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Functional Programming in Haskell

- Haskell Information Sources
- Fundamental concepts
- Functional Programming
- Sessions, modules, & scripts
- Polymorphic types
- Order of evaluation
- Patterns
- Lazy evaluation
- Side Effects

- Fundamental Data types
- Boolian
- Numbers
- Characters
- Compound data types
- Tuples
- Lists
- User Defined Types
- Enumerations
- Efficiency
- Evaluation order
- Lazy Evaluation
- Space
- Monads
- Examples

Sources of Information

- Book - “Introduction to Functional Programming Using Haskell”, Richard Bird, Pearson Education Limited, England, Prentice Hall Europe 1998
- Tutorial - http://www.cs.utah.edu/~hal/docs/daume02yaht.pdf
- For writing “real production code” see:

http://haskell.org/haskellwiki/How_to_write_a_Haskell_program

Three Haskell Systems

- Hugs – Haskell Users Gofer System
- Interpreter only
- http://www.haskell.org/hugs
- Faster than ghc
- GHC – Glasgow Haskell Compiler
- Both interpreter and compiler
- Slower , more complex, and bigger than hugs and nhc
- NHC - Nearly a Haskell Compiler
- Complier only
- http://www.haskell.org/nhc98/download.html

Functional Programming

- A functional program is a function that solves a problem
- That function may involve several subsidiary functions and is described in a notation that obeys normal mathematical principles
- The result of the function is the solution of the problem and is disjoint from the input to the function
- As in mathematics, once a function is “proven” correct, changes in the environment will not invalidate your “proof”
- Functions can be passed as arguments and are “first class”
- Functions do not change the global “state”
- Single assignment. Once a “variable” gets a value, it never changes.

Functional Programming in “C”

- Prohibit the use of pointers?
- Not likely!
- “Careful” use of pointers
- No modification of input parameters
- All “output” is clearly separated from the input
- Output = function_name(input)
- Subroutine_name (input; output)

Fundamental Concepts of Haskell

- Polymorphic Static types
- length list – The list can have elements of any type. So, length is polymorphic. It can be applied to lists of characters, numbers, tuples, lists, …

length [] = 0

length (x:xs) = 1+ length xs

- Where [] is a pattern that means the empty list
- And x:xs is a pattern that means x is the first element of the input list and xs is the rest of the list (“:” is the cons operator)
- Called “pattern matching”. And pattern matching is an important component of Haskell (more later)

Examples of Polymorphism

- head :: [a] -> a

head (x:xs) = x

- tail :: [a] -> [a]tail (x:xs) = xs
- Both fail if presented with an empty list
- Both work for lists of anything, even lists of empty lists and are Polymorphic
- Examples of the Hindley-Milner type system

Order of Evaluation

- Order of evaluation (simplification, or reduction) is not specified in a functional program
- Define: sq x = x*x
- sq(3+4) could be simplified as
- sq(7) 7*7 49
- (3+4)*(3+4) 7*(3+4) 7*749
- Both orders produce the same result
- The independence of the result from the order is a characteristic feature functional programs
- The OS is free to choose the “best” order

Lazy Evaluation

- let three x = 3
- let infinity = infinity +1
- Now simplify the expression
- three infinity
- Simplification of infinity first gives
- Three(infinity +1 +1 +1 and so on)
- which does not terminate
- Simplification of three first,
- three infinity = 3
- the expression terminates in one step
- Some simplification orders may terminate while others do not
- In GHCi three infinity =3
- In general, some simplification orders will be more efficient than others

Lazy Evaluation

- Guarantees termination whenever termination is possible
- Allows the OS to choose an “efficient” evaluation order

Side Effects

- A side effect is essentially something that happens in the course of executing a function that is not related to the output produced by that function.
- A pure function simply returns a value
- A pure function has no internal state
- A pure function cannot modify the input data
- Given the same arguments a pure function will always produce the same result
- In GHCi values may be displayed by the interactive environment
- Monadic programming allows functional programs to mimic imperative programs
- Monads provide a way to execute “Commands” and display values

Monads

- Haskell uses monads to isolate all impure (not functional) computations from the rest of the program and perform them in the “safe” way
- The execution order of a functional program is entirely determined by the operating system. And this applies to the order of execution of I/O as well
- Thus, the order of I/O can not be preserved by a functional program

