Basics

Functions

In type theory, functions are terms. Functions can either be lambda terms or defined "by rule".

Common "by rule" types and terms

Type theories are defined by their rules of inference. There are rules for a "§ functional core", described above, and rules that create types and terms. Below is a non-exhaustive list of common types and their associated terms.

The list ends with "§ inductive types", which is a powerful technique that is able to construct all the other ones in the list. The mathematical foundations used by the proof assistants "Coq" and "Lean" are based on the "Calculus for Inductive Constructions" which is the "Calculus of Constructions" (its "functional core") with inductive types.

Terms

A term in logic is recursively defined as a constant symbol, variable, or a function application, where a term is applied to another term. Some constant symbols will be "0" of the natural numbers, "true" of the Booleans, and functions like "S" and "if". Thus some terms are "0", "(S 0)", "(S (S x))", and "if true 0 (S 0)".

Judgements

Most type theories have 4 judgements:

• "${\displaystyle T}$ is a type"
• "${\displaystyle t}$ is a term of type ${\displaystyle T}$"
• "Type ${\displaystyle T_{1}}$ is equal to type ${\displaystyle T_{2}}$"
• "Terms ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ both of type ${\displaystyle T}$ are equal"

The judgements can be made under an assumption. For example, one might say "assuming ${\displaystyle x}$ is a term of type 'bool' and ${\displaystyle y}$ is a term of type 'nat' , (if x y y) is a term of type 'nat' ". The mathematical notation for assumptions is a comma-separated list of "term : type" that precedes the turnstile symbol '${\displaystyle \vdash }$'. Thus, the example statement is formally written:

• x:bool, y:nat ${\displaystyle \vdash }$ (if x y y) : nat

If there are no assumptions, there will be nothing to the left of the turnstile:

• ${\displaystyle \vdash }$ S : nat ${\displaystyle \to }$ nat

The list of assumptions is called the "context". It is very common to see the symbol ${\displaystyle \Gamma }$ used to represent some or all of the assumptions. Thus, the formal notation for the 4 different judgements is usually:

Formal notation for judgements	Description
${\displaystyle \Gamma \vdash T}$ Type	${\displaystyle T}$ is a type (under assumptions ${\displaystyle \Gamma }$).
${\displaystyle \Gamma \vdash t:T}$	${\displaystyle t}$ is a term of type ${\displaystyle T}$ (under assumptions ${\displaystyle \Gamma }$).
${\displaystyle \Gamma \vdash T_{1}=T_{2}}$	Type ${\displaystyle T_{1}}$ is equal to type ${\displaystyle T_{2}}$ (under assumptions ${\displaystyle \Gamma }$).
${\displaystyle \Gamma \vdash t_{1}=t_{2}:T}$	Terms ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ are both of type ${\displaystyle T}$ and are equal (under assumptions ${\displaystyle \Gamma }$).

(NOTE: The judgement of equality of terms is where the phrase "judgemental equality" comes from. )

The judgements enforce that every term has a type. The type will restrict which rules can be applied to a term.

Rules

A type theory's rules say what judgements can be made, based on the existence of other judgements. The rules are expressed using a horizontal line, with the required input judgements above the line and the resulting judgement below the line. The rule for creating a lambda term is:

$${\displaystyle {\begin{array}{c}\Gamma ,a:A\vdash b:B\\\hline \Gamma \vdash (\lambda a:A.b):A\to B\\\end{array}}}$$

The judgements required to create the lambda term go above the line. In this case, only one judgement is required: it is that there is some term "b" of some type "B", assuming there is some term "a" of some type "A" and some other assumptions "${\displaystyle \Gamma }$". (Note: "${\displaystyle \Gamma }$", "a", "A", "b", and "B" are all metavariables in the rule.) The resulting judgement goes below the line. This rule's resulting judgement states that the new lambda term has type "A ${\displaystyle \to }$ B" under the other assumptions ${\displaystyle \Gamma }$.

The rules are syntactic and work by rewriting. Thus, the metavariables like "${\displaystyle \Gamma }$", "a", "A", etc. may actually consist of complex terms that contain many function applications, not just single symbols.

To generate a particular judgement in type theory, there must be a rule to generate it. Then, there must be rules to generate all of that rule's required inputs. And then rules for all the inputs for those rules. The applied rules form a proof tree. This is usually drawn Gentzen-style,^[7] where the target judgement (root) is at the bottom, and rules that do not require any inputs (leaves) are at the top (see Natural deduction §Proofs and type theory). One example of a rule that does not require any inputs is one that states there is a term "0" of type "nat":

$${\displaystyle {\begin{array}{c}\hline \vdash 0:nat\\\end{array}}}$$

A type theory usually has several rules, including ones to:

• create a context
• add an assumption to the context ("weakening")
• rearrange the assumptions
• use an assumption to create a variable
• define reflexivity, symmetry and transitivity for judgmental equality
• define substitution for application of lambda terms
• list all the interactions of equality, substitution, etc.
• define universes

Also, for each "by rule" type, there are 4 different kinds of rules

• "type formation" rules say how to create the type
• "term introduction" rules define the canonical terms and constructor functions, like "pair" and "S".
• "term elimination" rules define the other functions like "first", "second", and "R".
• "computation" rules specify how computation is performed with the type-specific functions.

