[SOLVED] G6021 Comparative Programming

G6021 Comparative Programming

G6021 Comparative Programming

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Part 4 Types

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Typed -calculus

There are many variants of the -calculus (applied -calculi).
Here we briefly outline the simply typed -calculus;
and a minimal functional programming language.

Definition:
Types: type variables: , , and function types:
Typed terms: terms will now be annotated with types, which we
write as: t : (term t has type )

If M : and x : , then x .M :
If M : and N : then MN :

Does this look familiar?
Exercise: What are the differences between pure and typed
-calculi?

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Programming Concepts: week 10

Using Implications

P implies Q

We can conclude

Should be called Implication Elimination ImpElim
but Greeks got there first.

1 P implies (Q implies R) (P and Q) implies R
2 (P and Q) implies R P implies (Q implies R)

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Properties

Can every -term be given a type?
Strong Normalisation?
Confluence?

Exercise (for fun)
Consider the linear -calculus: each variable can occur exactly once:
i.e. x .x is linear, but x .xx is not. Now answer the above questions

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Extending the -calculus (PCF)

PCF: Programming language for Computable Functions.
A very simple functional programming language derived from the
-calculus:

Types: , ::= int | bool |
Typed terms: Same as the typed -calculus, with the addition of
constants:

n : int for n = 0,1,2, . . .
true, false : bool
succ,pred : int int
iszero : int bool
for each type , cond : bool ,
for each type , fix : ( )

Exercise: Write these in Haskell.

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succ 3 : int
condint(iszero(pred(pred 2)))1 : int int

and factorial

f x = if x==0 then 1 else x * f(x-1)

which we can code as

fixintint
f intint.xint.condint(iszero x) 1

mult x(f (pred x))

Exercise: Define mult.

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Where did that come from?

Here are several snap-shots of the transformation from Haskell to PCF:

f x = if x==0 then 1 else x*f(x-1)
f = x -> if x==0 then 1 else x*f(x-1)
f = x -> cond (x==0) 1 (x*f(x-1))
f = x -> cond (iszero x) 1 (x*f(pred x))

What next? PCF does not have recursion. Abstract that out:

F = f -> x -> cond (iszero x) 1 (x * f(pred x))

F is not recursive! But it does not compute factorial

Exercise: What are these:

F (x -> x)

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F = f -> x -> cond (iszero x) 1 (x*f(pred x))

None of the above give the factorial function. What we need is a
function fact, because:

F fact = fact

We dont have such a function But we do have a way of finding it!

What we need is the fixpoint of F, which we write as fix F.

fact = fix F

So to compute the factorial of a number, say 3, we write:

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Here are some snap-shots of the reduction of fix F 3 to
demonstrate how the computation works:

fix F 3 -> F (fix F) 3
-> (x -> cond (iszero x) 1 (x*(fix F (pred x)))) 3
-> cond (iszero 3) 1 (3*(fix F (pred 3)))
-> cond False 1 (3*(fix F 2))
-> 3*(fix F 2)

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Whats New?

true = xy .x
false = xy .y
n = fx .f nx
succ = abc.b(abc)
pred = z.fix H z S I false
cond = xyz.xyz
iszero = n.n S I true
fix = (xy .y(xxy))(xy .y(xxy))

where I = x .x , H and S are defined as:

H = hx .iszero x 0 (succ(h(x false)))
S = xy .y false x

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Operational Semantics of PCF

(x.M)N M{x 7 N}
fixM M(fixM)
cond true M N M
cond false M N N
succ n n + 1
pred (n + 1) n pred 0 0
iszero 0 true iszero (n + 1) false

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condMN1N2 condM N1N2

succ M succ M

pred M pred M

iszero M iszero M

Remark that:

The configurations are just terms.
Computation = evaluation = reduction:

M N means M reduces (evaluates) to N.
A final value is an irreducible (fully evaluated) term.

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Observations

Note that we do not have reductions in every context. Specifically, we
do not have:

x .N x .N

Strategies
Which strategy is being used here?
How can we change it to another strategy?