Example of Scrambled I/O Order

- “Thus, the order of I/O can not be preserved by a functional program”
- Suppose that your functional program wrote the words in the following order:
- “be preserved a functional program the order of I/O can not by Thus,”

Imperative Constructs are NOT Functional

- x=x+1 – is not allowed!
- All ghci commands are imperative.
- The interactive environment is imperative
- http://www.haskell.org/haskellwiki/Functional_programming
- http://www.haskell.org/all_about_monads/html/class.html

Haskell Fundamental Data Types

- Bool: True or False
- Char: ‘a’ , '\n', '\x05e0‘, ‘\122’

a newline α z

- Number:
- 1
- 2.718

Compound Data typesTuples

- (‘a’, “Daniel”, 3.14159) is valid
- (1, map) is a valid tuple. But you will have to define an I/O Monad to “Show” it.
- The functions for extracting the first and second element of a pair are defined in the standard Haskell environment
- fst(x,y) = x
- snd(x,y) = y
- fst(1,2,3) is not defined in the standard environment

Lists

- Lists – a list is enclosed in square brackets
- The empty list is []
- The cons operator is “:”
- 1:2:3:[] is [1,2,3]
- “Daniel” is ‘D’:’a’:’n’:’i’:’e’:’l’:[] =[‘D’,’a’,’n’,’i’,’e’,’l’]
- ‘D’:”an” = “Dan”
- All elements of a list must be of the same type
- [[1,2],[1]] is a valid list

Comments in Haskell Code

- Single line comments are preceded by ``--'' and continue to the end of the line. For example:

suc n = n + 1 -- this is a successor function

- Multiline and nested comments begin with {- and end with -}. Thus

{- can be used to inactivate a block of code -}

Literate Programming

- A “literate code file” is a file with suffix .lhs instead of .hs (Literate Haskell)
- Two styles for literate code:
- LaTeX Style : \begin{code} … \end{code}
- “Bird” Style: prefix code lines with the “>” character
- Compiler flags allow for reconfiguration of the literate style

Example: LaTeX Literate Style

Here is a simple example of a literate script

for defining the quicksort function:

\begin{code}

tsort [] = []

tsort (x:xs) = tsort [y | y<-xs, y<x] ++ [x] ++ tsort [y | y<-xs, y>=x]

\end{code}

Notice that this definition is very inefficient for a sorted list.

Example: Richard Bird Literate Style

In Bird-style a blank line is required before the code

>fact :: Integer -> Integer

> fact 0 = 1

> fact n = n * fact (n-1)

And a blank line is required after the code as well

Emacs Supports a Multi Mode Display

- One style for LaTeX
- And a second style for Haskell
- http://www.haskell.org/haskellwiki/Literate_programming#Haskell_and_literate_programming

Literate Programming in VIM

- http://www.haskell.org/haskellwiki/Literate_programming/Vim

Quick Sort Algorithm

qsort [] = []

qsort ( x:xs) = qsort (filter (< x) xs) ++

qsort (filter ( >= x) xs)

- Inefficient! Calls filter twice for xs
- Can use (length (x:xs))2memory

More Efficient quicksort

qsort [] = []

qsort x:xs = qsort ys ++ [x] ++ qsort zs

where (ys, zs) = partition (< x) xs

Avoids filtering xs twice

Still can use n2 memory!

Notice that the < is necessary in the comparison to preserve the original order of identical elements

Example of Enumerated Type

module Color

where

data Color

= Red | Orange | Yellow| Green| Blue| Purple | White

| Black

colorToRGB Red = (255,0,0)

colorToRGB Orange = (255,128,0)

colorToRGB Yellow = (255,255,0)

colorToRGB Green = (0,255,0)

colorToRGB Blue = (0,0,255)

colorToRGB Purple = (255,0,255)

colorToRGB White = (255,255,255)

colorToRGB Black = (0,0,0)

Example of Enumerated Types

- colorToRGB Red
- returns the value:

(255,0,0)

- Red == Blue fails because == is not defined for type Color
- colorToRGB Red == colorToRGB Blue
- Returns the value