Examples of rules:

• Rules to Martin-Löf's Intuitionistic Type Theory^[8]
• Appendix A.2 of the Homotopy Type Theory book^[9]

Properties of type theories

Terms usually belong to a single type. However, there are set theories that define "subtyping".

Computation takes place by repeated application of rules. Many types of theories are strongly normalizing, which means that any order of applying the rules will always end in the same result. However, some are not. In a normalizing type theory, the one-directional computation rules are called "reduction rules", and applying the rules "reduces" the term. If a rule is not one-directional, it is called a "conversion rule".

Some combinations of types are equivalent to other combinations of types. When functions are considered "exponentiation", the combinations of types can be written similarly to algebraic identities.^[10] Thus, ${\displaystyle {\mathbb {0} }+A\cong A}$, ${\displaystyle {\mathbb {1} }\times A\cong A}$, ${\displaystyle {\mathbb {1} }+{\mathbb {1} }\cong {\mathbb {2} }}$, ${\displaystyle A^{B+C}\cong A^{B}\times A^{C}}$, ${\displaystyle A^{B\times C}\cong (A^{B})^{C}}$.

Axioms

Most type theories do not have axioms. This is because a type theory is defined by its rules of inference. (See "Rules" above). This is a source of confusion for people familiar with Set Theory, where a theory is defined by both the rules of inference for a logic (such as first-order logic) and axioms about sets.

Sometimes, a type theory will add a few axioms. An axiom is a judgement that is accepted without a derivation using the rules of inference. They are often added to ensure properties that cannot be added cleanly through the rules.

Axioms can cause problems if they introduce terms without a way to compute on those terms. That is, axioms can interfere with the normalizing property of the type theory.^[11]

Some commonly encountered axioms are:

• "Axiom K" ensures "uniqueness of identity proofs". That is, that every term of an identity type is equal to reflexivity.^[12]
• "Univalence Axiom" holds that equivalence of types is equality of types. The research into this property led to cubical type theory, where the property holds without needing an axiom.^[13]
• "Law of Excluded Middle" is often added to satisfy users who want classical logic, instead of intuitionistic logic.

The Axiom of Choice does not need to be added to type theory, because in most type theories it can be derived from the rules of inference. This is because of the constructive nature of type theory, where proving that a value exists requires a method to compute the value. The Axiom of Choice is less powerful in type theory than most set theories, because type theory's functions must be computable and, being syntax-driven, the number of terms in a type must be countable. (See Axiom of choice § In constructive mathematics.)

Decision problems

A type theory is naturally associated with the decision problem of type inhabitation.^[14]

Research areas

Homotopy type theory differs from intuitionistic type theory mostly by its handling of the equality type. In 2016 cubical type theory was proposed, which is a homotopy type theory with normalization.^[16][13]

Types are propositions; terms are proofs

When used as a foundation, certain types are interpreted as propositions (statements that can be proven) and a term of the type is a proof of that proposition. Thus, the type "Π x:nat . x+1=1+x" represents that, for any "x" of type "nat", "x+1" and "1+x" are equal. And a term of that type represents its proof.

Curry-Howard correspondence

The Curry–Howard correspondence is the observed similarity between logics and programming languages. The implication in logic, "A ${\displaystyle \to }$ B" resembles a function from type "A" to type "B". For a variety of logics, the rules are similar to expressions in a programming language's types. The similarity goes farther, as applications of the rules resemble programs in the programming languages. Thus, the correspondence is often summarized as "proofs as programs".

The logic operators "for all" and "exists" led Per Martin-Löf to invent dependent type theory.

Intuitionistic logic

When some types are interpreted as propositions, there is a set of common types that can be used to connect them to make a logic out of types. However, that logic is not classical logic but intuitionistic logic. That is, it does not have the law of excluded middle nor double negation.

There is a natural relation of types to logical propositions. If "A" is a type representing a proposition, being able to create a function of type "${\displaystyle \top \to }$ A" indicates that A has a proof and being able to create the function "A ${\displaystyle \to \bot }$" indicates that A does not have a proof. That is, inhabitable types are proven and uninhabitable types are disproven.

