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Mathematical-logic system based on functions
In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and
Lambda_calculus
Formalism in computer science
and computer science, a typed lambda calculus is a typed formalism that uses the lambda symbol ( λ {\displaystyle \lambda } ) to denote anonymous function
Typed_lambda_calculus
Formal system in mathematical logic
simply typed lambda calculus ( λ → {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambda calculus with only
Simply_typed_lambda_calculus
Semi-fictional hacking organization
Knights of the Lambda Calculus is a semi-fictional organization of expert Lisp and Scheme hackers. The name refers to the lambda calculus, a mathematical
Knights of the Lambda Calculus
Knights_of_the_Lambda_Calculus
Mathematical formalism
The lambda calculus is a formal mathematical system consisting of constructing lambda terms and performing reduction operations on them. The definition
Lambda_calculus_definition
Simple Turing complete logic
version of the untyped lambda calculus. It was introduced by Moses Schönfinkel and Haskell Curry. All operations in lambda calculus can be encoded via abstraction
SKI_combinator_calculus
Logical formalism using combinators instead of variables
computation. Combinatory logic can be viewed as a variant of the lambda calculus, in which lambda expressions (representing functional abstraction) are replaced
Combinatory_logic
Extension of lambda calculus
mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by Michel Parigot. It introduces two
Lambda-mu_calculus
Relationship between programs and proofs
known as lambda calculus. Actually, Howard's first formulation of the isomorphism was referred to (a variant of) Gentzen's sequent calculus. The observation
Curry–Howard_correspondence
Framework in lambda calculus
(also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions
Lambda_cube
Typed lambda calculus
polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism
System_F
Mathematical paradox
language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic. The paradox is named after the logician Haskell
Curry's_paradox
Eleventh letter in the Greek alphabet
the concepts of lambda calculus. λ indicates an eigenvalue in the mathematics of linear algebra. In the physics of particles, lambda indicates the thermal
Lambda
Programming paradigm based on applying and composing functions
the lambda calculus and Turing machines are equivalent models of computation, showing that the lambda calculus is Turing complete. Lambda calculus forms
Functional_programming
Relation specifying a rewrite for each object, compatible with a reduction relation
z)((\lambda w.www)(\lambda w.www)(\lambda w.www)(\lambda w.www))\\\rightarrow &(\lambda x.z)((\lambda w.www)(\lambda w.www)(\lambda w.www)(\lambda w.www)(\lambda
Reduction_strategy
Symbolic description of a mathematical object
the basis for lambda calculus, a formal system used in mathematical logic and programming language theory. The equivalence of two lambda expressions is
Expression_(mathematics)
Type whose definition depends on a value
extensional. In 1934, Haskell Curry noticed that the types used in typed lambda calculus, and in its combinatory logic counterpart, followed the same pattern
Dependent_type
Dialect of Lisp
evaluation of "closed" Lambda expressions in LISP and ISWIM's Lambda Closures. van Tonder, André (1 January 2004). "A Lambda Calculus for Quantum Computation"
Scheme_(programming_language)
Graphical model of computation
Interaction nets are at the heart of many implementations of the lambda calculus, such as efficient closed reduction and optimal, in Lévy's sense, Lambdascope
Interaction_nets
Theoretical computer model
where the calculus is extended to numbers and addition (even though both numbers and addition can be encoded entirely in the lambda calculus). Each component
CEK_Machine
Type theory created by Thierry Coquand
predicative calculus of inductive constructions (which removes some impredicativity).[citation needed] The CoC is a higher-order typed lambda calculus, initially
Calculus_of_constructions
Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis). Lambda-calculus is thus effectively
Computable_topology
Subset of lambda calculus
computer science, kappa calculus is a formal system for defining first-order functions. Unlike lambda calculus, kappa calculus has no higher-order functions;
Kappa_calculus
Ability of a computing system to simulate Turing machines
contrast with Turing machines. Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not. AI-completeness Algorithmic information
Turing_completeness
Expression that cannot be rewritten further
systems of typed lambda calculus including the simply typed lambda calculus, Jean-Yves Girard's System F, and Thierry Coquand's calculus of constructions
Normal form (abstract rewriting)
