Download Lect_11_cont

Functional
Programming
Universitatea Politehnica Bucuresti
2007-2008
Adina Magda Florea
http://turing.cs.pub.ro/fp_08
Lecture No. 11 - cont
• How the "Countdown problem" functions
work
Evaluating expressions
Define an operator:
data Op = Add | Sub | Mul | Div
Main> :info Op
-- type constructor
data Op
-- constructors:
Add :: Op
Sub :: Op
Mul :: Op
Div :: Op
Evaluating expressions
Apply an operator:
apply
:: Op -> Int -> Int -> Int
apply Add x y = x + y
apply Sub x y = x-y
apply Mul x y = x*y
apply Div x y = x `div` y
Main> apply Div 10 2
5
Decide if the result of applying an operator to
two positive numbers is a natural number:
valid
:: Op -> Int -> Int -> Bool
valid Add _ _
= True
valid Sub x y
=x>y
valid Mul _ _
= True
valid Div x y
= x `mod` y == 0
Main> valid Sub 3 5
False
Main> valid Add 2 4
True
Define an expression:
data Expr = Val Int | App Op Expr Expr
Main> :info Expr
-- type constructor
data Expr
-- constructors:
Val :: Int -> Expr
App :: Op -> Expr -> Expr -> Expr
Evaluate an expression if the result is a natural
number:
eval
:: Expr -> [Int]
eval (Val n)
= [n | n>0]
eval (App o l r) = [apply o x y | x <- eval l
, y <- eval r
, valid o x y]
Main> eval (Val 1)
[1]
Main> eval (App Add (Val 1) (Val 2))
[3]
Return a list of all possible ways of choosing
zero or more elements from a list:
choices
:: [a] -> [[a]]
choices [ ] = [[ ]]
choices (x:xs) = [x:xs | xs <-l] ++ l
where l = choices xs
Main> choices [1,2]
[[1,2],[1],[2],[]]
Main> choices [1,2,3]
[[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]
Homework
Modify choices so that the result is:
Main> choices [1,2]
[[1,2],[2,1],[1],[2],[]]
In what follows this is the behaviour for
choices that we are relying on
Return a list of all values in an expressions:
values
:: Expr -> [Int]
values (Val n)
= [n]
values (App _ l r) = values l ++ values r
Main> values (App Add (Val 1) (Val 2))
[1,2]
Main> values (App Add (Val 1) (App Sub (Val 3) (Val 2)))
[1,3,2]
Main> values (App Add (Val 1) (App Sub (Val 3) (Val 20)))
[1,3,20]
Decide if an expression is a solution for a given list of
source numbers and a target number:
solution
:: Expr -> [Int] -> Int -> Bool
solution e ns n
= elem (values e) (choices ns)
&& eval e == [n]
Main> values (App Add (Val 1) (App Sub (Val 3) (Val 2)))
[1,3,2]
Main> choices [1,2,3]
[[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1],[1,2],[2,1],[1,3],[3,1],
[1],[2,3],[3,2],[2],[3],[]]
Main> elem [1,3,2] (choices [1,2,3])
True
Main> solution (App Add (Val 1) (App Sub (Val 3) (Val 2))) [1,3,2] 2
True
Return a list of all possible ways of
splitting a list into two non-empty lists:
splitaux
splitaux 0 l
splitaux n l
:: Int -> [a] -> [([a],[a])]
=[]
= [(x,y)] ++ splitaux (n-1) l
where x = take n l
y = drop n l
split l = splitaux (length l -1) l
Main> split [1,2,3,4]
[([1,2,3],[4]),([1,2],[3,4]),([1],[2,3,4])]
Combine two expressions using each
operator:
combine
:: Expr -> Expr -> [Expr]
combine l r = [App o l r | o <- [Add, Sub, Mul, Div]]
Main> combine (Val 1) (Val 2)
ERROR - Cannot find "show" function for:
*** Expression : combine (Val 1) (Val 2)
*** Of type : [Expr]
Homework: write a function that will display an expression
of type Expr so that it can be used to display the results
of combine
[(App Add (Val 1) (Val 2)), (App Sub (Val 1) (Val 2)), (App
Mul (Val 1) (Val 2)), (App Div (Val 1) (Val 2))]
Return a list of all possible expressions whose
values are a given list of numbers:
exprs
:: [Int] -> [Expr]
exprs [ ]
=[]
exprs [n]
= [Val n]
exprs ns
= [ e | (ls,rs) <- split ns
,l
<- exprs ls
,r
<- exprs rs
,e
<- combine l r ]
Return a list of all possible expressions that solve the countdown
problem:
solutions
solutions ns n
:: [Int] -> Int -> [Expr]
= [e | ns' <- choices ns
, e <- exprs ns'
, eval e == [n] ]
• choices returns the list ns' of all possible ways of choosing zero or
more elements from the list of numbers ns
• exprs returns the list e of all possible expressions whose values are
the list of numbers ns'
• eval evaluates the expression e (uses apply to apply the operator if
the application is validated by function valid)
• then check if the result e is equal to the natural number n
NB. In order to display this list a show function for the Expr type is
needed (see homework)