COMPUTER SCIENCE 123. Foundations of Computer Science. 6. Tuples

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1 COMPUTER SCIENCE 123 Foundtions of Computer Science 6. Tuples Summry: This lecture introduces tuples in Hskell. Reference: Thompson Sections R.L. While,

2 Tuples Most dt comes with structure e.g. mounts of money re expressed in dollrs nd cents, i.e. pir of numers In Hskell, this could e expressed s 2-tuple (or pir) (4, 73) A tuple is sequence of two or more vlues of ny type enclosed in rckets nd seprted y comms the type of tuple is determined y the types of its components For exmple: (4, 73) :: (Int, Int) (3, True) :: (Int, Bool) (25, "isn't", 's') :: (Int, String, Chr) ("", 1999) :: (String, Int) ("", "1999") :: (String, String) Note tht ech of these vlues hs different type the word tuples refers to group of types, not single type Also note tht the syntx for the vlues nd for the types is identicl in ech cse, the rckets nd comms mtch cs123 Foundtions of CS 1 of Tuples

3 Tuple opertions The only thing we cn do with tuple is to tke it prt! of course, once we hve tken it prt, we cn process the components using ll of the usul opertions We dissect tuple y plcing it on the left-hnd side of n eqution nd nming the components this is known techniclly s pttern-mtching Consider function ddcsh tht dds together two mounts of money e.g. ddcsh (2, 85) (1, 99) = (4, 84) ddcsh :: (Int, Int) -> (Int, Int) -> (Int, Int) -- ddcsh x y returns the normlised sum -- of csh mounts x nd y ddcsh (x, x') (y, y') = (x + y + z `div` 100, z `mod` 100) where z = x' + y' Consider function mulcsh tht multiplies mounts of money e.g. mulcsh (2, 99) 4 = (11, 96) mulcsh :: (Int, Int) -> Int -> (Int, Int) -- mulcsh x k returns the normlised -- product of csh mount x nd k mulcsh (x, x') k = (x * k + z `div` 100, z `mod` 100) where z = x' * k Note tht the first rgument to ech of these functions hs the type (Int, Int) we cn t cll either of them with (3, True) or (25, "isn't", 's') or ("", 1999) or ("", "1999") or cs123 Foundtions of CS 2 of Tuples

4 Tuples contining tuples A tuple cn contin vlues of ny type including other tuples For exmple: (4, 9, 0) :: (Int, Int, Int) (4, (9, 0)) :: (Int, (Int, Int)) ((4, 9), 0) :: ((Int, Int), Int) (3, True, "/", "[]") :: (Int, Bool, String, String) ((3, True), "/", "[]") :: ((Int, Bool), String, String) (3, (True, "/"), "[]") :: (Int, (Bool, String), String) (3, True, ("/", "[]")) :: (Int, Bool, (String, String)) ((3, True), ("/", "[]")) :: ((Int, Bool), (String, String)) ((3, True, "/"), "[]") :: ((Int, Bool, String), String) (3, (True, "/", "[]")) :: (Int, (Bool, String, String)) Agin, ech of these vlues hs different type Agin note tht the syntx for the vlues nd for the types is identicl in ech cse, the rckets nd comms mtch Exercise: write down the other four tuple-types tht contin the sme vlues s the seven ove cs123 Foundtions of CS 3 of Tuples

5 Exmples Consider function ddmrks tht tkes nme nd set of ssessment mrks nd clcultes finl mrk ddmrks :: (String, (Int, Int, Int)) -> (String, Int) -- ddmrks (n, s) returns -- finl mrk for n ddmrks (n, (1, 2, ex)) = (n, ex) Or suppose we dd the rule tht student fils overll if they fil to sumit ny piece of the ssessment ddmrks :: (String, (Int, Int, Int)) -> (String, Int) -- ddmrks (n, s) returns -- finl mrk for n, if n worked hrd ddmrks (n, (1, 2, ex)) 1==0 2==0 ex==0 = (n, 0) otherwise = (n, 1+2+ex) Tuples cn e tken prt in locl definitions mrk :: (String, (Int, Int, Int)) -> Int -- mrk (n, s) returns n's mrk mrk z = m where (n', m) = ddmrks z cs123 Foundtions of CS 4 of Tuples