Normal_form_(abstract_rewriting)
Mathematical theory of data types
conjunction with Alonzo Church's lambda calculus. One notable early example of type theory is Church's simply typed lambda calculus. Church's theory of types
Type_theory
Representation of natural numbers and other data types in lambda calculus
data types in the lambda calculus. In the untyped lambda calculus the only primitive data type are functions, represented by lambda abstraction terms
Church_encoding
Computer programming language
2023). "Functional Bits: Lambda Calculus based Algorithmic Information Theory" (PDF). tromp.github.io. John's Lambda Calculus and Combinatory Logic Playground
Binary_combinatory_logic
Various systems of symbolic logic
extended Curry–Howard correspondence between IPC and simply typed lambda calculus. BHK interpretation Computability logic Constructive analysis Constructive
Intuitionistic_logic
Theorem in theoretical computer science
In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does
Church–Rosser_theorem
Programming language for experimentation or art
being Befunge-93, named as such because of its release year. Binary lambda calculus is designed from an algorithmic information theory perspective to allow
Esoteric_programming_language
Branch of computer science
theory predates even the development of programming languages. The lambda calculus, developed by Alonzo Church and Stephen Cole Kleene in the 1930s, is
Programming_language_theory
Problem in computer science
Church published his proof of the undecidability of a problem in the lambda calculus. Turing's proof was published later, in January 1937. Since then, many
Halting_problem
lexical scope was similar to the lambda calculus. Sussman and Steele decided to try to model Actors in the lambda calculus. They called their modeling system
History of the Scheme programming language
History_of_the_Scheme_programming_language
Standard representation of a mathematical object
system. In the untyped lambda calculus, for example, the term ( λ x . ( x x ) λ x . ( x x ) ) {\displaystyle (\lambda x.(xx)\;\lambda x.(xx))} does not have
Canonical_form
Topics referred to by the same term
function, is a defined function not bound to an identifier. Lambda expression in lambda calculus, a formal system in mathematical logic and computer science
Lambda_expression
Branch of functional analysis
functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras
Borel_functional_calculus
Natural number
numerical value of true is equal to 1 in many programming languages. In lambda calculus and computability theory, natural numbers are represented by Church
1
Association of one output to each input
name of type in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus. History of the function
Function_(mathematics)
American mathematician and computer scientist (1903–1995)
foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem
Alonzo_Church
Control flow statement that branches according to a Boolean expression
people won!"); } else { console.log("It's a three-way tie!"); } In Lambda calculus, the concept of an if-then-else conditional can be expressed using
Conditional (computer programming)
Conditional_(computer_programming)
Higher-order function Y for which Y f = f (Y f)
\mathrm {Y} =\lambda f.\ (\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))} (Here using the standard notations and conventions of lambda calculus: Y is a function
Fixed-point_combinator
Symbols for constants, special functions
compensation for the risk borne in investment the α-conversion in lambda calculus the independence number of a graph a placeholder for ordinal numbers
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Process calculus
In theoretical computer science, the π-calculus (or pi-calculus) is a process calculus. The π-calculus allows channel names to be communicated along the
Π-calculus
In lambda calculus, a term is in beta normal form if no beta reduction is possible. A term is in beta-eta normal form if neither a beta reduction nor
Beta_normal_form
Extension of propositional modal logic
in the variable Z {\displaystyle Z} , much like in lambda calculus λ Z . ϕ {\displaystyle \lambda Z.\phi } is a function with formula ϕ {\displaystyle
Modal_μ-calculus
Topics referred to by the same term
to computational theory Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus Rho calculus, introduced as a general means
Calculus_(disambiguation)
Function definition that is not bound to an identifier
The names "lambda abstraction", "lambda function", and "lambda expression" refer to the notation of function abstraction in lambda calculus, where the
Anonymous_function
Formal study of linguistic meaning
semantics employs the typed lambda calculus to analyze the denotations of parts of sentences. Using the typed lambda calculus, one can formalize the denotation
Formal semantics (natural language)
