Initial algebra

In mathematics, an initial algebra is an initial object in the category of [[F-algebra|Template:Mvar-algebras]] for a given endofunctor Template:Mvar. This initiality provides a general framework for induction and recursion.

Examples

Functor Template:Math

Consider the endofunctor Template:Math, i.e. Template:Math sending Template:Mvar to Template:Math, where Template:Math is a one-point (singleton) set, a terminal object in the category. An algebra for this endofunctor is a set Template:Mvar (called the carrier of the algebra) together with a function Template:Math. Defining such a function amounts to defining a point Template:Math and a function Template:Math. Define

$\begin{array}{cc}\mathrm{zero}:1& ⟶𝐍\\ *& ⟼0\end{array}$

and

$\begin{array}{cc}\mathrm{succ}:𝐍& ⟶𝐍\\ n& ⟼n+1.\end{array}$

Then the set Template:Math of natural numbers together with the function Template:Math is an initial Template:Mvar-algebra. The initiality (the universal property for this case) is not hard to establish; the unique homomorphism to an arbitrary Template:Mvar-algebra Template:Math, for Template:Math an element of Template:Mvar and Template:Math a function on Template:Mvar, is the function sending the natural number Template:Mvar to Template:Math, that is, Template:Math, the Template:Mvar-fold application of Template:Mvar to Template:Mvar.

The set of natural numbers is the carrier of an initial algebra for this functor: the point is zero and the function is the successor function.

Functor Template:Math

For a second example, consider the endofunctor Template:Math on the category of sets, where Template:Math is the set of natural numbers. An algebra for this endofunctor is a set Template:Mvar together with a function Template:Math. To define such a function, we need a point Template:Math and a function Template:Math. The set of finite lists of natural numbers is an initial algebra for this functor. The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head.

In categories with binary coproducts, the definitions just given are equivalent to the usual definitions of a natural number object and a list object, respectively.

Final coalgebra

Dually, a final coalgebra is a terminal object in the category of [[F-coalgebra|Template:Mvar-coalgebras]]. The finality provides a general framework for coinduction and corecursion.

For example, using the same functor Template:Math as before, a coalgebra is defined as a set Template:Mvar together with a function Template:Math. Defining such a function amounts to defining a partial function Template:Math whose domain is formed by those $x\in X$ for which Template:Math does not belong to Template:Math. Having such a structure, we can define a chain of sets: Template:Math being a subset of Template:Math on which Template:Math is not defined, Template:Math which elements map into Template:Math by Template:Math, Template:Math which elements map into Template:Math by Template:Math, etc., and Template:Math containing the remaining elements of Template:Mvar. With this in view, the set $𝐍\cup \{\omega \}$, consisting of the set of natural numbers extended with a new element Template:Mvar, is the carrier of the final coalgebra, where $f^{\prime }$ is the predecessor function (the inverse of the successor function) on the positive naturals, but acts like the identity on the new element Template:Mvar: Template:Math, Template:Math. This set $𝐍\cup \{\omega \}$ that is the carrier of the final coalgebra of Template:Math is known as the set of conatural numbers.

For a second example, consider the same functor Template:Math as before. In this case the carrier of the final coalgebra consists of all lists of natural numbers, finite as well as infinite. The operations are a test function testing whether a list is empty, and a deconstruction function defined on non-empty lists returning a pair consisting of the head and the tail of the input list.

Theorems

• Initial algebras are minimal (i.e., have no proper subalgebra).
• Final coalgebras are simple (i.e., have no proper quotients).

Use in computer science

Various finite data structures used in programming, such as lists and trees, can be obtained as initial algebras of specific endofunctors. While there may be several initial algebras for a given endofunctor, they are unique up to isomorphism, which informally means that the "observable" properties of a data structure can be adequately captured by defining it as an initial algebra.

To obtain the type Template:Math of lists whose elements are members of set Template:Mvar, consider that the list-forming operations are:

• $\mathrm{n}\mathrm{i}\mathrm{l}:1\to \mathrm{L}\mathrm{i}\mathrm{s}\mathrm{t}(A)$
• $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}:A\times \mathrm{L}\mathrm{i}\mathrm{s}\mathrm{t}(A)\to \mathrm{L}\mathrm{i}\mathrm{s}\mathrm{t}(A)$

Combined into one function, they give:

• $[\mathrm{n}\mathrm{i}\mathrm{l},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}]:(1+A\times \mathrm{L}\mathrm{i}\mathrm{s}\mathrm{t}(A))\to \mathrm{L}\mathrm{i}\mathrm{s}\mathrm{t}(A),$

which makes this an Template:Mvar-algebra for the endofunctor Template:Mvar sending Template:Mvar to Template:Math. It is, in fact, the initial Template:Mvar-algebra. Initiality is established by the function known as foldr in functional programming languages such as Haskell and ML.

Likewise, binary trees with elements at the leaves can be obtained as the initial algebra

• $[\mathrm{t}\mathrm{i}\mathrm{p},\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}]:A+(\mathrm{T}\mathrm{r}\mathrm{e}\mathrm{e}(A)\times \mathrm{T}\mathrm{r}\mathrm{e}\mathrm{e}(A))\to \mathrm{T}\mathrm{r}\mathrm{e}\mathrm{e}(A).$

Types obtained this way are known as algebraic data types.

Types defined by using least fixed point construct with functor Template:Mvar can be regarded as an initial Template:Mvar-algebra, provided that parametricity holds for the type.

In a dual way, similar relationship exists between notions of greatest fixed point and terminal Template:Mvar-coalgebra, with applications to coinductive types. These can be used for allowing potentially infinite objects while maintaining strong normalization property. In the strongly normalizing (each program terminates) Charity programming language, coinductive data types can be used for achieving surprising results, e.g. defining lookup constructs to implement such “strong” functions like the Ackermann function.

See also

• Algebraic data type
• Catamorphism
• Anamorphism

External links

• Categorical programming with inductive and coinductive types by Varmo Vene
• Recursive types for free! by Philip Wadler, University of Glasgow, 1990-2014.
• Initial Algebra and Final Coalgebra Semantics for Concurrency by J.J.M.M. Rutten and D. Turi
• Initiality and finality from CLiki
• Typed Tagless Final Interpreters by Oleg Kiselyov