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Properties

Termination?
Subject Reduction: If M : and M M then M :
If M terminates, then:

if M : int then M n
if M : bool then either M true or M false

Otherwise: non-terminating (but still preserves the type)

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1 -calculus (pure, typed)
2 PCF: simple functional language

These languages are very primitive (as far as the programmer is
concerned)
However, they provide the basis of the functional paradigm
Many languages based on this:

Standard ML, CAML
Lisp, Scheme, . . .

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Type systems and Type Reconstruction

Type systems have become one of the most important theoretical
developments in programming languages
Here we will examine several key issues:

Type reconstruction (and unification)
Polymorphic types
Overloading
Intersection types
(System F)

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Proof Systems

We write M : A to mean that term M has type A using the context

, x : A x : A
, x : A M : B

x .M : A B

M : A B N : A

MN : B

Using these rules we can build derivations of typed terms

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x : A x : A

x .x : A A

x : A, y : B x : A

x : A y .x : B A

x .y .x : A B A

f : A B, x : A f : A B f : A B, x : A x : A

f : A B, x : A fx : B

f : A B x .fx : A B

f .x .fx : (A B) A B

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Type Reconstruction

Given a term M, can we find its type?
The proof system suggests an algorithm:

If M is a variable, then look up the type in the context
If M = x .M is an abstraction, find the type of M in the context
extended with x : A, then calculate the type of the result
If M is an application, find the type of the function, then the
argument, then calculate the type of the result

But how do we make the types fit?
E.g. M : A B and N : C. Can we give a type for MN?
(Can we make A and C the same type?)

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Polymorphism

A second question: can a term (program) have several types?

Example: P = x .1 : A int
Are both P true and P 3 possible?

It seems reasonable, but at what moment does type A become either
bool or int?

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is a mechanism which allows us to write functions
which can process objects of different types. It is a very powerful
programing technique.

len [] = 0
len (_:t) = 1+len t
len :: (Num t1) => [t] -> t1

len [1,2,3]

len G6021

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Another Example

map f [] = []
map f (h:t) = (f h): map f t
map :: (a -> b) -> [a] -> [b]

map (x -> x+1) [2,3,4]

map (x -> X) [x -> x, x -> x+1]

t, t1, a, b are type variables
len :: [t] -> Int means that len has type t .[t ] Int.
I.e. forall types t .
t is called a generic type

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Generalisation and Specialisation

The final machinery that we need are ways of introducing (and
eliminating) the .

M : .A

M : .A

M : [B/]A
Note: FV () for the GEN rule

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Reconstructing Polymorphic types

We give an algorithm which will find the most general type of a term
If M has a type, then we will find it, otherwise fail

Exercise: what could most general type mean?

Substitution (of types)
Unification
Type reconstruction

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Unification

There is an algorithm U , which given a pair of types either returns a
substitution V or fails; further:

If U(, ) = V then V = V (we say V unifies and ).
If S unifies and then U(, ) returns some V and there is
another substitution R such that S = RV (most general unifier).

Moreover, V only involves variables in and . Example:

U( , ( ) ) = [ /]

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Disagreement sets

The algorithm for unification is specified in terms of the notion of a
disagreement set. When unifying pairs of types we will have a
disagreement set that is either empty or cardinality 1.

D(, ) = (if = )
= {(, )}

where , are the first two subterms at which , disagree, using
depth first comparison. Some examples are in order:

D( , ) =
D( , ) = {(, )}
D(int, ) = {(int, )}
D((int int) int, ) = {(int int, )}

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Unification

U(, ) = iter(id, , )

iter(V , , ) = V , if V = V

= iter([b/a]V , , ), if a does not occur in b
= iter([a/b]V , , ), if b does not occur in a
= FAIL, otherwise.

where {(a,b)} = D(V ,V ).

