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Mathematical formalism
lambda calculus is a formal mathematical system consisting of constructing lambda terms and performing reduction operations on them. The definition of
Lambda_calculus_definition
Mathematical-logic system
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
Simple Turing complete logic
these definitions it can be shown that SKI calculus, while being a minimalistic system, can fully perform any computations of the lambda calculus. All
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
Higher-order function which returns some fixed point of the input function
be defined in the lambda calculus and in functional programming languages, and provide a means to allow for recursive definitions. Applied to a non-constant
Fixed-point_combinator
Function definition that is not bound to an identifier
programming, an anonymous function (function literal, lambda function, or block) is a function definition that is not bound to an identifier. Anonymous functions
Anonymous_function
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
Globalization meta-process
expression. The definition used is extended to cover the substitution of expressions, from the definition given on the Lambda calculus page. The matching
Lambda_lifting
Representation of data of various types in lambda calculus
types of data in the lambda calculus. In the untyped lambda calculus the only primitive data type are functions, represented by lambda abstraction terms
Church_encoding
Relation specifying a rewrite for each object, compatible with a reduction relation
with the same label, for a slightly different labelled lambda calculus. An alternate definition changes the beta rule to an operation that finds the next
Reduction_strategy
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
Branch of mathematical analysis
applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives
Fractional_calculus
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)
Branch of mathematics
propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term calculus has variously
Calculus
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
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
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
Tensor index notation for tensor-based calculations
used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro
Ricci_calculus
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
Mathematical techniques used in probability theory and related fields
related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic
Malliavin_calculus
Specialized notation for multivariable calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various
Matrix_calculus
0 {\displaystyle \lambda \to 0} has functions commuting with 1-forms, which is the special case of high school differential calculus. For A = C [ t , t
Quantum_differential_calculus
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
Differential calculus on function spaces
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and
Calculus_of_variations
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
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
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)
Way to represent data types in the lambda calculus
a way to represent algebraic data types in the lambda calculus, following their syntactic definition without regard whether they are recursive or not
Mogensen–Scott_encoding
Instantaneous rate of change of the function
In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given
Directional_derivative
Mathematical logic concept
reads "It is not the case that (R is true and S is false)", which is the definition of a material conditional. We can then make this substitution: R → S {\displaystyle
Contraposition
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)
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
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
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
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
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
Form of continuity for functions
as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. For an equivalent definition in terms of measures see the section Relation between
Absolute_continuity
Branch of algebraic geometry
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert in order to solve various
Schubert_calculus
Concept in computer science
recursion. Dana Scott's LCF language was a stage in the evolution of lambda calculus into modern functional languages. This language introduced the let
Let_expression
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
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
standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness"
Explicit_substitution
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
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
Mathematical operation
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative
Second_derivative
Thesis on the nature of computability
would disavow Herbrand–Gödel recursion and the λ-calculus in favor of the Turing machine as the definition of "algorithm" or "mechanical procedure" or "formal
Church–Turing_thesis
Unification of discrete and continuous theories of calculus
time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with
Time-scale_calculus
Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis). Lambda-calculus is thus effectively
Computable_topology
Pair of mathematical objects
of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered
Ordered_pair
Computer programming language
Curry's paradox#Lambda calculus — about inconsistency problems caused by combining (propositional) logic and untyped lambda calculus Comparison of Prolog
ΛProlog
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
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
Special forms of a typed lambda calculus
and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies
System_U
Basic framework of mathematics
progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the natural and real numbers. This led
Foundations_of_mathematics
Matrix of second derivatives
\mathbf {H} (\Lambda )={\begin{bmatrix}{\dfrac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf
