HOW TO BE
MORE PRODUCTIVE
Graham Hutton and Mauro Jaskelioff
Streams
A stream is an infinite sequence of values:
0  1  2  3  4 ...
The type of streams is co-inductively defined:
codata Stream A = A  Stream A
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Defining Streams
Streams can be defined by recursive equations:
ones :: Stream Nat
ones
= 1  ones
nats :: Stream Nat
nats
= 0  map (+1) nats
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This Talk
How do we ensure such equations make sense,
i.e. that they produce well-defined streams?
loop :: Stream A
loop
= tail loop
A new approach, based upon a representation
theorem for contractive functions.
3
Fixed Points
The starting point for our approach is the use of
explicit fixed points. For example:
ones = 1  ones
can be rewritten as:
ones
= fix body
body xs = 1  xs
fix f = f (fix f)
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The Problem
Given a function on streams
f :: Stream A  Stream A
when does
fix f :: Stream A
makes sense, i.e. produce a well-defined stream?
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Contractive Functions
Adapting an idea from topology, let us say that a
function f on streams is contractive iff:
xs =n ys
Equal for
the first n
elements.
f xs =n+1 f ys
Equal for
one further
element.
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Banach’s Theorem
Every contractive function
f :: Stream A c Stream A
has a unique fixed point
fix f :: Stream A
and hence produces a well-defined stream.
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This theorem provides
a semantic means of
ensuring that stream
definitions are valid.
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Example
The function (1 ) is contractive:
xs =n ys
1  xs =n+1 1  ys
Hence, it has a unique fixed point, and
ones = fix (1 )
is a valid definition for a stream.
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Example
The function tail is not contractive:
xs =n ys
tail xs =n+1 tail
ys
Hence, Banach’s theorem does not apply, and
loop = fix tail
is rejected as an invalid definition.
10
Questions
Does the converse also hold - every function
with a unique fixed point is contractive?
What does contractive actually mean?
What kind of functions are contractive?
11
Key Idea
If we view a stream as a time-varying value
x0  x1  x2  x3  x4 ...
then a function on streams is contractive iff
Its output value at any time only depends
on input values at strictly earlier times.
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This result simplifies the
process of deciding if a
function is contractive.
13
Examples
(1 )
tail
Each output depends on the
input one step earlier in time.
Each output depends on the
input one step later in time.
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Generating Functions
This idea is formalised using generating functions,
which map finite lists to single values:
[A]  B
All earlier
input values.
The next
output value.
15
Representation Theorem
Every contractive function can be represented by
a generating function, and vice versa:
rep
Stream A c Stream B
[A]  B
ge
n
Moreover, rep and gen form an isomorphism.
16
This theorem provides
a practical means of
producing streams
that are well-defined.
17
Example
g
g []
:: [Nat]  Nat
= 1
g (x:xs) = x
ones :: Stream Nat
ones
= fix (gen g)
Generator
for ones.
Guaranteed to
be well-defined.
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Example
g
g []
:: [Nat]  Nat
= 0
g (x:xs) = x+1
nats :: Stream Nat
nats
= fix (gen g)
Generator
for nats.
Guaranteed to
be well-defined.
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Example
g
:: [Nat]  Nat
g []
= 0
g [x]
= 1
Generator
for fibs.
g (x:y:xs) = x+y
fibs :: Stream Nat
fibs
= fix (gen g)
Guaranteed to
be well-defined.
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Summary
Generating functions are a sound and complete
representation of contractive functions;
Gives a precise characterisation of the class of
functions that are contractive;
Provides a simple but rather general means of
producing well-defined streams.
21
Ongoing and Further Work
Generalisation to final co-algebras;
Other kinds of generating functions;
Relationship to other techniques;
Improving efficiency.
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