WARNING: This interpretation can lead to a lot of confusion. A type theory may have terms "true" and "false" of type "bool", which act like a Boolean logic, and at the same time have types ${\displaystyle \top }$ and ${\displaystyle \bot }$ to represent "true" (provable) and "false" (disproven), as part of an intuitionistic logic for proposition.

Under this intuitionistic interpretation, there are common types that act as the logical operators:

Logic Name	Logic Notation	Type Notation	Type Name
True	${\displaystyle \top }$	${\displaystyle \top }$	Unit Type
False	${\displaystyle \bot }$	${\displaystyle \bot }$	Empty Type
Not	${\displaystyle \neg A}$	${\displaystyle A\to \bot }$	Function to Empty Type
Implication	${\displaystyle A\to B}$	${\displaystyle A\to B}$	Function
And	${\displaystyle A\land B}$	${\displaystyle A\times B}$	Product Type
Or	${\displaystyle A\lor B}$	${\displaystyle A+B}$	Sum Type
For All	${\displaystyle \forall a\in A,P(a)}$	Π a : A . P(a)	Dependent Function
Exists	${\displaystyle \exists a\in A,P(a)}$	Σ a : A . P(a)	Dependent Product Type

But under this interpretation, there is no law of excluded middle. That is, there is no term of type Π A . A + (A ${\displaystyle \to \bot }$).

Likewise, there is no double negation. There is no term of type Π A . ((A ${\displaystyle \to \bot }$) ${\displaystyle \to \bot }$) ${\displaystyle \to }$ A. (Note: Intuitionistic logic does allow ${\displaystyle \lnot \lnot \lnot A\to \lnot A}$ and there is a term of type (((A ${\displaystyle \to \bot }$) ${\displaystyle \to \bot }$) ${\displaystyle \to \bot }$) ${\displaystyle \to }$ (A ${\displaystyle \to \bot }$).)

Thus, the logic-of-types is an intuitionistic logic. Type theory is often cited as an implementation of the Brouwer–Heyting–Kolmogorov interpretation.

It is possible to include the law of excluded middle and double negation into a type theory, by rule or assumption. However, terms may not compute down to canonical terms and it will interfere with the ability to determine if two terms are judgementally equal to each other.

Constructive mathematics

Per Martin-Löf proposed his intuitionistic type theory as a foundation for constructive mathematics. Constructive mathematics requires when proving "There exists an ${\displaystyle x}$ with property P(${\displaystyle x}$)", there must be a particular ${\displaystyle x}$ and a proof that it has property "P". In type theory, existence is accomplished using the dependent product type and, its proof, requires a term of that type. For the term ${\displaystyle t}$, "first ${\displaystyle t}$" will produce the ${\displaystyle x}$ and "second ${\displaystyle t}$" will produce the proof of P(${\displaystyle x}$).

An example of a non-constructive proof is a "proof by contradiction". The first step is assuming that ${\displaystyle x}$ does not exist and refuting it by contradiction. The conclusion from that step is "it is not the case that ${\displaystyle x}$ does not exist". The last step is, by double negation, concluding that ${\displaystyle x}$ exists. To be clear, constructive mathematics still allows "refute by contradiction". It can prove that "it is not the case that ${\displaystyle x}$ does not exist". But constructive mathematics does not allow the last step of removing the double negation to conclude that ${\displaystyle x}$ exists.^[17]

Constructive mathematics has often used intuitionistic logic, as evidenced by the Brouwer–Heyting–Kolmogorov interpretation.

Most of the type theories proposed as foundations are constructive. This includes most of the ones used by proof assistants.

It is possible to add non-constructive features to a type theory, by rule or assumption. These include operators on continuations such as call with current continuation. However, these operators tend to break desirable properties such as canonicity and parametricity.

Category theory

Although the initial motivation for category theory was far removed from foundationalism, the two fields turned out to have deep connections. As John Lane Bell writes: "In fact categories can themselves be viewed as type theories of a certain kind; this fact alone indicates that type theory is much more closely related to category theory than it is to set theory." In brief, a category can be viewed as a type theory by regarding its objects as types (or sorts), i.e. "Roughly speaking, a category may be thought of as a type theory shorn of its syntax." A number of significant results follow in this way:^[18]

• cartesian closed categories correspond to the typed λ-calculus (Lambek, 1970);
• C-monoids (categories with products and exponentials and one non-terminal object) correspond to the untyped λ-calculus (observed independently by Lambek and Dana Scott around 1980);
• locally cartesian closed categories correspond to Martin-Löf type theories (Seely, 1984).

The interplay, known as categorical logic, has been a subject of active research since then; see the monograph of Jacobs (1999) for instance.

Homotopy type theory attempts to combine type theory and category theory. It focuses on equalities, especially equalities between types.

