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Dependent type, the Glossary

In computer science and logic, a dependent type is a type whose definition depends on a value.^[1]

69 relations: Ada (programming language), Agda (programming language), ATS (programming language), C (programming language), Calculus of constructions, Cartesian product, Codomain, Combinatory logic, Computer science, Constructive proof, Coq (software), Curry–Howard correspondence, Decision problem, Dependent ML, Dual (category theory), Epigram (programming language), Equality (mathematics), F Sharp (programming language), F* (programming language), First-order logic, Formal verification, Function (computer programming), Function type, Functional programming, Generalized quantifier, Generic programming, GitHub, Haskell, Haskell Curry, Henk Barendregt, Idris (programming language), If and only if, Intuitionistic logic, Intuitionistic type theory, Journal of Functional Programming, Kind (type theory), Lambda cube, Lean (proof assistant), Lennart Augustsson, Logic, Logic programming, Logical framework, Matita, Model theory, Natural number, Nicolaas Govert de Bruijn, Nuprl, OCaml, Oxford University Press, Polymorphism (computer science), ... Expand index (19 more) »
Dependently typed programming
Foundations of mathematics
Type systems

Ada (programming language)

Ada is a structured, statically typed, imperative, and object-oriented high-level programming language, inspired by Pascal and other languages.

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Agda (programming language)

Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis.

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ATS (programming language)

In computing, ATS (Applied Type System) is a multi-paradigm, general-purpose, high-level, functional programming language.

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C (programming language)

C (pronounced – like the letter c) is a general-purpose programming language.

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In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. Dependent type and calculus of constructions are dependently typed programming and type theory.

See Dependent type and Calculus of constructions

Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.

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Codomain

In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall.

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Combinatory logic

Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic.

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Computer science

Computer science is the study of computation, information, and automation.

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Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object.

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Coq (software)

Coq is an interactive theorem prover first released in 1989.

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Curry–Howard correspondence

In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. Dependent type and Curry–Howard correspondence are dependently typed programming and type theory.

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Decision problem

In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values.

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Dependent ML

Dependent ML is an experimental, multi-paradigm, general-purpose, high-level, functional programming language proposed by Hongwei Xi and Frank Pfenning.

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Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop.

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Epigram (programming language)

Epigram is a functional programming language with dependent types, and the integrated development environment (IDE) usually packaged with the language.

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Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

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F Sharp (programming language)

F# (pronounced F sharp) is a general-purpose, high-level, strongly typed, multi-paradigm programming language that encompasses functional, imperative, and object-oriented programming methods.

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F* (programming language)

F* (pronounced F star) is a high-level, multi-paradigm, functional and object-oriented programming language inspired by the languages ML, Caml, and OCaml, and intended for program verification.

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First-order logic

First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

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Formal verification

In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics.

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Function (computer programming)

In computer programming, a function, procedure, method, subroutine, routine, or subprogram is a callable unit of software logic that has a well-defined interface and behavior and can be invoked multiple times.

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Function type

In computer science and mathematical logic, a function type (or arrow type or exponential) is the type of a variable or parameter to which a function has or can be assigned, or an argument or result type of a higher-order function taking or returning a function. Dependent type and function type are type theory.

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Functional programming

In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions.

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Generalized quantifier

In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets.

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Generic programming

Generic programming is a style of computer programming in which algorithms are written in terms of data types to-be-specified-later that are then instantiated when needed for specific types provided as parameters.

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GitHub

GitHub is a developer platform that allows developers to create, store, manage and share their code.

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Haskell

Haskell is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation.

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Haskell Curry

Haskell Brooks Curry (September 12, 1900 – September 1, 1982) was an American mathematician and logician.

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Henk Barendregt

Hendrik Pieter (Henk) Barendregt (born 18 December 1947, Amsterdam) is a Dutch logician, known for his work in lambda calculus and type theory.

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Idris (programming language)

Idris is a purely-functional programming language with dependent types, optional lazy evaluation, and features such as a totality checker.

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If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

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Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

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Intuitionistic type theory

Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory, the latter abbreviated as MLTT) is a type theory and an alternative foundation of mathematics. Dependent type and Intuitionistic type theory are dependently typed programming, foundations of mathematics and type theory.

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Journal of Functional Programming

The Journal of Functional Programming is a peer-reviewed scientific journal covering the design, implementation, and application of functional programming languages, spanning the range from mathematical theory to industrial practice.