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Reconstruction of Types

Using unification, and the proof system as a guide, the algorithm is a
function which takes a set of assumptions () and a term to be typed
(e), and returns two things: a substitution (T ) and the type of the
whole term (): T (,e) = (T , )

1 T (, x) = (id, ) where = [1/1, . . . n/n] if
x : 1 . . . n. . Otherwise, = x

2 T (,MN) = (USR,U) where (R, ) = T (,M),
(S, ) = T (R,N) and U = U(S, ) ( new)

3 T (, x .M) = (R,R ) where (R, ) = T ( x : ,M) ( new)
4 T (, let x = M in N) = (SR, ) where (R, ) = T (,M),

(S, ) = T (R x : ,N), = 1 . . . n. and
{1, . . . , n} = FV () FV ()

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Reconstruction of types in functional languages

We can add a number of extra rules for the built-in types. For example,
something like this:

n :: Integer True :: Bool False :: Bool

P :: Bool Q :: Bool

P&&Q :: Bool

P :: Int Q :: Int

P + Q :: Int

[] :: [a]

h :: a t :: [a]

h : t :: [a]
Type reconstruction can be extended in a straightforward way.

Question: What about user defined types?

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Type checking versus type inference

Type checking refers to the process of checking that the types
declared in a program are compatible with the use of the functions
and variables.
Type inference (or type reconstruction) is the process of inferring
types for the elements of the program (where type declarations
might be present, optionally).

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Other notions of type

Overloading: 1 + 4, 1.2 + 3.7,
this + is a + string
Also known as ad hoc polymorphism.
Intersection types

M : (1 2)

M : i

M : M :

M :
System F: types as terms, dependent types, etc.

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Type classes in: same code executed
Overloaded: different code executed
Haskell has an intermediate notion: type classes:
len :: (Num t1) => [t] -> t1

Num is a typeclass: all things like numbers. So, len takes a list of
anything (really anything) and produces a number of some kind (but
might be Int, Integer, etc.). Saying that the type is in this class
groups all these functions together.

Another example: Eq class defines equality (==) and inequality (/=).
All the basic datatypes exported by the Prelude are instances of
Eq, and Eq may be derived for any datatype whose constituents are
also instances of Eq.

Not to be confused with classes in Java.

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A < A (reflexivity)A < B and B < C then A < C (transitivity)a : A and A < B then a : B (subsumption)We also add a top type , which is above everything else: A < Can you give examples of these from Java?Objects: A larger type is a subtype of a smaller typePart 4 – Types G6021 Comparative Programming 33 / 63Types of polymorphismParametric polymorphism: operates uniformly across differentSubtype polymorphism: operates through an inclusion relation.Ad-hoc polymorphism is another name for overloading and isabout the use of the same name for different functions.Part 4 – Types G6021 Comparative Programming 34 / 63Object-Oriented LanguagesMany modern programming languages are based around theobject model: Java, Eiffel, C++, Smalltalk, Self, etc.Naive understanding: object = (pointer to a) recordBasic features: Object creation, Field selection, Field update, andMethod invocationWe could study an object calculus which allows us to understand thebasic elements of object-oriented programming in the same spirit asthe -calculus for functional programming.However, we want to focus now on comparing the paradigms.Question: Functions vs. objects?Part 4 – Types G6021 Comparative Programming 35 / 63Object Oriented Programmingpublic data: methods (member functions), public variablesprivate data: instance variables, hidden methodsObject-Oriented Program:Send messages to objectsObject-Oriented ProgrammingProgramming methodology: organise concepts into objects andConcepts: encapsulate data, subtyping (extensions ofdata-types), inheritance (reuse of implementation)Part 4 – Types G6021 Comparative Programming 36 / 63Four Basic ConceptsDynamic Lookup – when a message is sent to an object, themethod executed is determined by the object implementation.Different objects can respond differently to the same message.The response is not based on the static property of the variable orAbstraction – implementation details are hidden inside a programunit and exposed via a specific interface. Usually a set of publicmethods manipulate private data.Subtyping – if object A has all the functionality of another object B,we can use A in place of B in contexts expecting A. Subtypingmeans that the subtype has at least as much functionality as thebase type.Inheritance – reuse definition of one type of object when defininganother object.Part 4 – Types G6021 Comparative Programming 37 / 63Aside: delegation-based languagesIn these languages, objects are defined directly from other objects byadding methods and replacing methods (rather then from classes).Part 4 – Types G6021 Comparative Programming 38 / 63Dynamic LookupA method is selected dynamically (at run time) according to theimplementation of the object that receives the message: Differentobjects may implement the same operation differently.x.add(y) means send the message add(y) to the object x. If x isan integer, then we may perform usual addition; if x is a string, thenconcatenation; if x is a set, then we add the element y to the set, etc.etc. Thus:while(c) {may perform a different operation each time we enter the loop.Part 4 – Types G6021 Comparative Programming 39 / 63Dynamic lookup, continuedIn functional languages, x.add(y) would be written as add(x,y):the meaning of the operation stays the same.Exercise: does dynamic lookup = overloading ?Answer: To some extent: however, overloading is a static concept: it isthe static type information that dictates which code is used.Dynamic lookup is an important part of Java, C++ and Smalltalk. (It isthe default in Java and Smalltalk, in C++ only virtual member functionsare dynamic).Part 4 – Types G6021 Comparative Programming 40 / 63Abstraction (encapsulation)Programmer has a detailed view of programUser has an abstract viewEncapsulation is a mechanism for separating these two viewsSML has a notion of abstraction:abstype Set withempty : unit -> Set
isEmpty : Set -> boolean
add : int * Set -> Set
union : Set * Set -> Set

