SEUJP in Foundations of Computer Science Advanced Functional Programming

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SEUJP in Foundations of Computer
ScienceAdvanced Functional Programming

• 30-31 October 2001
• Tony Daniels Keith Hanna Stefan Kahrs
• Claus Reinke Simon Thompson

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Introduction

• Welcome
• Why SEUJP?
• Introductions
• Arrangements
• SE14 and the multimedia lab
• Coffee machine water
• We'll take you to lunch
• and we'll got somewhere for dinner curry?
• Anything else?

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The course

• Introduction Practical Simon Team
• The lambda calculus Stefan Tony
• Types Keith Stefan
• Domain Specific Embedded Languages Claus
• Verification Simon
• Wrap-up Team
• You have the detailed timetable.

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Functional programming

• Computation evaluation of expressions.
• Programming defining your own functions.
• Function

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Modern FP Haskell, ML, ...

• Functions as data
• higher-order programming.
• Advanced type systems
• polymorphism, overloading, generics, dependent
  types,
• Pure or pureish
• flexibility in implementation
• verification
• Interoperability
• controlled side-effects

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Why FP for SEUJP?

• A perspective on the fundamentals of programming.
• Consequences in related fields
• OO types, process algebra, domains,
• Implementation language of choice for many
  theorists.

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Haskell and Hugs

• Haskell Brooks Curry mathematician and logician
  inventor of the ?-calculus.
• Haskell 98 standard.
• Various implementations Hugs (interpreter for
  Windows, Mac, Unix) GHC, NHC, HBC
  (compilers).
• http//www.haskell.org/

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Examples, examples, examples.

• To familiarise you with Haskell syntax.
• To introduce the case studies.

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Definitions in Haskell

• name Type
• name expression
• size Int
• size 1213
• square Int -gt Int
• square n nn

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Evaluation in Haskell

• square n nn
• double n 2n

6 66
3 23
square (double 3)
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How many pieces with n cuts?
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How many pieces with n cuts?

• No cuts 1 piece.
• With the nth cut, you get n more pieces
• cuts Int -gt Int
• cuts n
• n0 1
• ngt0 cuts (n-1) n
• otherwise 0

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Functions over pictures

• A function to flip a picture in a vertical mirror

flipV
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Functions over pictures

• A function to invert the colours in a picture

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Functions over pictures

• A function to put one picture above another

above
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A naïve implementation

• type Picture String
• type String Char
• A Picture is a list of Strings.
• A String is a list of Char (acters).

.......... ......... ......... .........
. ......... ....... ........ .......
... .......... .......... .......... ......
....
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How are they implemented?

• flipH Reverse the list of strings.
• flipV Reverse each string.
• rotate flipH then flipV (or v.versa).
• above Join the two lists of strings.
• sideBySide Join corresponding lines.
• invertColour Change each Char and each line.
• superimpose Join each Char join each line.

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How are they implemented?

• flipH reverse
• flipV map reverse
• rotate flipV . flipH
• above
• sideBySide zipWith ()
• invertColour map (map invertChar)
• superimpose zipWith (zipWith combine)

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Reversing a list

• A list is either empty, , or is built by adding
  an element to an existing list, (xxs). is
  CONS.
• Convention variables xs, ys, over lists.
• reverse
• reverse (xxs) reverse xs x
• reverse 2,4,1

reverse 4,1 2
joins two lists.
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The type of reverse

• Haskell is strongly typed detect all type errors
  before evaluation.
• reverse
• reverse (xxs) reverse xs x
• reverse a -gt a
• a is a type variable reverse works over any list
  type, returning a list of the same type.

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Implementing the mapping pattern

• map (a -gt b) -gt a -gt b
• map f
• map f (xxs) f x map f xs
• Examples over pictures
• flipV pic map reverse pic
• invertColour pic map invertLine pic

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Partial application functions as results

• map isEven 1,2 False, True
• map isEven ??
• It is a partial application, which gives a
  function
• give it a Int and it will give you back a
  Bool
• map isEven Int -gt Bool

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Another pattern zipping together
zipWith f xs ys zipWith f (xxs) (yys)
f x y zipWith f xs ys
zipWith (a-gtb-gtc) -gt a -gt b -gt c
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Highly generic functions

• Not only apply to a whole family of types, as in
• concat a -gt a
• but also apply to any methods over those types.
• zipWith (a-gtb-gtc) -gt a -gt b -gt c
• Supports reuse in many (unforeseen) contexts.
• When you define a new type, ask what are the HOT
  (higher-order, polymorphic type) functions.

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Semantic tableaux an example
?((A?C)?((A?B)?C))
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Semantic tableaux implementation

• data Formula A B C D E F G H
• Not Formula
• Or Formula Formula
• And Formula Formula
• Implies Formula Formula
• Iff Formula Formula
• deriving (Show, Eq, Ord)
• type Tableau Set (Set Formula)

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Sets

• An abstract data type, using ordered lists
  without repetitions as the representation.
• empty Set a
• sing a -gt Set a
• memSet Ord a gt Set a -gt a -gt Bool
• union,inter, Ord a gt Set a -gt Set a -gt Set a
• eqSet Eq a gt Set a -gt Set a -gt Bool
• makeSet Ord a gt a -gt Set a
• mapSet Ord b gt (a -gt b) -gt Set a -gt
  Set b
• etc.

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Expanding formulas

• expandFormula Formula -gt Set (Set Formula)
• expandFormula (Or a b)
• makeSet sing a, sing b
• expandFormula (And a b)
• sing (makeSet a,b)
• expandFormula (Not (Or a b))
• sing (makeSet Not a, Not b)
• expandFormula (Not (And a b))
• makeSet sing (Not a), sing (Not b)

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Algorithm

• A single step expands one formula in each branch,
  if it's possible
• and removes contradictory branches.
• Iterate until reach a fixed point.

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Practical session

• Getting started with Hugs.
• Picture and tableau exercises of varying
  difficulty
• more than enough for this session.