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Formalism in computer science
mathematics 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
Mathematical-logic system
cube: Typed lambda calculus – Lambda calculus with typed variables (and functions) System F – A typed lambda calculus with type-variables Calculus of constructions
Lambda_calculus
Framework in lambda calculus
\;\vdash \;\lambda x.t:\sigma \to \tau }}} In System F (also named λ2 for the "second-order typed lambda calculus") there is another type of abstraction
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
Type whose definition depends on a value
\mathbb {N} \to \mathbb {R} } in typed lambda calculus. For a more concrete example, taking A {\displaystyle A} to be the type of unsigned integers from 0
Dependent_type
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)
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
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
Extension of lambda calculus
rules are the same as simply typed lambda calculus. The next 2 are new to the lambda-mu calculus. Simply typed lambda calculus, by the Curry–Howard correspondence
Lambda-mu_calculus
Subset of lambda calculus
first class objects. Kappa-calculus can be regarded as "a reformulation of the first-order fragment of typed lambda calculus". Because its functions are
Kappa_calculus
Feature of a typed formal language that builds new types from old ones
applications of unary type operators. Therefore, we can view the type operators as a simply typed lambda calculus, which has only one basic type, usually denoted
Type_constructor
Form of typed lambda calculus
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
Pure_type_system
Relationship between programs and proofs
deduction and typed combinatory logic, Howard made explicit in 1969 a syntactic analogy between the programs of simply typed lambda calculus and the proofs
Curry–Howard_correspondence
Mathematical theory of data types
typed lambda calculus. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus
Type_theory
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
described 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
uninhabited types. For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for simply typed lambda calculus the type inhabitation
Type_inhabitation
semantics Typed lambda calculus Typed and untyped languages Type signature Type inference Datatype Algebraic data type (generalized) Type variable First-class
List of functional programming topics
List_of_functional_programming_topics
Inference rule defining when a program phrase has a type
later used in proofs about the type system. Consider a version of the simply typed lambda calculus with booleans. Its types and expressions can be described
Typing_rule
Special forms of a typed lambda calculus
mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and
System_U
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
theories with simply typed lambda calculus at the lowest corner and the calculus of constructions at the highest. Prior to 1994, many type theorists thought
History_of_type_theory
Higher-order function which returns some fixed point of the input function
number of different areas: General mathematics Untyped lambda calculus Typed lambda calculus Functional programming Imperative programming Fixed-point
Fixed-point_combinator
Logical formalism using combinators instead of variables
reduction of a typed lambda term, and conversely. Moreover, theorems can be identified with function type signatures. Specifically, a typed combinatory logic
Combinatory_logic
Computer science concept
under the slogan: "Abstract [data] types have existential type". The theory is a second-order typed lambda calculus similar to System F, but with existential
Type_system
System of formal mathematical logic
Q0 is Peter Andrews' formulation of the simply typed lambda calculus, and provides a foundation for mathematics comparable to first-order logic plus set
Q0_(mathematical_logic)
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
Representation of data of various types in lambda calculus
various types of data in the lambda calculus. In the untyped lambda calculus the only primitive data type are functions, represented by lambda abstraction
Church_encoding
Mathematical methods
formulas. Kreisel introduced modified realizability, which uses typed lambda calculus as the language of realizers. Modified realizability is one way
Realizability
same type system. A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax
Logical_framework
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
Type of category in category theory
language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable
Cartesian_closed_category
American computer scientist (born 1946)
proof that the type inhabitation problem in simply typed lambda calculus is PSPACE-complete, lower bounds on simply typed lambda calculus, logical relations
Richard_Statman
Computational problem with high complexity
first-order logic β-convertibility of two closed terms in simply typed lambda calculus ACK-complete problems: reachability in vector addition systems (VAS)
Nonelementary_problem
axioms 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
Value-level_programming
The simply typed lambda-calculus, on the other hand, has both of these properties. Generally speaking, type systems based on intersection types also have
Principal_type
theorem Simply typed lambda calculus Typed lambda calculus Curry–Howard isomorphism Calculus of constructions Constructivist analysis Lambda cube System
List of mathematical logic topics
List_of_mathematical_logic_topics
American mathematician (1926–2026)
demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He
William_Alvin_Howard
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)
Family of type systems based on substructural logic
typed lambda calculus is the language of Cartesian closed categories. More precisely, one may construct functors between the category of linear type systems
Substructural_type_system
Typed functional language
be considered as an extended version of the typed lambda calculus, or a simplified version of modern typed functional languages such as ML or Haskell.
