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Mathematical theory of data types
science, type theory is the study of formal systems that classify expressions or mathematical objects by their types. Roughly speaking, a type plays a
Type_theory
Alternative foundation of mathematics
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of
Intuitionistic_type_theory
Type theory in logic and mathematics
science, homotopy type theory (HoTT) includes various lines of development of intuitionistic type theory, based on the interpretation of types as objects to
Homotopy_type_theory
Personality hypothesis which describes two contrasting personality types
The Type A and Type B personality theory associates two contrasting personality types with different incidence of coronary heart disease. According to
Type A and Type B personality theory
Type_A_and_Type_B_personality_theory
Pseudoscientific personality questionnaire
Jung's book Psychological Types (first published in German as Psychologische Typen in 1921), Briggs recognized that Jung's theory resembled, but went far
Myers–Briggs_Type_Indicator
Logical system in mathematics
cubical type theory is a flavor of type theory which gives a computational interpretation to univalent foundations (also known as homotopy type theory). In
Cubical_type_theory
3-volume treatise on mathematics, 1910–1913
set theory at the turn of the 20th century, like Russell's paradox. This third aim motivated the adoption of the theory of types in PM. The theory of types
Principia_Mathematica
The type theory was initially created to avoid paradoxes in a variety of formal logics and rewrite systems. Later, type theory referred to a class of formal
History_of_type_theory
Pseudoscience linking character and blood type
The blood type personality theory is a pseudoscientific belief prevalent in East Asia that a person's blood type is predictive of a person's personality
Blood_type_personality_theory
Theory in the philosophy of mind
Type physicalism (also known as reductive materialism, type identity theory, mind–brain identity theory, and identity theory of mind) is a physicalist
Type_physicalism
Topics referred to by the same term
mathematical structure might behave Type, the subject of type theory Type, proposition or set in intuitionistic type theory Type, a numeric property of an entire
Type
Concept in model theory
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements
Type_(model_theory)
Type of types in a type system
science known as type theory, a kind is the type of a type constructor or, less commonly, the type of a higher-order type operator (type constructor). A
Kind_(type_theory)
Type whose definition depends on a value
dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent
Dependent_type
Aspect of theoretical physics
theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for
Type_II_string_theory
following system is Mendelson's (1997, 289–293) ST type theory. ST is equivalent with Russell's ramified theory plus the Axiom of reducibility. The domain of
ST_type_theory
of type theory involves several closely related kinds of models, which are constructed and studied in order to justify axioms and new type theories, and
Semantics_of_type_theory
In type theory, a discipline within mathematical logic, containers are abstractions which permit various "collection types", such as lists and trees,
Container_(type_theory)
Type theory with records is a formal semantics representation framework, using records to express type theory types. It has been used in natural language
Type_theory_with_records
Branch of mathematics that studies sets
First Introduction to Topos Theory, Springer-Verlag, ISBN 978-0-387-97710-2 homotopy type theory at the nLab Homotopy Type Theory: Univalent Foundations of
Set_theory
Computer science concept
a type. Even a type can become associated with a type. An implementation of a type system could in theory associate identifications called data type (a
Type_system
Area of mathematical logic
continuum). A theory of the first type is called unstable, a theory of the second type is called strictly stable and a theory of the third type is called
Model_theory
Formal system in mathematical logic
The simply typed lambda calculus ( λ → {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambda calculus with
Simply_typed_lambda_calculus
Extent to which a programming language discourages type errors
In computer science, type safety is the extent to which a programming language discourages or prevents type errors.[vague] Type-safe languages are sometimes
Type_safety
Branch of computer science
abstract typed functional language. In 1978, Robin Milner introduces the Hindley–Milner type system inference algorithm for ML language. Type theory became
Programming_language_theory
Linked node hierarchical data structure
value(node(e, f)) = e children(node(e, f)) = f In terms of type theory, a tree is an inductive type defined by the constructors nil (empty forest) and node
Tree_(abstract_data_type)
Aspect of theoretical physics
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings
Type_I_string_theory
Mathematical constructs and creation rules
In type theory, a system has inductive types if it has facilities for creating a new type from constants and functions that create terms of that type. The
Inductive_type
In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept of
Polynomial functor (type theory)
Polynomial_functor_(type_theory)
Formal system of logic
"simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types. Leon Chwistek and Frank P. Ramsey
Higher-order_logic
Notion of equality in type theory
In type theory, a branch of mathematics, the identity type represents the concept of equality. It is also known as propositional equality to differentiate
Identity_type
Classification of individuals based on personality traits
According to type theories, for example, introverts and extraverts are two fundamentally different categories of people. According to trait theories, introversion
Personality_type
Family of type systems based on substructural logic
Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only
Substructural_type_system
Theory of subatomic structure
string theory to another type of physical theory called a quantum field theory. One of the shortcomings of string theory is that the full theory does not
