Stop thinking about bottoms when writing programs . . . Thorsten - - PowerPoint PPT Presentation

Stop thinking about bottoms when writing programs . . .

Thorsten Altenkirch University of Nottingham

BCTCS 06 – p.1/19

Trouble with

BCTCS 06 – p.2/19

Trouble with

• ✁

BCTCS 06 – p.2/19

Trouble with

• ✁

BCTCS 06 – p.2/19

Trouble with

• ✁

No, because

BCTCS 06 – p.2/19

Trouble with . . .

BCTCS 06 – p.3/19

Trouble with . . .

Many useful algebraic properties do not hold.

BCTCS 06 – p.3/19

Trouble with . . .

Many useful algebraic properties do not hold.

Correctness proofs get obliterated with reasoning about

.

BCTCS 06 – p.3/19

Trouble with . . .

Many useful algebraic properties do not hold.

Correctness proofs get obliterated with reasoning about

.

Do we actually care about non-terminating programs?

BCTCS 06 – p.3/19

Trouble with . . .

Many useful algebraic properties do not hold.

Correctness proofs get obliterated with reasoning about

.

Do we actually care about non-terminating programs?

Programs are not natural phenomena. . .

BCTCS 06 – p.3/19

Trouble with . . .

Many useful algebraic properties do not hold.

Correctness proofs get obliterated with reasoning about

.

Do we actually care about non-terminating programs?

Programs are not natural phenomena. . .

Programs are constructed!

BCTCS 06 – p.3/19

Do we need to be lazy?

BCTCS 06 – p.4/19

Do we need to be lazy?

• ✒

• ✞

BCTCS 06 – p.4/19

Do we need to be lazy?

• ✒

• ✞

is total, if we interpret lists as a terminal coalgebra.

BCTCS 06 – p.4/19

data vs codata

BCTCS 06 – p.5/19

data vs codata

• ✩

• ✣

BCTCS 06 – p.5/19

data vs codata

• ✩

• ✣

is total, . . .

BCTCS 06 – p.5/19

data vs codata

• ✩

• ✣

is total, . . .

if we interpret lists as initial algebra:

BCTCS 06 – p.5/19

data vs codata

• ✩

• ✣

is total, . . .

if we interpret lists as initial algebra:

Problem:

• ✒

BCTCS 06 – p.5/19

data vs codata

BCTCS 06 – p.6/19

data vs codata

Finite lists

BCTCS 06 – p.6/19

data vs codata

Finite lists

Potentially infinite lists:

BCTCS 06 – p.6/19

data vs codata

Finite lists

Potentially infinite lists:

Better types

BCTCS 06 – p.6/19

data vs codata

Finite lists

Potentially infinite lists:

Better types

• ✒

doesn’t typecheck.

BCTCS 06 – p.6/19

Can we always avoid ?

BCTCS 06 – p.7/19

Can we always avoid ?

• ✧

• ✞

• ✞

• ❁

• ❀

• ✿

• ✿

• ✿

• ✧

• ❂

BCTCS 06 – p.7/19

Computational Reals

BCTCS 06 – p.8/19

Computational Reals

Define computational reals (

) using Cauchy sequences.

BCTCS 06 – p.8/19

Computational Reals

Define computational reals (

) using Cauchy sequences.

We cannot implement

BCTCS 06 – p.8/19

Computational Reals

Define computational reals (

) using Cauchy sequences.

We cannot implement

Indeed, all total computable functions of type

are constant (Brouwer).

BCTCS 06 – p.8/19

Computational Reals

Define computational reals (

) using Cauchy sequences.

We cannot implement

Indeed, all total computable functions of type

are constant (Brouwer).

However, there are perfectly reasonable partial implementations

• f

.

BCTCS 06 – p.8/19

We need for:

BCTCS 06 – p.9/19

We need for:

Interpreters.

BCTCS 06 – p.9/19

We need for:

Interpreters.

Functions on

.

BCTCS 06 – p.9/19

We need for:

Interpreters.

Functions on

.

more examples ?

BCTCS 06 – p.9/19

Epigram

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

All Epigram programs are total (i.e. no

).

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

All Epigram programs are total (i.e. no

).

It is not a programming language in Peter Mosses sense.

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

All Epigram programs are total (i.e. no

).

It is not a programming language in Peter Mosses sense.

because not all computable functions can be expressed.

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

All Epigram programs are total (i.e. no

).

It is not a programming language in Peter Mosses sense.

because not all computable functions can be expressed.

I am going to show how we can fix this. . .

BCTCS 06 – p.10/19

Epigram

Epigram is a dependently typed programming language...

All Epigram programs are total (i.e. no

).

It is not a programming language in Peter Mosses sense.

because not all computable functions can be expressed.

I am going to show how we can fix this. . .

without making Epigram partial.

BCTCS 06 – p.10/19

• Monads. . .