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Kind (type theory)

In the area of mathematical logic and computer 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. Dependent type and kind (type theory) are type theory.

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Lambda cube

In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Dependent type and lambda cube are type theory.

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Lean (proof assistant)

Lean is a proof assistant and a functional programming language.

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Lennart Augustsson

Lennart Augustsson is a Swedish computer scientist.

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Logic

Logic is the study of correct reasoning.

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Logic programming

Logic programming is a programming, database and knowledge representation paradigm based on formal logic.

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Logical framework

In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory. Dependent type and logical framework are dependently typed programming and type theory.

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Matita

Matita is an experimental proof assistant under development at the Computer Science Department of the University of Bologna.

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Model theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

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Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

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Nicolaas Govert de Bruijn

Nicolaas Govert "Dick" de Bruijn (9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic.

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Nuprl

Nuprl is a proof development system, providing computer-mediated analysis and proofs of formal mathematical statements, and tools for software verification and optimization.

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OCaml

OCaml (formerly Objective Caml) is a general-purpose, high-level, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features.

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Oxford University Press

Oxford University Press (OUP) is the publishing house of the University of Oxford.

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Polymorphism (computer science)

In programming language theory and type theory, polymorphism is the use of a single symbol to represent multiple different types. Dependent type and polymorphism (computer science) are type theory.

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Predicate (mathematical logic)

In logic, a predicate is a symbol that represents a property or a relation.

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Product type

In programming languages and type theory, a product of types is another, compounded, type in a structure. Dependent type and product type are type theory.

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Proof assistant

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration.

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Proof-carrying code

Proof-carrying code (PCC) is a software mechanism that allows a host system to verify properties about an application via a formal proof that accompanies the application's executable code. Dependent type and proof-carrying code are dependently typed programming.

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Propositional calculus

The propositional calculus is a branch of logic.

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Prototype Verification System

The Prototype Verification System (PVS) is a specification language integrated with support tools and an automated theorem prover, developed at the Computer Science Laboratory of SRI International in Menlo Park, California.

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Pure type system

In the branches of mathematical logic known as proof theory and type theory, a pure type system (PTS), previously known as a generalized type system (GTS), is a form of typed lambda calculus that allows an arbitrary number of sorts and dependencies between any of these. Dependent type and pure type system are type theory.

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Purely functional programming

In computer science, purely functional programming usually designates a programming paradigm—a style of building the structure and elements of computer programs—that treats all computation as the evaluation of mathematical functions.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Scheme (programming language)

Scheme is a dialect of the Lisp family of programming languages.

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Simply typed lambda calculus

The simply typed lambda calculus (\lambda^\to), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor (\to) that builds function types. Dependent type and simply typed lambda calculus are type theory.

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System F

System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. Dependent type and System F are type theory.

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Termination analysis

In computer science, termination analysis is program analysis which attempts to determine whether the evaluation of a given program halts for each input.

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Twelf

Twelf is an implementation of the logical framework LF developed by Frank Pfenning and Carsten Schürmann at Carnegie Mellon University. Dependent type and Twelf are type theory.

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Type system

In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a ''type'' (for example, integer, floating point, string) to every term (a word, phrase, or other set of symbols). Dependent type and type system are type systems and type theory.

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Type theory

In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system.

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Typed lambda calculus

A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. Dependent type and typed lambda calculus are type theory.

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Universe (mathematics)

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.

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William Alvin Howard

William Alvin Howard (born 1926) is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence.

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References

[1] https://en.wikipedia.org/wiki/Dependent_type

Also known as Comparision of programming languages with dependent types, Comparison of dependently typed programming languages, Comparison of programming languages with dependent types, Dependent type theory, Dependent types, Dependent typing, Dependently typed, Dependently typed language, Dependently typed programming, Dependently typed programming language, Dependently-typed, Dependently-typed language, Dependently-typed programming language, List of dependently-typed programming languages, Sigma type, Type-level programming, ΛΠ-calculus.

, Predicate (mathematical logic), Product type, Proof assistant, Proof-carrying code, Propositional calculus, Prototype Verification System, Pure type system, Purely functional programming, Real number, Scheme (programming language), Simply typed lambda calculus, System F, Termination analysis, Twelf, Type system, Type theory, Typed lambda calculus, Universe (mathematics), William Alvin Howard.

Dependent type, the Glossary

Table of Contents

See also

Dependently typed programming

Foundations of mathematics

Type systems

References