# Hask

### From HaskellWiki

m (→The seq problem: Format) |
(Rewrite) |
||

Line 1: | Line 1: | ||

− | '''Hask''' | + | '''Hask''' refers to a [[Category theory|category]] with types as objects and functions between them as morphisms. However, its use is ambiguous. Sometimes it refers to Haskell (''actual Hask''), and sometimes it refers to some subset of Haskell where no values are bottom and all functions terminate (''platonic Hask''). The reason for this is that platonic Hask has lots of nice properties that actual Hask does not, and is thus easier to reason in. |

− | + | == Hask == | |

− | + | Actual Hask does not have sums, products, or an initial object, and <hask>()</hask> is not a terminal object. The Monad identities fail for almost all instances of the Monad class. | |

− | + | {| class="wikitable" | |

+ | |+ Why '''Hask''' isn't as nice as you'd thought. | ||

+ | ! scope="col" | | ||

+ | ! scope="col" | Initial Object | ||

+ | ! scope="col" | Terminal Object | ||

+ | ! scope="col" | Sum | ||

+ | ! scope="col" | Product | ||

+ | |- | ||

+ | ! scope="row" | Definition | ||

+ | | There is a unique function | ||

+ | <hask>u :: Empty -> r</hask> | ||

+ | | There is a unique function | ||

+ | <hask>u :: r -> ()</hask> | ||

+ | | For any functions | ||

+ | <hask>f :: a -> r</hask> | ||

+ | <hask>g :: b -> r</hask> | ||

− | + | there is a unique function | |

+ | <hask>u :: Either a b -> r</hask> | ||

+ | such that: | ||

+ | <hask>u . Left = f</hask> | ||

+ | <hask>u . Right = g</hask> | ||

+ | | For any functions | ||

+ | <hask>f :: r -> a</hask> | ||

+ | <hask>g :: r -> b</hask> | ||

+ | there is a unique function | ||

+ | <hask>u :: r -> (a,b)</hask> | ||

− | == | + | such that: |

+ | <hask>fst . u = f</hask> | ||

+ | <hask>snd . u = g</hask> | ||

+ | |- | ||

+ | ! scope="row" | Platonic candidate | ||

+ | | <hask>u1 r = case r of {}</hask> | ||

+ | | <hask>u1 _ = ()</hask> | ||

+ | | <hask>u1 (Left a) = f a</hask> | ||

+ | <hask>u1 (Right b) = g b</hask> | ||

+ | | <hask>u1 r = (f r,g r)</hask> | ||

+ | |- | ||

+ | ! scope="row" | Example failure condition | ||

+ | | <hask>r ~ ()</hask> | ||

+ | | <hask>r ~ ()</hask> | ||

+ | | <hask>r ~ ()</hask> | ||

+ | <hask>f _ = ()</hask> | ||

+ | <hask>g _ = ()</hask> | ||

+ | | <hask>r ~ ()</hask> | ||

+ | <hask>f _ = undefined</hask> | ||

+ | <hask>g _ = undefined</hask> | ||

+ | |- | ||

+ | ! scope="row" | Alternative u | ||

+ | | <hask>u2 _ = ()</hask> | ||

+ | | <hask>u2 _ = undefined</hask> | ||

+ | | <hask>u2 _ = ()</hask> | ||

+ | | <hask>u2 _ = undefined</hask> | ||

+ | |- | ||

+ | ! scope="row" | Difference | ||

+ | | <hask>u1 undefined = undefined</hask> | ||

+ | <hask>u2 undefined = ()</hask> | ||

+ | | <hask>u1 _ = ()</hask> | ||

+ | <hask>u2 _ = undefined</hask> | ||

+ | | <hask>u1 undefined = undefined</hask> | ||

+ | <hask>u2 undefined = ()</hask> | ||

+ | | <hask>u1 _ = (undefined,undefined)</hask> | ||

+ | <hask>u2 _ = undefined</hask> | ||

+ | |- style="background: red;" | ||

+ | ! scope="row" | Result | ||

+ | ! scope="col" | FAIL | ||

+ | ! scope="col" | FAIL | ||

+ | ! scope="col" | FAIL | ||

+ | ! scope="col" | FAIL | ||

+ | |} | ||

− | + | == Platonic '''Hask''' == | |

− | + | Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottoms. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products. Instances of Functor and Monad are really endofunctors and monads. | |

− | + | == Links == | |

+ | * [http://www.cs.gunma-u.ac.jp/~hamana/Papers/cpo.pdf Makoto Hamana: ''What is the category for Haskell?''] | ||

+ | * [http://www.cs.nott.ac.uk/~nad/publications/danielsson-popl2006-tr.pdf Nils A. Danielsson, John Hughes, Patrik Jansson, and Jeremy Gibbons. ''Fast and loose reasoning is morally correct.''] | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

[[Category:Mathematics]] | [[Category:Mathematics]] | ||

[[Category:Theoretical foundations]] | [[Category:Theoretical foundations]] |

## Revision as of 10:23, 7 August 2012

**Hask** refers to a category with types as objects and functions between them as morphisms. However, its use is ambiguous. Sometimes it refers to Haskell (*actual Hask*), and sometimes it refers to some subset of Haskell where no values are bottom and all functions terminate (*platonic Hask*). The reason for this is that platonic Hask has lots of nice properties that actual Hask does not, and is thus easier to reason in.

## 1 Hask

Actual Hask does not have sums, products, or an initial object, andInitial Object | Terminal Object | Sum | Product | |
---|---|---|---|---|

Definition | There is a unique function
u :: Empty -> r |
There is a unique function
u :: r -> () |
For any functions
f :: a -> r g :: b -> r there is a unique function u :: Either a b -> r such that: u . Left = f u . Right = g |
For any functions
f :: r -> a g :: r -> b there is a unique function u :: r -> (a,b) such that: fst . u = f snd . u = g |

Platonic candidate | u1 r = case r of {} |
u1 _ = () |
u1 (Left a) = f a u1 (Right b) = g b |
u1 r = (f r,g r) |

Example failure condition | r ~ () |
r ~ () |
r ~ () f _ = () g _ = () |
r ~ () f _ = undefined g _ = undefined |

Alternative u | u2 _ = () |
u2 _ = undefined |
u2 _ = () |
u2 _ = undefined |

Difference | u1 undefined = undefined u2 undefined = () |
u1 _ = () u2 _ = undefined |
u1 undefined = undefined u2 undefined = () |
u1 _ = (undefined,undefined) u2 _ = undefined |

Result | FAIL | FAIL | FAIL | FAIL |

## 2 Platonic **Hask**

Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottoms. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products. Instances of Functor and Monad are really endofunctors and monads.