Formal_semantics_(natural_language)
Computer programming for quantum computers
Maymin, "Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms", 1996 van Tonder, André. "A lambda calculus for quantum computation
Quantum_programming
Computation model defining an abstract machine
(UTM, or simply a universal machine). Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church
Turing_machine
Esoteric programming languages
simpler than other more popular alternatives, such as lambda calculus and SKI combinator calculus. Thus, they can also be considered minimalist computer
Iota_and_Jot
Globalization meta-process
untyped lambda calculus. See also intensional versus extensional equality. The reverse operation to lambda lifting is lambda dropping. Lambda dropping
Lambda_lifting
Transforming a function in such a way that it only takes a single argument
functions have exactly one argument. This property is inherited from lambda calculus, where multi-argument functions are usually represented in curried
Currying
Programming style in which control is passed explicitly
a Yoneda embedding. It is also similar to the embedding of lambda calculus in π-calculus. Outside of computer science, CPS is of more general interest
Continuation-passing_style
Software optimization technique
most[quantify] programming languages. Lazy evaluation was introduced for lambda calculus by Christopher Wadsworth. For programming languages, it was independently
Lazy_evaluation
Mathematical notation in lambda calculus
mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices
De_Bruijn_index
General purpose functional programming language
conceptually a combination of the first-order predicate calculus and the simply typed polymorphic lambda calculus, was the underlying language that theorem statements
ML_(programming_language)
Way to represent data types in the lambda calculus
Scott encoding is a way to represent algebraic data types in the lambda calculus, following their syntactic definition without regard whether they are
Mogensen–Scott_encoding
Category of formal programming language semantics
first formal incarnation of operational semantics was the use of the lambda calculus to define the semantics of Lisp. Abstract machines in the tradition
Operational_semantics
and algebraic laws, that is, to the algebraic study of data types. Lambda calculus-based languages (such as Lisp, ISWIM, and Scheme) are in actual practice
Value-level_programming
response, Gödel's incompleteness proof, Turing's machine and Church's Lambda calculus showed that there were, in fact, limits to what formal mathematics
History of artificial intelligence
History_of_artificial_intelligence
Academic subfield of computer science
Church–Turing thesis) models of computation are in use. Lambda calculus A computation consists of an initial lambda expression (or two if you want to separate the
Theory_of_computation
Evaluation of a function on its argument
function abstraction. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has
Function_application
Dutch computer scientist
combinatory logic (binary lambda calculus) [citation needed] and lambda diagrams that supply a graphical way of representing lambda calculus expressions. Shotwell
John_Tromp
Impossible task in computing
by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis. The origin
Entscheidungsproblem
satisfiability and model checking Type inhabitation problem for simply typed lambda calculus Integer circuit evaluation Word problem for linear bounded automata
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Combinatory logic system
the propositional axiom F → A. Combinatory logic SKI combinator calculus Lambda calculus To Mock a Mockingbird Raymond Smullyan (1994) Diagonalization and
B,_C,_K,_W_system
Function that takes one or more functions as an input or that outputs a function
Functor (disambiguation). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming
Higher-order_function
Concept in mathematics or computer science
x_{n})\mapsto t\right]} is directly analogous to lambda expressions in lambda calculus, where the λ {\displaystyle \lambda } symbol is the fundamental variable-binding
Free variables and bound variables
Free_variables_and_bound_variables
American mathematician (1900-1982)
systems, including one proposed by Alonzo Church (a system that had the lambda calculus as a consistent subsystem) and Curry's own system. However, unlike
Haskell_Curry
Form of typed lambda calculus
as is the case with the calculus of constructions, but this is not generally the case, e.g. the simply typed lambda calculus allows only terms to depend
Pure_type_system
Mathematical function that can be computed by a program
proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very
Computable_function
Programming language family
(though not originally derived from) the notation of Alonzo Church's lambda calculus. It quickly became a favored programming language for artificial intelligence