String_theory
Data type defined by combining other types
programming and type theory, an algebraic data type (ADT) is a composite data type, i.e. a type formed by combining other types. An algebraic data type is defined
Algebraic_data_type
Type that allows only one value
area of mathematical logic and computer science known as type theory, a unit type is a type that allows only one value (and thus can hold no information)
Unit_type
Axiomatic set theory devised by W.V.O. Quine
non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. The well-formed
New_Foundations
Style of dynamic typing in object-oriented programming
Structural typing is a static typing system that determines type compatibility and equivalence by a type's structure, whereas duck typing is dynamic and
Duck_typing
Transforming a function in such a way that it only takes a single argument
function calls. In type theory, the general idea of a type system in computer science is formalized into a specific algebra of types. For example, when
Currying
Universal subtype in logic and computer science
type theory, a theory within mathematical logic, the bottom type of a type system is the type that is a subtype of all other types. Where such a type
Bottom_type
Automatic detection of the type of an expression in a formal language
In type theory, type inference (sometimes called type reconstruction) is the automatic detection of the type of an expression. These include programming
Type_inference
Data type in type theory
the field of type theory in computer science, a quotient type is a data type that respects a user-defined equality relation. A quotient type defines an
Quotient_type
In type theory, a type with no terms
In type theory, an empty type or absurd type, typically denoted 0 {\displaystyle \mathbb {0} } is a type with no terms. Such a type may be defined as the
Empty_type
German professor of computer science
University of Nottingham known for his research on logic, type theory, and homotopy type theory. Altenkirch was part of the 2012/2013 special year on univalent
Thorsten_Altenkirch
Theory of strings with supersymmetry
superstring theories (Type I, Type IIA, Type IIB, HO and HE) are regarded as different limits of a single theory tentatively called M-theory. One of the
Superstring_theory
Mathematical concept
version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory. Univalent foundations
Univalent_foundations
Swedish logician, philosopher, and mathematical statistician
in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science
Per_Martin-Löf
Using one interface or symbol with regards to multiple different types
In programming language theory and type theory, polymorphism allows a value or variable to have more than one type and allows a given operation to be performed
Polymorphism (computer science)
Polymorphism_(computer_science)
Association of atoms to form chemical compounds
matter. All bonds can be described by quantum theory, but, in practice, simplified rules and other theories allow chemists to predict the strength, directionality
Chemical_bond
Axiom of set theory
of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does. The type theoretical context
Axiom_of_choice
Basic framework of mathematics
axiomatic method and on set theory, specifically Zermelo–Fraenkel set theory with the axiom of choice. Foundations based on type theory have also gained prevalence
Foundations_of_mathematics
Feature of a typed formal language that builds new types from old ones
science known as type theory, a type constructor is a feature of a typed formal language that builds new types from old ones. Basic types are considered
Type_constructor
Attribute of data
types in a library. C data types Data dictionary Kind Type (model theory) Type theory for the mathematical models of types Type conversion ISO/IEC 11404
Data_type
Formalism in computer science
also be considered the more fundamental theory and untyped lambda calculus a special case with only one type. Typed lambda calculi are foundational programming
Typed_lambda_calculus
Types constrained by a predicate
In type theory, a refinement type is a type endowed with a predicate which is assumed to hold for any element of the refined type. Refinement types can
Refinement_type
Supposition or system of ideas intended to explain something
theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of
Theory
Branch of mathematical logic
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating
Proof_theory
Programming language concept
though that could violate type safety. These terms come from the notion of covariant and contravariant functors in category theory. Consider the category
Type_variance
Data type that refers to itself in its definition
equal – that is, those two type expressions are understood to denote the same type. In fact, most theories of equirecursive types go further and essentially
Recursive_data_type
All-encompassing set or class
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains
Universe_(mathematics)
Proof assistant and programming language
constructions with inductive types (specifically, the Calculus of Inductive Constructions), the foundational type theory developed with the Coq theorem
Lean_(proof_assistant)
American mathematician (1937–2025)
Transfinite Type Theory with Type Variables. North Holland Publishing Company, Amsterdam. Andrews, Peter B. (1971). "Resolution in type theory". Journal
Peter_B._Andrews
In type theory, a type system is said to have the principal type property if, given a term and an environment, there exists a principal type for this
Principal_type
epimorphism). Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf. topos). Allegory theory provides a generalization
List_of_types_of_functions
Mathematical technique used in proof theory
a theory T {\displaystyle T} is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can
Ordinal_analysis
Branch of psychology focused on personality
in degree. For example, in type theory, there are two types of people: introverts and extroverts. According to trait theories, introversion and extroversion
Personality_psychology
Form of logic that allows quantification over predicates
logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals
