HOW TO BE MORE PRODUCTIVE

1. HOW TO BE MORE PRODUCTIVE Graham Hutton and Mauro Jaskelioff

2. 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

3. Defining Streams Streams can be defined by recursive equations: ones :: Stream Nat ones = 1  ones nats :: Stream Nat nats = 0  map (+1) nats

4. 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.

5. 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)

6. 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?

7. Contractive Functions Adapting an idea from topology, let us say that a function f on streams is contractive iff: xs =n ys f xs =n+1 f ys Equal for the first n elements. Equal for one further element.

8. Banach’s Theorem Every contractive function f :: Stream A cStream A has a unique fixed point fix f :: Stream A and hence produces a well-defined stream.

9. This theorem provides a semantic means of ensuring that stream definitions are valid.

10. Example The function (1 ) is contractive: 1  xs =n+1 1  ys xs =n ys Hence, it has a unique fixed point, and ones = fix (1 ) is a valid definition for a stream.

11. Example The function tail is not contractive: tail xs =n+1 tail ys xs =n ys Hence, Banach’s theorem does not apply, and loop = fix tail is rejected as an invalid definition.

12. 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?

13. 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.

14. This result simplifies the process of deciding if a function is contractive.

15. Examples Each output depends on the input one step earlier in time. (1 ) Each output depends on the input one step later in time. tail

16. Generating Functions This idea is formalised using generating functions, which map finite lists to single values: [A]  B The next output value. All earlier input values.

17. Representation Theorem Every contractive function can be represented by a generating function, and vice versa: rep Stream A cStream B [A]  B gen Moreover, rep and gen form an isomorphism.

18. This theorem provides a practical means of producing streams that are well-defined.

19. Example g :: [Nat]  Nat g [] = 1 g (x:xs) = x Generator for ones. Guaranteed to be well-defined. ones :: Stream Nat ones = fix (gen g)

20. Example g :: [Nat]  Nat g [] = 0 g (x:xs) = x+1 Generator for nats. Guaranteed to be well-defined. nats :: Stream Nat nats = fix (gen g)

21. Example g :: [Nat]  Nat g [] = 0 g [x] = 1 g (x:y:xs) = x+y Generator for fibs. Guaranteed to be well-defined. fibs :: Stream Nat fibs = fix (gen g)

22. 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.

23. Ongoing and Further Work • Generalisation to final co-algebras; • Other kinds of generating functions; • Relationship to other techniques; • Improving efficiency.