False

User Defined Types

- User defined data types are done via a ``data'' declaration having the general form:

data T u1 ... un = C1 t11 ... t1k1| ... | Cn tn1 ... Tnkn

- where T is a type constructor; the ui are type variables; the Ci are (data) constructors; and the tij are the constituent types (possibly containing some ui). The presence of the ui implies that the type is polymorphic --- it may be instantiated by substituting specific types for the ui

User Defined Types

- data Bool = True | False
- Bool is the “type” constructor
- True and False are the “data” constructors
- data Color = Red | Green | Blue | Indigo
- data Point a = Pt a a
- “a” on the lhs is a “type” variable
- data Tree a = Branch (Tree a) (Tree a) | Leaf a
- “a” is a “constituent type” on the rhs

Type Synonyms

- General Definition Unknown (examples only)
- type String = [Char]
- type Person = (Name, Address)
- type Name = String
- data Address = None | Addr String

Pythagorian Triads

module PythagorianTriads

where

triples :: Int -> [(Int, Int, Int)]

triples n = [(x, y, z) | x <- [1..n], y <- [1..n], z <- [1..n]]

pyth (x, y, z) = (x*x + y*y == z*z)

ptriads n = filter pyth (triples n)

ptriads 13 returns [[3,4,5], [4,3,5], [5,12,13] ,[6,8,10] ,[12,5,13]]

More Efficient Version

module PythagorianTriads

where

triples :: Int -> [(Int, Int, Int)]

triples n = [(x, y, z) | x <- [1..n], y <- [x..n], z <- [y..n]]

pyth (x, y, z) = (x*x + y*y == z*z)

ptriads n = filter pyth (triples n)

ptriads 13 returns [[3,4,5], [5,12,13], [6,8,10]]

Overloading Operators

class Eq α where

(==) :: α -> α -> Bool

instance Eq Color where

(x == y) = ((colorToRGB x) == (colorToRGB y))

Unfortunately, this does not compile!

Unicode in Haskell

- Haskell 98 specification says that Haskell supports Unicode
- http://blog.kfish.org/2007/10/survey-haskell-unicode-support.html
- http://code.haskell.org/utf8-string/

Unicode Experiment

- Create a list of byte codes for some Hebrew characters:
- hebrew = ['\n', '\x05d0', '\x05d1', '\x05d2', '\x05d3', '\x05d4', '\x05d5', '\x05d6', '\x05d7', '\x05d8', '\x05d9','\x5da','\x5db','\x5dc','\x5de','\x5df', '\x05e0', '\x0e1', '\x05e2', '\x05e3', '\x05e4', '\x05e5', '\x05e6', '\x05e7', '\x05e8', '\x05e9' , '\x05ea', '\x05eb', '\x05ec', '\x05ed', '\x05ee', '\x05ef' , '\n','\n‘]
- putStrhebrew
- Result on next slide

Unicode Greek

The letters printed by my program are in the order

αβΓΠΣσμτΦΘΩδ

And this does not agree with the order in the

above table. Therefore, my environment is not using

this table.

Encoding Problem

- Hexadecimal ‘\x05d0’ = ‘\1488’ decimal
- So, my coding is not the problem

Begin Appendix

- Details of available modules
- Comparison to other languages
- List of some Haskell functions

List of Packages

- http://hackage.haskell.org/packages/archive/pkg-list.html

Example: Algorithm package

- binary-search library: Binary and exponential searches
- Binpack library: Common bin-packing heuristics.
- DecisionTree library: A very simple implementation of decision trees for discrete attributes.
- Diff library: O(ND) diff algorithm in haskell.
- dom-lt library: The Lengauer-Tarjan graph dominators algorithm.
- edit-distance library and programs: Levenshtein and restricted Damerau-Levenshtein edit distances
- funsat library and program: A modern DPLL-style SAT solver
- garsia-wachs library: A Functional Implementation of the Garsia-Wachs Algorithm
- Graphalyze library: Graph-Theoretic Analysis library.
- GraphSCC library: Tarjan's algorithm for computing the strongly connected components of a graph.