Major

• Simply typed lambda calculus which is a higher-order logic
• intuitionistic type theory
• system F
• LF is often used to define other type theories
• calculus of constructions and its derivatives

Minor

• Automath
• ST type theory
• UTT (Luo's Unified Theory of dependent Types)
• some forms of combinatory logic
• others defined in the lambda cube (also known as pure type systems)
• others under the name typed lambda calculus

Active research

• Homotopy type theory explores equality of types
• Cubical Type Theory is an implementation of homotopy type theory

Mathematical foundations

The first computer proof assistant, called Automath, used type theory to encode mathematics on a computer. Martin-Löf specifically developed intuitionistic type theory to encode all mathematics to serve as a new foundation for mathematics. There is ongoing research into mathematical foundations using homotopy type theory.

Mathematicians working in category theory already had difficulty working with the widely accepted foundation of Zermelo–Fraenkel set theory. This led to proposals such as Lawvere's Elementary Theory of the Category of Sets (ETCS).^[19] Homotopy type theory continues in this line using type theory. Researchers are exploring connections between dependent types (especially the identity type) and algebraic topology (specifically homotopy).

Proof assistants

Much of the current research into type theory is driven by proof checkers, interactive proof assistants, and automated theorem provers. Most of these systems use a type theory as the mathematical foundation for encoding proofs, which is not surprising, given the close connection between type theory and programming languages:

• LF is used by Twelf, often to define other type theories;
• many type theories which fall under higher-order logic are used by the HOL family of provers and PVS;
• computational type theory is used by NuPRL;
• calculus of constructions and its derivatives are used by Coq, Matita, and Lean;
• UTT (Luo's Unified Theory of dependent Types) is used by Agda which is both a programming language and proof assistant

Many type theories are supported by LEGO and Isabelle. Isabelle also supports foundations besides type theories, such as ZFC. Mizar is an example of a proof system that only supports set theory.

Programming languages

Any static program analysis, such as the type checking algorithms in the semantic analysis phase of compiler, has a connection to type theory. A prime example is Agda, a programming language which uses UTT (Luo's Unified Theory of dependent Types) for its type system.

The programming language ML was developed for manipulating type theories (see LCF) and its own type system was heavily influenced by them.

Linguistics

Type theory is also widely used in formal theories of semantics of natural languages,^[20][21] especially Montague grammar^[22] and its descendants. In particular, categorial grammars and pregroup grammars extensively use type constructors to define the types (noun, verb, etc.) of words.

The most common construction takes the basic types ${\displaystyle e}$ and ${\displaystyle t}$ for individuals and truth-values, respectively, and defines the set of types recursively as follows:

• if ${\displaystyle a}$ and ${\displaystyle b}$ are types, then so is ${\displaystyle \langle a,b\rangle }$;
• nothing except the basic types, and what can be constructed from them by means of the previous clause are types.

A complex type ${\displaystyle \langle a,b\rangle }$ is the type of functions from entities of type ${\displaystyle a}$ to entities of type ${\displaystyle b}$. Thus one has types like ${\displaystyle \langle e,t\rangle }$ which are interpreted as elements of the set of functions from entities to truth-values, i.e. indicator functions of sets of entities. An expression of type ${\displaystyle \langle \langle e,t\rangle ,t\rangle }$ is a function from sets of entities to truth-values, i.e. a (indicator function of a) set of sets. This latter type is standardly taken to be the type of natural language quantifiers, like everybody or nobody (Montague 1973, Barwise and Cooper 1981).

Social sciences

Gregory Bateson introduced a theory of logical types into the social sciences; his notions of double bind and logical levels are based on Russell's theory of types.

Introductory material

• Type Theory at nLab, which has articles on many topics.
• Intuitionistic Type Theory article at the Stanford Encyclopedia of Philosophy
• Lambda Calculi with Types book by Henk Barendregt
• Calculus of Constructions / Typed Lambda Calculus textbook style paper by Helmut Brandl
• Intuitionistic Type Theory notes by Per Martin-Löf
• Programming in Martin-Löf's Type Theory book
• Homotopy Type Theory book, which proposed homotopy type theory as a mathematical foundation.

Advanced material

• Robert L. Constable (ed.). "Computational type theory". Scholarpedia.
• The TYPES Forum — moderated e-mail forum focusing on type theory in computer science, operating since 1987.
• The Nuprl Book: "Introduction to Type Theory."
• Types Project lecture notes of summer schools 2005–2008
  • The 2005 summer school has introductory lectures
• Oregon Programming Languages Summer School, many lectures and some notes.
  • Summer 2013 lectures including Robert Harper's talks on YouTube
  • Summer 2015 Types, Logic, Semantics, and Verification
• Andrej Bauer's blog