Lisp_(programming_language)
functions originate in the work of Alonzo Church in his invention of the lambda calculus, in which all functions are anonymous, in 1936, before electronic computers
Examples of anonymous functions
Examples_of_anonymous_functions
Symbol in mathematical logic
n {\displaystyle B_{1},\,\dots ,B_{n}} must be true. In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing
Turnstile_(symbol)
Formal system of logic
Higher-order logic programming HOL (proof assistant) Many-sorted logic Typed lambda calculus Modal logic Jacobs, 1999, chapter 5 Shapiro 1991, p. 87. Menachem Magidor
Higher-order_logic
English computer scientist (1912–1954)
(as indeed could Church's lambda calculus). According to the Church–Turing thesis, Turing machines and the lambda calculus are capable of computing anything
Alan_Turing
Branch of mathematical analysis
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number
Fractional_calculus
for their applications: e.g., Alonzo Church was able to express the lambda calculus in a formulaic way, and the Turing machine was an abstraction of the
History of programming languages
History_of_programming_languages
Type of interpreter in computing
self-evaluator for the λ {\displaystyle \lambda } calculus. The abstract syntax of the λ {\displaystyle \lambda } calculus is implemented as follows in OCaml
Meta-circular_evaluator
Programming language evaluation rules
have terminated without error. The name "normal order" comes from the lambda calculus, where normal order reduction will find a normal form if there is one
Evaluation_strategy
Type system used in computer programming and mathematics
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
logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility
Scott–Curry_theorem
Branch of mathematics relating to posets
the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus
Domain_theory
Expression denoting a set of sets in formal semantics
write complex functions is the lambda calculus. For example, one can write the meaning of sleeps as the following lambda expression, which is a function
Generalized_quantifier
Theoretical model of computation
concepts related to lambda calculus, namely head reduction and call by name. A redex (one says also β-redex) is a term of the lambda calculus of the form (λ
Krivine_machine
Ability to solve a problem by an effective procedure
computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of
Computability
treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per
Logical_framework
interpretation Curry–Howard correspondence Linear logic Game semantics Typed lambda calculus Typed and untyped languages Type signature Type inference Datatype
List of functional programming topics
List_of_functional_programming_topics
included some existing theories with simply typed lambda calculus at the lowest corner and the calculus of constructions at the highest. Prior to 1994,
History_of_type_theory
Decision problem pertaining to equivalence of expressions
Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm that can discern whether
Word_problem_(mathematics)
inhabitant of τ {\displaystyle \tau } . In the case of simply typed lambda calculus, a type has an inhabitant if and only if its corresponding proposition
Type_inhabitation
Variable that represents an argument to a function
lambda calculus, each function has exactly one parameter. What is thought of as functions with multiple parameters is usually represented in lambda calculus
Parameter (computer programming)
Parameter_(computer_programming)
Type of category in category theory
of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal
Cartesian_closed_category
Computer programming language
Curry's paradox#Lambda calculus — about inconsistency problems caused by combining (propositional) logic and untyped lambda calculus Comparison of Prolog
ΛProlog
to computational theory Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus Rho calculus, introduced as a general means
List_of_formal_systems
Type of logical system
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system used in mathematics, philosophy
First-order_logic
the lambda calculus were expressed using variable substitution in which the values of parameters were substituted into the body of an invoked lambda expression
History_of_the_Actor_model
Rules to verify computer program correctness
calculus for a simple while language. j-Algo Hoare Calculus module (j-Algo on GitHub, j-Algo on SourceForge) – A visualisation of the Hoare calculus in
Hoare_logic
LAMBDA CALCULUS
LAMBDA CALCULUS
Girl/Female
Indian
Ambitious
Girl/Female
Indian
Praiseworthy, Praiser of Allah
Female
Greek
(Λαμία) Greek myth name of an evil spirit who abducts and devours children, LAMIA means "large shark." The name means "vampire" in Latin and "fiend" in Arabic.