Default Packages – provided by the downloaded system (283 functions)

- ghc-prim
- integer - Arbitrary Precision Integer Arithmetic
- base – basic data types and functions
- 31 data types
- rts

More Algorithms

- hgal library: library for computation automorphism group and canonical labelling of a graph
- hmm library: Hidden Markov Model algorithms
- incremental-sat-solver library: Simple, Incremental SAT Solving as a Library
- infinite-search library: Exhaustively searchable infinite sets.
- iproute library: IP Routing Table
- kmeans library: K-means clustering algorithm
- ListTree library: Combinatoric search using ListT
- markov-chain library: Markov Chains for generating random sequences with a user definable behaviour.
- Munkres library: Munkres' assignment algorithm (hungarian method)
- natural-sort library: User-friendly text collation
- Numbers library: An assortment of number theoretic functions
- NumberSieves library: Number Theoretic Sieves: primes, factorization, and Euler's Totient
- palindromes library and program: Finding palindromes in strings
- pqueue-mtl library: Fully encapsulated monad transformers with queuelike functionality.
- presburger library: Cooper's decision procedure for Presburger arithmetic.
- primes library: Efficient, purely functional generation of prime numbers
- queuelike library: A library of queuelike data structures, both functional and stateful.
- rangemin library: Linear range-min algorithms.
- sat programs: CNF SATisfier
- sat-micro-hs program: A minimal SAT solver
- satchmo library: SAT encoding monad
- satchmo-examples programs: examples that show how to use satchmo
- satchmo-funsat library: funsat driver as backend for satchmo
- teams library: Graphical modeling tools for sequential teams
- TrieMap library: Automatic type inference of generalized tries.
- union-find library: Efficient union and equivalence testing of sets.

Modules in the Default Package

array

bytestring

Cabal

containers

directory

editline

filepath

haskell98

hpc

old-locale

old- time

packedstring

pretty

process

random

readline

syb

template-haskell

unix

Win32

Some Haskell Functions (From the appendix of the book)

- (.) – Functional Composition
- (.) :: (βγ) (αβ) (αγ)
- (f.g)x =f(g x)
- (++) Concatenation of two lists
- (++) :: [α] [α] [α]
- [] ++ ys = ys
- (x:xs) ++ ys = x: (xs ++ ys)

More Functions

- (^) Conjunction
- (^) :: Bool Bool Bool
- True ^ x = x
- False ^ x = False
- (v) Disjunction
- (v) :: Bool Bool Bool
- True v x = True
- False v x = x

More Functions

- (!!) List indexing
- (!!) :: [a] Int a
- []!!n = error “(!!): Index too large”
- (x:xs)!!0 = x
- (x:xs)!!(n+1) = xs!!n
- and returns the conjunction of a list of booleans
- and :: [Bool] Bool
- and = foldr (^) True

More Functions

- concat Concatineates a list of lists
- concat :: [[a]] [a]
- concat = foldr (++) []
- const creates a constant valued function
- const :: a b a
- const (x,y) = x

More Functions

- cross Applies a pair of functions to the corresponding elements of a pair
- cross :: (a b, c d) (a,c) (b,d)
- cross (f,g) = pair(f.fst, g.snd)
- curry converts a non-curried function into a curried one
- curry :: ((a,b)c) (abc)
- curry f x y = f(x,y)

List Functions documented at:

- http://www.cs.chalmers.se/Cs/Grundutb/Kurser/d1pt/d1pta/ListDoc/

Comparison to other languages

- Haskell separates the definition of a type from the definition of the methods associated with that type. A class in C++ or Java usually defines both a data structure (the member variables) and the functions associated with the structure (the methods). In Haskell, these definitions are separated.
- The class methods defined by a Haskell class correspond to virtual functions in a C++ class. Each instance of a class provides its own definition for each method; class defaults correspond to default definitions for a virtual function in the base class.
- Haskell classes are roughly similar to a Java interface. Like an interface declaration, a Haskell class declaration defines a protocol for using an object rather than defining an object itself.
- Haskell does not support the C++ overloading style in which functions with different types share a common name.
- The type of a Haskell object cannot be implicitly coerced; there is no universal base class such as Object which values can be projected into or out of.
- C++ and Java attach identifying information (such as a VTable) to the runtime representation of an object. In Haskell, such information is attached logically instead of physically to values, through the type system.
- There is no access control (such as public or private class constituents) built into the Haskell class system. Instead, the module system must be used to hide or reveal components of a class.

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