Programming Computable Functions
Programming_Computable_Functions
Ability of a computing system to simulate Turing machines
Turing-complete. The untyped lambda calculus is Turing-complete, but many typed lambda calculi, including System F, are not. The value of typed systems is based in
Turing_completeness
Subdiscipline of proof theory
intuitionistic logic and types in typed lambda calculus. In this correspondence, every proposition can be viewed as a type, and a proof of that proposition
Structural_proof_theory
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
Formal semantics (natural language)
Formal_semantics_(natural_language)
Topics referred to by the same term
STLC may refer to: Simply typed lambda calculus Software testing life cycle (disambiguation) The St. Louis Cardinals, a professional baseball team based
STLC
Globalization meta-process
. In the untyped lambda calculus, where the basic types are functions, lifting may change the result of beta reduction of a lambda expression. The resulting
Lambda_lifting
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
Standard representation of a mathematical object
(\lambda x.(xx)\;\lambda x.(xx))} does not have a normal form. In the typed lambda calculus, every well-formed term can be rewritten to its normal form. In
Canonical_form
Association of one output to each input
under the 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_(mathematics)
Type of types in a type system
essentially a simply typed lambda calculus "one level up", endowed with a primitive type, usually denoted ∗ {\displaystyle *} and called "type", which is the
Kind_(type_theory)
Automatic detection of the type of an expression in a formal language
Is there any example of a T? This is known as type inhabitation. For the simply typed lambda calculus, all three questions are decidable. The situation
Type_inference
Basis of generic programming
extends simply typed lambda calculus with quantification over types. It is possible to write functions that do not depend on the types of their arguments
Parametric_polymorphism
Various systems of symbolic logic
an 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
the lambda calculus, such as the simply typed lambda calculus, many calculi with advanced type systems, and Gordon Plotkin's beta-value calculus. Plotkin
Church–Rosser_theorem
Interactive theorem prover software
calculus of inductive constructions. Theorem Proving System (TPS) and ETPS – Interactive theorem provers also based on simply typed lambda calculus,
Proof_assistant
Programming paradigm based on applying and composing functions
simply typed lambda calculus, which extended the lambda calculus by assigning a data type to all terms. This forms the basis for statically typed functional
Functional_programming
Transforming a function in such a way that it only takes a single argument
typed lambda calculus is the internal language of cartesian closed categories; and it is for this reason that pairs and lists are the primary types in
Currying
influenced by the functional style of programming. Combinatory logic Typed lambda calculus Cartesian closed category Applicative computing systems Anonymous
Categorical_abstract_machine
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
higher-kinded type). In theoretical settings and programming languages where functions are defined in curried form, such as the simply typed lambda calculus, a function
Function_type
Mathematical paradox
}}X{\mbox{ and }}((mX)Z)\\\end{array}}} In simply typed lambda calculus, fixed-point combinators cannot be typed and hence are not admitted. Curry's paradox
Curry's_paradox
Branch of type theory
{\displaystyle (\vdash _{\text{CD}})} extends the simply typed λ-calculus by allowing multiple types to be assumed for a term variable. The term language
Intersection_type_discipline
Belgian logician and philosopher (1889-1961)
1958 Feys and Haskell B. Curry devised the type inference algorithm for the simply typed lambda calculus (Combinatory Logic). Haskell B. Curry, Robert
Robert_Feys
In functional programming
in a partial function application. In the simply typed lambda calculus with function and product types (λ→,×) partial application, currying and uncurrying
Partial_application
Form of type polymorphism
allow the subtyping of records. Consequently, simply typed lambda calculus extended with record types is perhaps the simplest theoretical setting in which
Subtyping
Concept in functional programming
syntax in a type safe fashion. Here is an embedding of the simply typed lambda calculus with an arbitrary collection of base types, product types (tuples)
Generalized algebraic data type
Generalized_algebraic_data_type
Formal languages for expressing mathematical theories
adopted and/or reinvented in areas such as typed lambda calculus and explicit substitution. Dependent types is one outstanding example. Automath was also
Automath
Function definition that is not bound to an identifier
function type as literals do for other data types. Anonymous functions originate in the work of Alonzo Church in his invention of the lambda calculus, in which
