Intuitionistic type theory
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory is a type theory and an alternative foundation of mathematics based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972. Martin-Löf has modified his proposal a few times; his 1971 impredicative formulation was inconsistent as demonstrated by Girard's paradox. Later formulations were predicative. He proposed both intensional[disambiguation needed] and extensional variants of the theory.
Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs. This identification is usually called the Curry–Howard isomorphism, which was originally formulated for intuitionistic logic and simply typed lambda calculus. Type Theory extends this identification to predicate logic by introducing dependent types, that is types which contain values. Type Theory internalizes the interpretation of intuitionistic logic proposed by Brouwer, Heyting and Kolmogorov, the so-called BHK interpretation. The types of Type Theory play a similar role to sets in set theory but functions definable in Type Theory are always computable.
Connectives of type theory 
In the context of Type Theory a connective is a way of constructing types, possibly using already given types. The basic connectives of Type Theory are:
-types, also called dependent product types, are analogous to the indexed products of sets. As such, they generalize the normal function space to model functions whose result type may vary on their input. E.g. writing for the type of n-tuples of real numbers and for the type of natural numbers,
stands for the type of a function that, given a natural number n, returns an n-tuple of real numbers. The usual function space arises as a special case when the range type does not actually depend on the input, e.g., is the type of functions from natural numbers to the real numbers, which is also written as .
Using the Curry–Howard isomorphism -types also serve to model implication and universal quantification: e.g., a term inhabiting is a function which assigns to any pair of natural numbers a proof that addition is commutative for that pair and hence can be considered as a proof that addition is commutative for all natural numbers. (Here we have used the equality type explained below.)
-types, also called dependent sum types, are analogous to the indexed disjoint unions of sets. As such, they generalize the usual Cartesian product to model pairs where the type of the second component depends on the first. For example, the type stands for the type of pairs of a natural number and an -tuple of real numbers, i.e., this type can be used to model sequences of arbitrary but finite length (usually called lists). The conventional Cartesian product type arises as a special case when the type of the second component doesn't actually depend on the first, e.g., is the type of pairs of a natural number and a real number, which is also written as .
Finite types 
Of special importance are 0 or ⊥ (the empty type), 1 or ⊤ (the unit type) and 2 (the type of Booleans or classical truth values). Invoking the Curry–Howard isomorphism again, ⊥ stands for False and ⊤ for True.
Using finite types we can define negation as .
Equality type 
Given then is the type of equality proofs that is equal to . There is only one (canonical) inhabitant of and this is the proof of reflexivity .
Examination of the properties of the equality type lead to homotopy type theory.
Inductive types 
A prime example of an inductive type is the type of natural numbers which is generated by and . An important application of the propositions as types principle is the identification of (dependent) primitive recursion and induction by one elimination constant: for any given type indexed by . In general inductive types can be defined in terms of W-types, the type of well-founded trees.
An important class of inductive types are inductive families like the type of vectors mentioned above, which is inductively generated by the constructors and . Applying the Curry–Howard isomorphism once more, inductive families correspond to inductively defined relations.
Dybjer suggested extending inductive types with induction-recursion.
An example of a universe is , the universe of all small types, which contains names for all the types introduced so far. To every name we associate a type , its extension or meaning. It is standard to assume a predicative hierarchy of universes: for every natural number , where the universe contains a code for the previous universe, i.e., we have with . (A hierarchy with this property is called "cumulative".)
Stronger universe principles have been investigated, i.e., super universes and the Mahlo universe. In 1992 Huet and Coquand introduced the calculus of constructions, a type theory with an impredicative universe, thus combining Type Theory with Girard's System F. This extension is not universally accepted by Intuitionists since it allows impredicative, i.e., circular, constructions, which are often identified with classical reasoning.
Formalisation of type theory 
This formalization is based on the discussion in Nordström, Petersson, and Smith.
The formal theory works with types and objects.
A type is declared by:
An object exists and is in a type if:
Objects can be equal
and types can be equal
A type that depends on an object from another type is declared
and removed by substitution
- , replacing the variable with the object in .
An object that depends on an object from another type can be done two ways. If the object is "abstracted", then it is written
and removed by substitution
- , replacing the variable with the object in .
The object-depending-on-object can also be declared as a constant as part of a recursive type. An example of a recursive type is:
Here, is a constant object-depending-on-object. It is not associated with an abstraction. Constants like can be removed by defining equality. Here the relationship with addition is defined using equality and using pattern matching to handle the recursive aspect of .
is manipulated as an opaque constant - it has no internal structure for substitution.
