Download Defining Functions

DefiningFunctions
YanHuang
[email protected]
TheVeryBasicSyntax
double :: Int ® Int
double x = x + x
ConditionalExpressions
Asinmostprogramminglanguages,functions
canbedefinedusingconditionalexpressions.
takesanintegernandreturnsnif
itisnon-negativeand-notherwise.
abs
ConditionalExpressions
Asinmostprogramminglanguages,functions
canbedefinedusingconditionalexpressions.
abs :: Int ® Int
abs n = if n ≥ 0 then n else -n
takesanintegernandreturnsnif
itisnon-negativeand-notherwise.
abs
Conditionalexpressionscanbenested:
Definethefunctionsignum,whichreturns-1when
givenanegativeinteger;returns1whengivena
positiveinteger;and0ifgiven0.
Conditionalexpressionscanbenested:
signum :: Int ® Int
signum n = if n < 0 then -1 else
if n == 0 then 0 else 1
• InHaskell,conditionalexpressionsmustalways
haveanelsebranch,whichavoidsanypossible
ambiguityproblemswithnestedconditionals.
GuardedEquations
Asanalternativetoconditionals,functionscan
alsobedefinedusingguardedequations.
GuardedEquations
Asanalternativetoconditionals,functionscan
alsobedefinedusingguardedequations.
abs n | n ≥ 0
=n
| otherwise = -n
Aspreviously,butusingguardedequations.
Thecatchallconditionotherwise isdefinedin
the“Prelude”byotherwise=True.
Guardedequationscanbeusedtomakedefinitions
involvingmultipleconditionseasiertoread.E.g.,
Trydefinesignum usingguardedequations.
Guardedequationscanbeusedtomakedefinitions
involvingmultipleconditionseasiertoread:
signum n | n < 0
= -1
| n == 0 = 0
| otherwise = 1
PatternMatching
Manyfunctionshaveaparticularlycleardefinition
usingpatternmatchingontheirarguments.
not
:: Bool ® Bool
not False = True
not True = False
notmapsFalsetoTrue,andTruetoFalse.
Functionscanoftenbedefinedinmanydifferent
waysusingpatternmatching.Forexample
(&&)
True
True
False
False
:: Bool ® Bool ® Bool
&& True = True
&& False = False
&& True = False
&& False = False
canbedefinedmorecompactlyby
True && True = True
_ && _ = False
Theunderscoresymbol_isa
wildcard patternthatmatches
anyargumentvalue.
However,thefollowingdefinitionismoreefficient,
becauseitavoidsevaluatingthesecondargumentif
thefirstargumentisFalse:
True && b = b
False && _ = False
z Patternsarematchedinorder.Forexample,the
followingdefinitionalwaysreturnsFalse:
_ && _ = False
True && True = True
z Youmaynotrepeat variablesinthesamepattern.
Forexample,thefollowingdefinitiongivesanerror:
b && b = b
_ && _ = False
ListPatterns
Internally,everynon-emptylistisconstructed
byrepeateduseofanoperator(:)called“cons”
thataddsanelementtothestartofalist.
[1,2,3,4]
Means1:(2:(3:(4:[]))).
Functionsonlistscanbedefinedusingx:xs pattern.
head
:: [a] ® a
head (x:_) = x
tail
:: [a] ® [a]
tail (_:xs) = xs
headandtailmapanynon-emptylist
toitsfirstandremainingelements.
z x:xs patternsonlymatchnon-empty lists:
> head []
ERROR
z x:xspatternsmustbeparenthesised,because
applicationhaspriorityover(:).Forexample,
thefollowingdefinitiongivesanerror:
head x:_ = x
Scala
• Fullsupportforfunctionalprogrammingwithaverystrongstatictype
system,heavilyinfluencedbyHaskell
- Currying
- Typeinference
- Immutability
- Lazyevaluation
- Patternmatching
- Algebraicdatatypes
• CompilestoJVMbytecode
LambdaExpressions
Functionscanbeconstructedwithoutnaming
thefunctionsbyusinglambdaexpressions.
lx®x+x
Thenamelessfunctionthattakesa
numberxandreturnstheresultx+x.
z Thesymboll istheGreekletterlambda,andistypedatthe
keyboardasabackslash“\”.
z Inmathematics,namelessfunctionsareusuallydenoted
usingthea symbol, asinxa x+x.
z InHaskell,theuseofthel symbolfornamelessfunctions
comesfromthelambdacalculus,thetheoryoffunctionson
whichHaskellisbased.
WhyAreLambda's Useful?
Lambdaexpressionscanbeusedtogiveaformal
meaningtofunctionsdefinedusingcurrying.
Forexample:
add x y = x + y
means
add = l x ® (l y ® x + y)
Lambdaexpressionsarealsousefulwhendefining
functionsthatreturnfunctionsasresults.
Forexample:
const :: a ® b ® a
const x _ = x
ismorenaturallydefinedby
const :: a ® (b ® a)
const x = \_ ® x
Lambdaexpressionscanbeusedtoavoid
namingfunctionsthatareonlyreferencedonce.
Forexample:
odds n = map f [0..n-1]
where
f x = x*2 + 1
canbesimplifiedto
odds n = map (\x ® x*2 + 1) [0..n-1]
Sections
Anoperatorwrittenbetween itstwoargumentscanbe
convertedintoacurriedfunctionwrittenbefore itstwo
argumentsbyusingparentheses.
Forexample:
> 1+2
3
> (+) 1 2
3
Thisconventionalsoallowsoneoftheargumentsof
theoperatortobeincludedintheparentheses.
Forexample:
> (1+) 2
3
> (+2) 1
3
Ingeneral,ifÅ isanoperatorthenfunctionsoftheform
(Å),(xÅ)and(Åy)arecalledsections.
WhyAreSections Useful?
Usefulfunctionscansometimesbeconstructed
inasimplewayusingsections.Forexample:
(1+)
- successorfunction
(1/)
- reciprocationfunction
(*2)
- doublingfunction
(/2)
- halvingfunction
Exercises
(1) Considerafunctionsafetail thatbehavesinthesamewayas
tail,exceptthatsafetail mapstheemptylisttotheemptylist,
whereastailgivesanerrorinthiscase.Definesafetail using:
(a) aconditionalexpression;
(b) guardedequations;
(c) patternmatching.
Hint:thelibraryfunctionnull::[a]® Boolcanbeusedto
testifalistisempty.
(2) Givethreepossibledefinitionsforthelogicalor
operator(||)usingpatternmatching.
(3) Redefinethefollowingversionof(&&)using
conditionalsratherthanpatterns:
True && True = True
_ && _ = False
(4) Dothesameforthefollowingversion:
True && b = b
False && _ = False