6 Type synonyms Redility is the key to good progrmming the esier nd more quickly someone else cn understnd your progrm, the etter One importnt spect is your choice of nmes the nme of function should lwys reflect wht the function does Another key spect is the nmes of types e.g. wht does (Int, Int) men? Hskell llows us to renme type using type synonym declrtion for exmple type Csh = (Int, Int) type Nme type Mrks type StudentEntry type StudentTotl = String = (Int, Int, Int) = (Nme, Mrks) = (Nme, Int) This is lmost lwys good ide The type declrtions of the functions from ove ecome ddcsh :: Csh -> Csh -> Csh mulcsh :: Csh -> Int -> Csh ddmrks :: StudentEntry -> StudentTotl Their specifictions nd equtions re unchnged cs123 Foundtions of CS 5 of Tuples

7 An extended exmple rtionl numers A rtionl numer is frction often vulgr frction, i.e. > e.g Rtionl numers re used in some lnguges for doing exct (or infinite-precision) rithmetic with rtionl representtion, ll rithmetic is performed exctly, nd only output involves ny inccurcy using floting-point representtion for rithmetic cn led to unpredictle inccurcies A rtionl numer cn e represented s pir of integers type Rt = (Int, Int) cs123 Foundtions of CS 6 of Tuples

8 Normlistion A normlised rtionl numer oeys three rules the denomintor is positive negtive rtionl is represented s e.g = is represented in the form 0 1 zerort :: Rt -- zerort returns normlised 0 zerort = (0, 1) it is in its lowest terms, i.e. nd hve no common fctors = z where z = gcd(, ) z e.g In Hskell: = 9 10 normlisert :: Rt -> Rt -- normlisert r returns normlised r normlisert (, ) == 0 = zerort /= 0 = ( * signum `div` z, * signum `div` z) where z = gcd the gurds tke cre of 0 dividing top nd ottom y gcd ensures tht the Rt is in its lowest terms multiplying top nd ottom y signum ensures tht the denomintor is positive cs123 Foundtions of CS 7 of Tuples

9 Adding nd sutrcting rtionl numers Addition is defined y the eqution + c d = d + c d In Hskell: ddrt :: Rt -> Rt -> Rt -- ddrt r r' returns r + r' ddrt (, ) (c, d) = normlisert ( * d + * c, * d) Sutrction is defined y the eqution c d = d c d In Hskell: surt :: Rt -> Rt -> Rt -- surt r r' returns r - r' surt (, ) (c, d) = normlisert ( * d - * c, * d) cs123 Foundtions of CS 8 of Tuples

10 Multiplying nd dividing rtionl numers Multipliction is defined y the eqution c d = c d In Hskell: mulrt :: Rt -> Rt -> Rt -- mulrt r r' returns r * r' mulrt (, ) (c, d) = normlisert ( * c, * d) Division is defined y the eqution c d = d c of course the divisor must not e zero In Hskell: divrt :: Rt -> Rt -> Rt -- pre: r' isn't zero -- divrt r r' returns r / r' divrt (, ) (c, d) = normlisert ( * d, * c) cs123 Foundtions of CS 9 of Tuples

11 Compring rtionl numers Equlity is defined y the eqution == c d iff d == c In Hskell: eqrt :: Rt -> Rt -> Bool -- eqrt r r' returns r == r' eqrt (, ) (c, d) = * d == * c Orderings re defined y the eqution < c d iff d < c In Hskell: ltrt :: Rt -> Rt -> Bool -- ltrt r r' returns r < r' ltrt (, ) (c, d) = * d < * c cs123 Foundtions of CS 10 of Tuples

12 Displying rtionl numers If we just pply show to Rt, it will pper in tuple nottion Prelude> show zerort "(0,1)" (132 reductions, 229 cells) Better is to define function showrt tht returns string using the / formt or just for integer Rts showrt :: Rt -> String -- showrt r returns string contining r showrt (, ) == 1 = show > 1 = show ++ " / " ++ show cs123 Foundtions of CS 11 of Tuples