Second-order_logic
Result of multiplying types in type theory
programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure
Product_type
Symbol used in mathematics and logic
element in wheel theory and lattice theory, which also represents absurdum when used for logical semantics The bottom type in type theory, which is the bottom
Up_tack
Wiki for mathematics, physics, and philosophy
physics, and philosophy with a focus on methods from type theory, category theory, and homotopy theory. The nLab espouses the "n-point of view" (a deliberate
NLab
Distinguishing objects and classes of objects
between using a word and mentioning it Type theory – Mathematical theory of data types Type physicalism – Theory in the philosophy of mind Brekle, Herbert
Type–token_distinction
Data type for values having two types
In type theory, an intersection type can be allocated to values that can be assigned both the type σ {\displaystyle \sigma } and the type τ {\displaystyle
Intersection_type
Academic subfield of computer science
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation
Theory_of_computation
Topics referred to by the same term
domain Top type, in computer science type theory, the data type containing all others Top of stack, the first element of a Stack (abstract data type) The Opportunity
Top
Universal type in logic and computer science
In type theory and computer science, type systems include a top, universal, or any type (often represented with the down tack (⊤) symbol), which includes
Any_type
Computer programming function
collection, e.g. a list or set, returning the results in a collection of the same type. It is often called apply-to-all when considered in functional form. The
Map_(higher-order_function)
Sequence of data items available over time
starting condition of the stream. Streams can be used as the underlying data type for channels in interprocess communication. The term stream is also applied
Stream_(computing)
Type system used in computer programming and mathematics
A Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or
Hindley–Milner_type_system
Finite ordered list of elements
record type. Both of these types can be defined as simple extensions of the simply typed lambda calculus. The notion of a tuple in type theory and that
Tuple
Functional programming language
is based on Zhaohui Luo's unified theory of dependent types (UTT), a type theory similar to Martin-Löf type theory. Agda is named after the Swedish song
Agda_(programming_language)
Topics referred to by the same term
applied to time series data Exponential type Exponential type or function type, in type theory Exponential type in complex analysis Topics listed at list
Exponential
Mathematical construction of a set with an equivalence relation
set, or extensional set. Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one
Setoid
Typological term
of abstract, hypothetical concepts. The "ideal type" is therefore a subjective element in social theory and research, and one of the subjective elements
Ideal_type
P\to Q} is a method turning any proof of P into a proof of Q. Used in: type theory, constructive mathematics. Founder(s): K F. Gödel (1933) showed that
Constructive_logic
Branch of type theory
logic, the intersection type discipline is a branch of type theory encompassing type systems that use the intersection type constructor ( ∩ ) {\displaystyle
Intersection_type_discipline
Field theory involving topological effects in physics
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes
Topological quantum field theory
Topological_quantum_field_theory
Axiomatic set theories based on the principles of mathematical constructivism
classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed
Constructive_set_theory
Feature of some programming languages
values of the data type. In statically typed languages, a nullable type is an option type,[citation needed] while in dynamically typed languages (where
Nullable_type
Topics referred to by the same term
a scientific journal Environment (type theory), the association between variable names and data types in type theory Deployment environment, in software
Environment
American mathematician (born 1959)
category theory and logic, and has also written on the philosophy of mathematics. He is one of the originators of the field of homotopy type theory. He was
Steve_Awodey
System of formal mathematical logic
formulation of the simply typed lambda calculus, and provides a foundation for mathematics comparable to first-order logic plus set theory. It is a form of higher-order
Q0_(mathematical_logic)
Defines the inputs and outputs for a function, subroutine or method
science, a type signature or type annotation defines the inputs and outputs of a function, subroutine or method.[citation needed] A type signature includes
Type_signature
Kind of polymorphism
In programming language type theory, row polymorphism is a kind of polymorphism that allows one to write programs that are structurally (rather than nominally)
Row_polymorphism
Basis of generic programming
languages and type theory, parametric polymorphism allows a single piece of code to be given a "generic" type, using variables in place of actual types, and then
Parametric_polymorphism
Relationship between programs and proofs
proof system and as a typed programming language based on functional programming. This includes Martin-Löf's intuitionistic type theory and Coquand's calculus
Curry–Howard_correspondence
Study of computable functions and Turing degrees
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated
Computability_theory
Topics referred to by the same term
consequences of an action. Tree (set theory), historically called a ramification system Type theory, Ramified Theory of Types by mathematician Bertrand Russell
Ramification
Concept in mathematical logic
intuitionistic type theory (ITT), a discipline within mathematical logic, induction-recursion is a feature for simultaneously declaring a type and function
Induction-recursion
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
The Type 0 string theory is a less well-known model of string theory. It is a superstring theory in the sense that the worldsheet theory is supersymmetric
Type_0_string_theory
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