Girl/Female
Indian
Flame
Girl/Female
Muslim
Dark lipped
Girl/Female
Indian
Soft to touch
Surname or Lastname
English
English : from Middle English lamb, a nickname for a meek and inoffensive person, or a metonymic occupational name for a keeper of lambs. See also Lamm.English : from a short form of the personal name Lambert.Irish : reduced Anglicized form of Gaelic Ó Luain (see Lane 3). MacLysaght comments: ‘The form Lamb(e), which results from a more than usually absurd pseudo-translation (uan ‘lamb’), is now much more numerous than O’Loan itself.’Possibly also a translation of French agneau.
Girl/Female
Muslim
Soft to touch
Female
Native American
Native American Indian name ALAMEDA means "grove of cottonwood."
Surname or Lastname
English
English : from a pet form of Lamb 1 and 2.English : from an Old Norse personal name Lambi, from lamb ‘lamb’.
Girl/Female
Indian
Dark lipped
Girl/Female
Arabic, Indian, Muslim, Pashtun, Sanskrit
Flame; Large; Spacious; Tall; Another Name for Durga and Lakshmi
Female
Italian
Italian form of English Amber, AMBRA means "amber."
Female
Spanish
Feminine form of Spanish Amado, AMADA means "beloved."
Boy/Male
Hindu
Lord Ganesh, The huge bellied Lord
Girl/Female
Muslim
Ambitious
Boy/Male
Indian
Jaws.
Girl/Female
Muslim
Flame
Girl/Female
Muslim
Praiseworthy, Praiser of Allah
Surname or Lastname
English
English : habitational name from Lambden in Berwickshire.
LAMBDA CALCULUS
LAMBDA CALCULUS
Boy/Male
Indian, Sanskrit
With a Beautiful Form
Surname or Lastname
English
English : apparently a habitational name, probably a variant of Tenley.
Male
Norse
Old Norse name derived from a diminutive form of *anuR, ÃKI means "father."
Boy/Male
Hindu
Small
Boy/Male
Indian, Sanskrit
Incarnation of God; The Descent
Boy/Male
Scottish
Son of the smith.
Surname or Lastname
English (Northumberland)
English (Northumberland) : variant of Blackie.
Girl/Female
Tamil
Boat
Girl/Female
Arabic, Australian, Farsi, Hebrew, Muslim
Life; Found; Evident; Few
Boy/Male
Australian, British, English
Light
LAMBDA CALCULUS
LAMBDA CALCULUS
LAMBDA CALCULUS
LAMBDA CALCULUS
LAMBDA CALCULUS
a.
Shaped like the Greek letter lambda (/); as, the lambdoid suture between the occipital and parietal bones of the skull.
n.
A lamb.
n.
A viola da gamba.
n.
A thin plate or lamina.
n.
A thin plate or scale; a layer or coat lying over another; -- said of thin plates or platelike substances, as of bone or minerals.
n.
The blade of a leaf; the broad, expanded portion of a petal or sepal of a flower.
n.
Any person who is as innocent or gentle as a lamb.
n.
A monster capable of assuming a woman's form, who was said to devour human beings or suck their blood; a vampire; a sorceress; a witch.
n.
A lamb.
p. pr. & vb. n.
of Lamb
v. i.
To bring forth a lamb or lambs, as sheep.
a.
Lamed; lame; disabled; impeded.
pl.
of Lamina
n.
The name of the Greek letter /, /, corresponding with the English letter L, l.
n.
The lamb's-quarters (Chenopodium album).
imp. & p. p.
of Lamb
n.
A thin plate or scale; specif., one of the thin, flat processes composing the vane of a feather.
n.
A lamp or candlestick.
pl.
of Lamina
n.
The point of junction of the sagittal and lambdoid sutures of the skull.