BCTCS 06 – p.11/19

• Monads. . .

A monad

is given by

• ✠

• ✩

subject to some equations.

BCTCS 06 – p.11/19

• Monads. . .

A monad

is given by

• ✠

• ✩

subject to some equations.

We can use monads to encapsulate effects (e.g. state)

• ❍
• ❍

• ❍

• ❍

• ❍

• ❍
• ✂

BCTCS 06 – p.11/19

• Monads. . .

A monad

is given by

• ✠

• ✩

subject to some equations.

We can use monads to encapsulate effects (e.g. state)

• ❍
• ❍

• ❍

• ❍

• ❍

• ❍
• ✂

and to model effects (e.g. state) :

• ✳

• ✩◗P

• ✽

• ❩

• ✩

• ✽

• ❩

• ✩

BCTCS 06 – p.11/19

The Delay monad

BCTCS 06 – p.12/19

The Delay monad

• ❫

BCTCS 06 – p.12/19

The Delay monad

• ❫

• ▲

BCTCS 06 – p.12/19

The Delay monad

• ❫

• ▲

BCTCS 06 – p.12/19

Iteration with Delay

• ✩

• ❢

• ✓

BCTCS 06 – p.13/19

Fixpoints with Delay

BCTCS 06 – p.14/19

Fixpoints with Delay

• ✩

• ✩

BCTCS 06 – p.14/19

Fixpoints with Delay

• ✩

• ✩

• ❩

• ✩

• ❡

• ❜

• ✩

• ♥

BCTCS 06 – p.14/19

Fixpoints with Delay

• ✩

• ✩

• ❩

• ✩

• ❡

• ❜

• ✩

• ♥

• ❫

• ❫

• ❫

• ❴

• ❜

• ❴

• ❜

• ❜

• ✓

BCTCS 06 – p.14/19

From Delay to Partial

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

Hence

• ✓

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

Hence

• ✓

We define

where

identifies values with different finite delay.

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

Hence

• ✓

We define

where

identifies values with different finite delay.

We have to show that

,

,

preserve

.

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

Hence

• ✓

We define

where

identifies values with different finite delay.

We have to show that

,

,

preserve

.

We have

• ✓

BCTCS 06 – p.15/19

From Delay to Partial

is too intensional. . .

We can observe how fast a function terminates.

Hence

• ✓

We define

where

identifies values with different finite delay.

We have to show that

,

,

preserve

.

We have

• ✓

if

is

• continuous,

however all definable

are.

BCTCS 06 – p.15/19

Defining

BCTCS 06 – p.16/19

Defining

• ②

is defined inductively.

BCTCS 06 – p.16/19

Defining

• ②

is defined inductively.

BCTCS 06 – p.16/19

Defining

• ②

is defined inductively.

BCTCS 06 – p.16/19

Deja vu ?

BCTCS 06 – p.17/19

Deja vu ?

Constructive Domain Theory!

BCTCS 06 – p.17/19

Deja vu ?

Constructive Domain Theory!

BCTCS 06 – p.17/19

Deja vu ?

Constructive Domain Theory!

Note that constructively

because we cannot observe non-termination.

BCTCS 06 – p.17/19

Deja vu ?

Constructive Domain Theory!

Note that constructively

because we cannot observe non-termination.

and hence

are

CPOs.

BCTCS 06 – p.17/19

Deja vu ?

Constructive Domain Theory!

Note that constructively

because we cannot observe non-termination.

and hence

are

CPOs.

the code before constructs

in

.

BCTCS 06 – p.17/19

Conclusions and further work

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

Partiality is an effect

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

Partiality is an effect

We can reason about partial programs at compile time using the definition of

.

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

Partiality is an effect

We can reason about partial programs at compile time using the definition of

.

and we can execute non-terminating programs at run-time.

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

Partiality is an effect

We can reason about partial programs at compile time using the definition of

.

and we can execute non-terminating programs at run-time.

In future Epigram could support partiality without giving up the advantages of having a total language for most programs.

BCTCS 06 – p.18/19

Conclusions and further work

Using the partiality monad we can encapsulate partial programs in a total language.

Partiality is an effect

We can reason about partial programs at compile time using the definition of

.

and we can execute non-terminating programs at run-time.

In future Epigram could support partiality without giving up the advantages of having a total language for most programs.

Still to do: recursive datatypes by a constructive implementation

• f the standard domain-theoretic construction.

BCTCS 06 – p.18/19

Thank you

Thanks to Conor McBride & the Epigram Team (James Chapman, Peter Morris, Wouter Swierstra) see www.e-pig.org for more information on Epigram.

Acknowledgements to Tarmo Uustalu and Venanzio Capretta for joint work on a partial paper. . .

Looking for my papers? Type “Thorsten” into google. . .

BCTCS 06 – p.19/19