Hessian_matrix
Kind of proof calculus
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to
Natural_deduction
Non-commutative algebraic structure
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In modern terms, he
Icosian_calculus
Expression that may be integrated over a region
df(x)=f'(x)\,dx} ). This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special
Differential_form
Hilbert-style deductive systems for propositional logics. Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
Discrete analog of a derivative
including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals. Three basic types
Finite_difference
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
Recursion without calling a function by name
recursive definition of the factorial in lambda calculus (anonymous functions of a single variable): (lambda f: (lambda x: f(lambda v: x(x)(v))) (lambda x: f(lambda
Anonymous_recursion
Certain vector fields are the sum of an irrotational and a solenoidal vector field
the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the
Helmholtz_decomposition
Type of mathematical space
c_{\lambda }(x,c_{\mu }(y,z))=c_{\lambda \mu }\left(c_{\frac {\lambda (1-\mu )}{1-\lambda \mu }}(x,y),z\right)} (for λ μ ≠ 1 {\displaystyle \lambda \mu
Convex_space
Mathematical object for the lambda calculus
In the study of denotational semantics of the lambda calculus, Böhm trees, Lévy-Longo trees, and Berarducci trees are (potentially infinite) tree-like
Böhm_tree
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
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
Result about when a matrix can be diagonalized
{\displaystyle V_{\lambda }=\{v\in V:Av=\lambda v\}} be the eigenspace corresponding to an eigenvalue λ {\displaystyle \lambda } . Note that the definition does not
Spectral_theorem
for the simply typed lambda calculus. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic
Normalisation_by_evaluation
Field-equations in general relativity
0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.} Our simplifying assumptions
Einstein_field_equations
Syntactically correct logical formula
propositional calculus, also called propositional formulas, are expressions such as ( A ∧ ( B ∨ C ) ) {\displaystyle (A\land (B\lor C))} . Their definition begins
Well-formed_formula
Method to solve constrained optimization problems
( x ) + ⟨ λ , g ( x ) ⟩ {\displaystyle {\mathcal {L}}(x,\lambda )\equiv f(x)+\langle \lambda ,g(x)\rangle } for functions f , g {\displaystyle f,g} ;
Lagrange_multiplier
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
Complexity class used to classify decision problems
Turing machine. The first definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because
NP_(complexity)
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
Calculus of functions generalization
In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean
Calculus_on_Euclidean_space
Infinite cardinal number
the infinity ( ∞ {\displaystyle \infty } ) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly
Aleph_number
Technique for creating lexically scoped first class functions
interpreter for extended lambda calculus". "... a data structure containing a lambda expression, and an environment to be used when that lambda expression is applied
Closure (computer programming)
Closure_(computer_programming)
Algebraic manipulation of "true" and "false"
propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed
Boolean_algebra
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
programming, an anonymous function (function literal, lambda function, or block) is a function definition that is not bound to an identifier. Anonymous functions
Examples of anonymous functions
Examples_of_anonymous_functions
Programming style in which control is passed explicitly
the (=& n 0 (lambda (b) (if b ...))) call inside f-aux& definition above would be written instead as (=& n 0 (lambda () (k a)) (lambda () (-& n 1 ..
Continuation-passing_style
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
functional calculus, a − m = − ( λ − T ) m − 1 e λ ( T ) {\displaystyle a_{-m}=-(\lambda -T)^{m-1}e_{\lambda }(T)} where e λ {\displaystyle e_{\lambda }} is
Jordan_normal_form
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)
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
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
Family of linear transformations
Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}{\vphantom {{x'}^{0}}}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda
Lorentz_transformation
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
Tensor having both covariant and contravariant indices
vectors Einstein notation Ricci calculus Tensor (intrinsic definition) Two-point tensor D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill
Mixed_tensor
Exterior algebraic map taking tensors from p forms to n-p forms
{\displaystyle \lambda >0} , then the induced Hodge stars ⋆ g , ⋆ λ g : Λ n V → Λ n V {\displaystyle {\star }_{g},{\star }_{\lambda g}:\Lambda ^{n}V\to \Lambda ^{n}V}
Hodge_star_operator
All-encompassing set or class
decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine
Universe_(mathematics)
Area of mathematical logic
\lambda } such that T is λ {\displaystyle \lambda } -stable. T is λ {\displaystyle \lambda } -stable if and only if λ ℵ 0 = λ {\displaystyle \lambda ^{\aleph
Model_theory
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
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
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)
Specification of a derivative along a tangent vector of a manifold
\lambda ^{a}} , we have: λ a ; b ≡ ∂ b λ a + Γ a b c λ c {\displaystyle {\lambda ^{a}}_{;b}\equiv \partial _{b}\lambda ^{a}+{\Gamma ^{a}}_{bc}\lambda ^{c}}
Covariant_derivative
Indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function
Antiderivative
Intermediate language
constructs varies by author and desired use for the calculus, but the following constructs are typical: Lambdas λx.M are computations of type A → B _ {\displaystyle
Call-by-push-value
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
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LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
LAMBDA CALCULUS-DEFINITION
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