High, Linlithgow and Edinburgh Haymarket. These stations might berepresented by the data type:datatype station = Queens_Street | Bishopbriggs |Lenzie | Croy |Polmont | Falkirk_High |Linlithgow | Haymarket | Waverly;Write functions which return the station to the east or the west of a given station, for example:6) The data type:- east Croy;> Polmont : station- west Croy;> Lenzie : stationdatatype exp = add of exp * exp |diff of exp * exp |mult of exp * exp |quot of exp * exp |numb of int;might be used to represent strictly bracketed integer arithmetic expressions:so: ::= ( + ) |( - ) |( * ) |( / ) |
- 187 -( + )( - )( * )( / ) == numb()== add(,)== diff(,)== mult(,)== quot(,)For example:1 == numb(1)(1 + 2) == add(numb(1),numb(2))((1 * 2) + 3) == add(mult(numb(1),numb(2)),numb(3))((1 * 2) + (3 - 4)) == add(mult(numb(1),numb(2)),diff(numb(3),numb(4)))Write a function which evaluates an arithemtic expression in this representation, for example:- eval (numb(1));> 1 : exp- eval (add(numb(1),numb(2)));> 3 : exp- eval (add(mult(numb(1),numb(2)),numb(3)));> 5 : exp- eval (add(mult(numb(1),numb(2)),diff(numb(3),numb(4))));> 1 : exp10. FUNCTIONAL PROGRAMMING AND LISP10.1. IntroductionLISP(LISt Processor) is a widely used artificial intelligence language. It is weakly typed with run-time type checking.Functions calls are evaluated in applicative order. LISP lacks structure and pattern matching.Although LISP is not a pure functional language, it has a number of functional features. Like the λ calculus, LISP hasan incredibly simple core syntax. This is based on bracketed symbol sequences which may be interpreted as list datastructures or as function calls. The shared representation of data and program reputedly makes LISP particularlyappropriate for artificial intelligence applications.The original LISP programming system was devised by McCarthy in the late 1950’s as an aid to the Advice Takerexperimental artificial intelligence system. McCarthy’s early description of LISP was based on a functional formalisminfluenced by λ calculus, known as M-expressions(Meta expressions). These were represented in an extremelysimple S-expression’(Symbolic expression) format for practical programming. Contemporary LISP systems arebased solely on the S-expression format although other functional languages are reminiscent of the richer M-expressions. We won’t consider M-expressions here.LISP, like BASIC, is not a unitary language and is available in a number of widely differing dialects. The heart ofthese differences, as we shall see, lies in the way that name/object associations are treated. Here we will considerCOMMON LISP which is a modern standard. We will also look briefly at Scheme, a fully functional LISP. OtherLISPs include FRANZ LISP which is widely available on UNIX systems, MACLISP and INTERLISP whichCOMMON LISP subsumes, and Lispkit Lisp which is another fully functional LISP.It is important to note that LISP has been refined and developed for many years and so is not a very ‘clean’ language.LISP systems differ in how some aspects of LISP are implemented. Some aspects of LISP are extremely arcane andsubject to much disputation amongst the cognoscenti. Furthermore, while LISP has much in common with functional
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
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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
Page 17 and 18:
- 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
Page 25 and 26:
- 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
Page 31 and 32:
- 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
Page 37 and 38:
- 37 -Bound variablesi) is bound i
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- 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
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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
Page 61 and 62:
- 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)) -
Page 67 and 68:
- 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
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
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- 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
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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:[,,]
Page 129 and 130:
- 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
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
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: - 185 -iv)Join strings s1 and s2 to
Page 189 and 190: - 189 -10.4. Logictis the primitive
Page 191 and 192: - 191 -(> 9 8 7 6 5) ->tThe primiti
Page 193 and 194: - 193 -It is important to note that
Page 195 and 196: - 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
Page 203 and 204: - 203 -10.17. Symbols, quoting and
Page 205 and 206: - 205 -Note once again that there m
Page 207 and 208: - 207 -iv) find the sum of applying
Page 209 and 210: - 209 -ii) Write a function which i
Page 211 and 212: - 211 -Chapter 3 - Conditions, bool
Page 213 and 214: - 213 -11. R. M. Burstall, J. S. Co
Page 215 and 216: - 215 -55. P. Wegner, Programming L
Page 217 and 218: - 217 - - (g h) - - g - - h - f
Page 219 and 220: - 219 -λb.b3)i) a) (identity ) =>
Page 221 and 222: - 221 -λb.(λa.{a} b))λa.{a}a fre
Page 223 and 224: - 223 -not (or false false) => ...
Page 225 and 226: - 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
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- 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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