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 propositional 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.
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 -tuples of real numbers, stands for the type of functions which given a natural number returns a tuple of real numbers of this size. 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.
-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 2nd component depends on the first. E.g. the type stands for the type of pairs of a natural number and a tuple of real numbers of that size, i.e. this type can be used to model sequences of arbitrary length (usually called lists). The conventional Cartesian product type arises as a special case when the type of the 2nd 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 . Again, using the Curry-Howard isomorphism, -types also serve to model conjunction and existential quantification.
Using finite types we can define negation as .
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 .
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.
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 . This hierarchy is often assumed to be cumulative, that is the codes from are embedded in .
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.
Type Theory is usually presented as a dependently typed lambda calculus, using the judgements:
Of special importance is the conversion rule, which says that given and then .
Using the language of category theory, 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 : C^op -> 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.
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 contrast in intensional Type Theory type checking is decidable, but the representation of many mathematical concepts is non-standard due to a lack of extensional reasoning. 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.
Type Theory has been the base of a number of proof assistants, such as NuPRL, LEGO, Coq, and Agda. Recently, dependent types also featured in the design of programming languages such as Dependent ML, Cayenne and Epigram.