So, objects and types and these relations are used to express formulae in the theory. The following styles of judgements are used to create new objects, types and relations from existing ones.
- , is a well-formed type in context .
- , is a well-formed term of type in context .
- , and are equal types in context .
- , and are equal terms of type in context .
- , is a well-formed context of typing assumptions.
By convention, there is a type that represents all other types. It is called (or ). Since is a type, the member of it are objects. There is a dependent type that maps each object to its corresponding type. In most texts is never written. From the context of the statement, a reader can almost always tell whether refers to a type, or whether it refers to the object in that corresponds to the type.
This is the complete foundation of the theory. Everything else is derived.
To implement logic, each proposition is given its own type. The objects in those types represent the different possible ways to prove the proposition. Obviously, if there is no proof for the proposition, then the type has no objects in it. Operators like "and" and "or" that work on propositions introduce new types and new objects. So is a type that depends on the type and the type . The objects in that dependent type are defined to exist for every pair of objects in and . Obviously, if or has no proof and is an empty type, then the new type representing is also empty.
This can be done for other types (booleans, natural numbers, etc.) and their operators.
Categorical models of type theory 
Using the language of category theory, R.A.G. Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of Type Theory. This has been refined by Hofmann and Dybjer to Categories with Families or Categories with Attributes based on earlier work by Cartmell.
A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : Cop → Fam(Set).
Fam(Set) is the category of families of Sets, in which objects are pairs (A,B) of an "index set" A and a function B: X → A, and morphisms are pairs of functions f : A → A' and g : X → X' , such that B' ° g = f ° B - in other words, f maps Ba to B'g(a).
The functor T assigns to a context G a set Ty(G) of types, and for each A : Ty(G), a set Tm(G,A) of terms. The axioms for a functor require that these play harmoniously with substitution. Substitution is usually written in the form Af or af, where A is a type in Ty(G) and a is a term in Tm(G,A), and f is a substitution from D to G. Here Af : Ty(D) and af : Tm(D,Af).
The category C must contain a terminal object (the empty context), and a final object for a form of product called comprehension, or context extension, in which the right element is a type in the context of the left element. If G is a context, and A : Ty(G), then there should be an object (G,A) final among contexts D with mappings p : D → G, q : Tm(D,Ap).
A logical framework, such as Martin-Löf's takes the form of closure conditions on the context dependent sets of types and terms: that there should be a type called Set, and for each set a type, that the types should be closed under forms of dependent sum and product, and so forth.
A theory such as that of predicative set theory expresses closure conditions on the types of sets and their elements: that they should be closed under operations that reflect dependent sum and product, and under various forms of inductive definition.
Extensional versus intensional 
A fundamental distinction is extensional vs intensional Type Theory. In extensional Type Theory definitional (i.e., computational) equality is not distinguished from propositional equality, which requires proof. As a consequence type checking becomes undecidable in extensional type theory. This is because relying on computational equality means that the equality depends on computations that could be Turing complete in general and thus the equality itself is undecidable due to the halting problem. Some type theories enforce the restriction that all computations be decidable so that definitional equality may be used.
In contrast in intensional Type Theory type checking is decidable, but the representation of standard mathematical concepts is somewhat complex, since extensional reasoning requires using setoids or similar constructions. It is a subject of current discussion whether this tradeoff is unavoidable and whether the lack of extensional principles in intensional Type Theory is a feature or a bug.
Implementations of type theory 
Type Theory has been the base of a number of proof assistants, such as NuPRL, LEGO and Coq. Recently, dependent types also featured in the design of programming languages such as ATS, Cayenne, Epigram and Agda.
See also 
Further reading 
- Bengt Nordström; Kent Petersson; Jan M. Smith (1990). Programming in Martin-Löf's Type Theory. Oxford University Press. The book is out of print, but a free version can be picked up from here.
- Thompson, Simon (1991). Type Theory and Functional Programming Addison-Wesley. ISBN 0-201-41667-0.
- Granström, Johan G. (2011). Treatise on Intuitionistic Type Theory Springer. ISBN 978-94-007-1735-0.
- EU Types Project: Tutorials - lecture notes and slides from the Types Summer School 2005
- n-Categories - Sketch of a Definition - letter from John Baez and James Dolan to Ross Street, November 29, 1995