Maths - W-types

See ncatlab and Wikipedia.

Not sure what the 'W' stands for, could be from the Martin-Löf concept of 'well orderings'. I think its a 'W' and not omega. Though it seems related to 'ω-complete partial order' (see cpo) from denotational semantics/domain theory.

In category theory this is an initial algebra for a polynomial endofunctor.

In type theory it is a free term algebra.

Examples

An important example is the natural numbers.

They can be defined in terms of Peano algebra. This is not very practical for everyday use but has nice properties for proving things about natural numbers.

Idris Code

data Nat =
  Z // Zero
  | S Nat // Successor.
The natural number example has two constructors, Zero (arity 0) and Successor (arity 1). Lets explore what happens when we add more constructors. For instance lets have two constructors of arity 1.

Idris Code

data MyType =
  Z |
  X MyType |
  Y MyType

This generates 'words', starting with Z and then any sequence of X and Y.

So a W-type is a tree structure. There is no restriction on the order or length of the terms.

This is known as a 'free' algebra.

xyz diagram

We could add additional constructors of arity 1 to increase the types of characters in the words.

Another thing we could do is add constructors of arity 2 and above.

Idris Code

data MyType =
  Z |
  X MyType |
  Y MyType |
  U MyType MyType

This seems to allow us to join arrows. So we appear to be able to construct lattices although I cant see a way to limit the construction to a lattice because we can always have paths going off in all directions.

uxyz diagram

Logic of W-types

The terms/elements of a W-type can be considered to be “rooted well-founded trees” with a certain branching type.

In the table below:

  • A is the set of constructor names (such as Z and S for natural numbers).
  • B is a set of maps from these constructor names to their arity.
diagram

So if a is an element of A, that is an individual constructor label, then B(a) is the arity for that constructor.

Introduction

In a programming language a constructor would look something like this:
name(parameter1, parameter2)
or in a functional language like this:
(name parameter1 parameter2)

In the case of a W-type the parameters are all the same as the type being constructed so all we need to know is the number of parameters.

So (W a : A)B(a) is a W-type.

Elimination

Elimination is usually a way to prove properties of W.

For example, for natural numbers:

Idris2

nat_induction :
    (prop : Nat -> Type) ->                -- Property to show
    (prop Z) ->                            -- Base case
    ((k : Nat) -> prop k -> prop (S k)) -> -- Inductive step
    (x : Nat) ->                           -- Show for all x
    prop x
nat_induction prop p_Z p_S Z = p_Z
nat_induction prop p_Z p_S (S k) = p_S k (nat_induction prop p_Z p_S k)

That is, for all properties on natural numbers, if it holds for zero and P n implies P (S n) then the property holds for all numbers. (proof by induction).

Table

in extensional type theory:

 

W-Type

W-formation rule

A:Type   a:A   right adjoint   B(a):Type
(W a : A)B(a) : Type

W-introduction rule

where W means (W a : A)B(a)

a: A t : B(a) -> W
sup(a,t) : W

W-elimination rule

Usually a way to prove properties of W

w:Wright adjoint C(w):Type   x:A,u:B(x)-<W,v:(Πy : B(x))C(u(y))   right adjoint   c:(x,u,v):C(sup(x,u))
w:W right adjoint wrec(w,c) : C(w)

W-computation rule

w:Wright adjoint C(w):Type   x:A,u:B(x)-<W,v:(Πy : B(x))C(u(y))   right adjoint   c:(x,u,v):C(sup(x,u))
x:A,u:B(x) -> W right adjoint wrec(sup(x,u),c) = c(x,u, λy.wrec(u(y),c)) : C(sup(x,u))

nlab page

Intensional and Extensional TypeTheories

Intensional and Extensional TypeTheories are discussed on page here. One destinction is:

In category theory the locally closed Cartesian category is the counterpart of extensional type theory.

For intensional type theory we need to add quotient completions - used for modeling extensional type theories into intensional ones.

wtype

Setoid

Setoids are discussed on this page.

The notion of setoid allows the intensional type theory to represent extensional concepts. Setoids provide a model of extensional type theory within the intensional one

In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~.

Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set). - From Wiki

One way to define a setoid is define a tuple consisting of:

For an Idris implementation of setoids see this github site

Extensionality of functions

if f(x) = g(x) for all x

then f = g

Example - consider the two functions f and g mapping from and to natural numbers, defined as follows:

These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same.

Functions between Setoids

Extensional Functions between Setoids

A function between setoids is a function between the underlying types (sets) together with a proof of extensionality:

A function |f| : |X|->|Y| is extensional, with respect to the equalities of X and Y,if there is a term of type:

f ~X=>Y g: Π|x|,|x'|:|X| x ~X x' -> |f|(x) ~Y |f|(y)

The bars, as in |x|, indicate x is an element of the underlying type. This distinction will usually be dropped if extensionality applies.

Algebra of Extensional Trees

 

Topological Space and W-types

Are the structures of topological Spaces (especially Alexandrov topology) and W-types related in some way?

They both seem to be able to code linear structures and trees but not loops (circles).

Category Theory and W-types

In category theory W-types are initial algebras for polynomial endofunctors.

Doctrine

a categorification of the concept of “theory”.

Concepts

See:

For more information about the relationship between type theory and category theory see this page.

Related Topics


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see also:

Setoids - Idris proofs for extensional equalities

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Computation and Reasoning - This book is about type theory. Although it is very technical it is aimed at computer scientists, so it has more discussion than a book aimed at pure mathematicians. It is especially useful for the coverage of dependant types.

 

Terminology and Notation

Specific to this page here:

 

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