is (* detailed implementation *)
in (* program *) end

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Abstraction (Haskell example)

module Stack (Stack,empty,isEmpty,push,top,pop)

empty :: Stack a
isEmpty :: Stack a -> Bool
push :: a -> Stack a -> Stack a
top :: Stack a -> a
pop :: Stack a -> (a,Stack a)

newtype Stack a = StackImpl [a]
empty = StackImpl []
isEmpty (StackImpl s) = null s
push x (StackImpl s) = StackImpl (x:s)
top (StackImpl s) = head s
pop (StackImpl (s:ss)) = (s,StackImpl ss)

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Encapsulation

Guarantee invariants of data structure: only functions of the data
type have access to the internal representation of the data
Limited Reuse: cannot reuse code

Exercise: What is the essential difference between functional style
abstraction and OO abstraction?

Object-oriented languages allow encapsulation in an extensible form.

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Subtyping and Inheritance

Interface: The external view of an object; messages accepted by
an object; the type
Subtyping: relation between interfaces
Implementation: internal representation of an object
Inheritance: relation between implementations

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interface Point: x, y, move
interface ColouredPoint: x, y, move, colour, changeColour.

If interface A contains all of interface B, then A objects can also be
used as B objects.

ColouredPoint interface contains Point: ColouredPoint is a subtype of

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Inheritance

Implementation mechanism
New objects may be defined by reusing implementations of other

class Point
float x,y; Point move(float dx, dy)

class ColouredPoint
float x,y; colour c; Point move(float dx, dy)
Point changeColour(colour newc)

Subtyping: ColouredPoints can be used in place of Points:
property used by the client
Inheritance: ColouredPoints can be implemented by reusing the
implementation of Point: property used by the programmer

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Multiple Inheritance

A controversial aspect of Object-oriented programming
Should we be allowed to build a class by inheriting from more than
one base class?

Name clashes: if class C inherits from classes A and B, where A
and B have members of the same name, then we have a name
solutions:

Implicit resolution: arbitrary way defined by the language
Explicit resolution: programmer decides
Disallow name clashes: programs are not allowed to contain name

Exercise: can you give an example of name clashes using a Java-like

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Case Study: Java

Design Goals
Portability
Reliability
Simplicity
Efficient (secondary)

Almost everything in Java is an object; Does not allow multiple
inheritance; statically typed.

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Java Classes and Objects

Syntax similar to C++
Objects: fields, methods
Dynamic lookup: similar behaviour to other languages, static
typing (more efficient than some other languages, e.g. Smalltalk)
Dynamic linking (slower than C++)

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Terminology

Class, Object (as usual)
Field: data member
Method: data function
Static members: class fields and methods
this: self
Package: se

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