Anonymous_function
From the formal point of view, TIL is a hyperintensional, partial, typed lambda calculus. TIL applications cover a wide range of topics from formal semantics
Transparent_intensional_logic
French computer scientist
unification algorithm for simply typed lambda calculus, and of a complete proof method for Church's theory of types (constrained resolution). He worked
Gérard_Huet
Symbols for constants, special functions
shear stress in continuum mechanics a type variable in type theories, such as the simply typed lambda calculus path tortuosity in reservoir engineering
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Symbolic logic system
formula of this restricted minimal logic corresponds to a type in the simply typed lambda calculus, see Curry–Howard correspondence. This implicational fragment
Minimal_logic
Way to represent data types in the lambda calculus
science, Scott encoding is a way to represent algebraic data types in the lambda calculus, following their syntactic definition without regard whether
Mogensen–Scott_encoding
Alternative foundation of mathematics
of choices and so there is no specific type theory associated with it. Intuitionistic logic Typed lambda calculus Bertot, Yves; Castéran, Pierre (2004)
Intuitionistic_type_theory
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
temporal logic satisfiability and model checking Type inhabitation problem for simply typed lambda calculus Integer circuit evaluation Word problem for linear
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
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
Family of formalisms in natural language syntax
shares some features with the simply typed lambda calculus. Whereas the lambda calculus has only one function type A → B {\displaystyle A\rightarrow B}
Categorial_grammar
Mathematical object for the lambda calculus
of the lambda calculus, such as a typed lambda calculus. This naive assignment of meaning is however inadequate for the full lambda calculus. The term
Böhm_tree
binding, ranging in complexity from the simple binders of simply typed lambda calculus to complex, potentially infinite binders needed in the treatment
POPLmark_challenge
List of unsolved computational problems
397–405. The RTA list of open problems – Open problems in rewriting. The TLCA List of Open Problems – Open problems in the area of typed lambda calculus.
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
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
Symbol in mathematical logic
B_{n}} must be true. In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing judgment. In category theory
Turnstile_(symbol)
Type of interpreter in computing
in any of the typed lambda calculi such as the simply typed lambda calculus, Jean-Yves Girard's System F, or Thierry Coquand's calculus of constructions
Meta-circular_evaluator
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
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
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
Theorem in formal logic
the appropriate system. For proof systems based on higher-order typed lambda calculus through a Curry–Howard isomorphism, cut elimination algorithms correspond
Cut-elimination_theorem
Logical generalization for symbolic expressions
Simply typed lambda calculus (Input: Terms in the eta-long beta-normal form. Output: Various fragments of the simply typed lambda calculus including
Anti-unification
higher-order abstract syntax, where the latter uses the simply typed lambda calculus as a metalanguage. Many interesting calculi, logics and programming
Nominal terms (computer science)
Nominal_terms_(computer_science)
Functor mapping hom objects to an underlying category
famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal
Hom_functor
the meta-theory: there is no realizer in the language of simply typed lambda calculus as this language is not Turing-complete and arbitrary loops cannot
Markov's_principle
Finite ordered list of elements
has a 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
Tuple
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
{\displaystyle f:A\rightarrow B} . These notions extend directly to lambda calculus and type theory, respectively. These are functions that operate on functions
List_of_types_of_functions
Formal system of logic
Second-order logic Type theory Higher-order grammar Higher-order logic programming HOL (proof assistant) Many-sorted logic Typed lambda calculus Modal logic
Higher-order_logic
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
Academic discipline
and programs. In particular it showed that terms in the simply typed lambda calculus correspond to proofs of intuitionistic propositional logic. Category
Logic_in_computer_science
Logical quantification that ranges over a subset of the universe of discourse
grounds. Subtyping — bounded quantification in type theory System F<: — a polymorphic typed lambda calculus with bounded quantification Hinman, P. (2005)
Bounded_quantifier
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