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simply typed lambda calculus
A Wisdom Archive on simply typed lambda calculus

simply typed lambda calculus A selection of articles related to simply typed lambda calculus

More material related to Simply Typed Lambda Calculus can be found here:

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simply typed lambda calculus

ARTICLES RELATED TO simply typed lambda calculus

simply typed lambda calculus: Encyclopedia II - Fixed point combinator - Existence of fixed point combinators

In certain formalizations of mathematics, such as the untyped lambda calculus and combinatorial calculus, every expression can be considered a higher-order function. In these formalizations, the existence of a fixed-point combinator means that every function has at least one fixed point; a function may have more than one distinct fixed point. In some other systems, for example the simply typed lambda calculus, a well-typed fixed-point combinator cannot be written -- in those systems any support for recursion must be explicitly ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Existence of fixed point combinators

simply typed lambda calculus: Encyclopedia II - Intuitionistic Type Theory - Connectives of Type Theory

In the context of Type Theory a connective is a way of constructing types, possibly using already given types. The basic connectives of Type Theory are: Intuitionistic Type Theory - Π-types. Π-types, also called dependent function types, generalize the normal function space to model functions whose result type may vary on their input. E.g. writing for n-tuples of real numbers, stands for the type of functions wh ...

See also:

Intuitionistic Type Theory, Intuitionistic Type Theory - Connectives of Type Theory, Intuitionistic Type Theory - $Π$ -types, Intuitionistic Type Theory - $Σ$ -types, Intuitionistic Type Theory - Finite types, Intuitionistic Type Theory - Equality type, Intuitionistic Type Theory - Inductive types, Intuitionistic Type Theory - Universes, Intuitionistic Type Theory - Formalisation of Type Theory, Intuitionistic Type Theory - Categorical models of Type Theory, Intuitionistic Type Theory - Extensional versus intensional, Intuitionistic Type Theory - Implementations of Type Theory

Read more here: » Intuitionistic Type Theory: Encyclopedia II - Intuitionistic Type Theory - Connectives of Type Theory

simply typed lambda calculus: Encyclopedia II - Type inference - Hindley-Milner type inference algorithm

The common algorithm used to perform the type inference is the one now commonly referred to as Hindley-Milner or Damas-Milner algorithm. The origin of this algorithm is the type inference algorithm for the simply typed lambda calculus, which was devised by Haskell B. Curry and Robert Feys in 1958. In 1969 Roger Hindley extended this work and proved that their algorithm always inferred the most general type. In 1978 Robin Milner, independently of Hindley's work, provided an equivalent algorithm, In 1985 Luis Damas finally proved that Milner's algorithm i ...

See also:

Type inference, Type inference - Example, Type inference - Hindley-Milner type inference algorithm, Type inference - The Algorithm, Type inference - External link

Read more here: » Type inference: Encyclopedia II - Type inference - Hindley-Milner type inference algorithm

simply typed lambda calculus: Encyclopedia II - Calculus of constructions - The basics of the calculus of constructions

The Calculus of Constructions can be considered an extension of the Curry-Howard isomorphism. The Curry-Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will a ...

See also:

Calculus of constructions, Calculus of constructions - The basics of the calculus of constructions, Calculus of constructions - Terms, Calculus of constructions - Judgements, Calculus of constructions - Inference rules for calculus of constructions, Calculus of constructions - Defining logical operators, Calculus of constructions - Defining data types

Read more here: » Calculus of constructions: Encyclopedia II - Calculus of constructions - The basics of the calculus of constructions

simply typed lambda calculus: Encyclopedia II - Type inference - Example

For example, let us consider the Haskell function length, which may be defined as: length [] = 0 length (first:rest) = 1 + length rest From this, it is evident that the function handles lists as inputs, and the base case of this recursive function returns an integer (Haskell "Int"). So we can reliably construct a type signature length :: [a] -> Int Since there are no ad-hoc polymorphic subfunctions in the function definition, we can declare ...

See also:

Type inference, Type inference - Example, Type inference - Hindley-Milner type inference algorithm, Type inference - The Algorithm, Type inference - External link

Read more here: » Type inference: Encyclopedia II - Type inference - Example

simply typed lambda calculus: Encyclopedia II - Fixed point combinator - Other fixed point combinators

A version of the Y combinator that can be used in call-by-value (applicative-order) evaluation is given by η-expansion of part of the ordinary Y combinator: Z = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) The Y combinator can be expressed in the SKI-calculus as Y = S (K (S I I)) (S (S (K S) K) (K (S I I))) The simplest fixed point combinator in the SK-calculus, found by John Tromp, is Y = S S K (S (K (S S (S (S S K)))) K) which corresponds to the lambda expres ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Other fixed point combinators

simply typed lambda calculus: Encyclopedia II - Intuitionistic Type Theory - Formalisation of Type Theory

Type Theory is usually presented as a dependently typed lambda calculus, using the judgements: , Γ is a well-formed context of typing assumptions. , σ is a well-formed type in context Γ. , t is a well-formed term of type σ in context Γ. , σ and ...

See also:

Intuitionistic Type Theory, Intuitionistic Type Theory - Connectives of Type Theory, Intuitionistic Type Theory - $Π$ -types, Intuitionistic Type Theory - $Σ$ -types, Intuitionistic Type Theory - Finite types, Intuitionistic Type Theory - Equality type, Intuitionistic Type Theory - Inductive types, Intuitionistic Type Theory - Universes, Intuitionistic Type Theory - Formalisation of Type Theory, Intuitionistic Type Theory - Categorical models of Type Theory, Intuitionistic Type Theory - Extensional versus intensional, Intuitionistic Type Theory - Implementations of Type Theory

Read more here: » Intuitionistic Type Theory: Encyclopedia II - Intuitionistic Type Theory - Formalisation of Type Theory

simply typed lambda calculus: Encyclopedia II - Fixed point combinator - Example

Consider the factorial function (under Church encoding). The usual recursive mathematical equation is fact(n) = if n=0 then 1 else n * fact(n-1) We can express a "single step" of this recursion in lambda calculus as F = λf. λx. (ISZERO x) 1 (MULT x (f (PRED x))), where "f" is a place-holder argument for the factorial function to be passed to itself. The function F performs a single step in the evaluation of the recursive formula. A ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Example

simply typed lambda calculus: Encyclopedia II - Fixed point combinator - Other fixed point combinators

A version of the Y combinator that can be used in call-by-value (applicative-order) evaluation is given by η-expansion of part of the ordinary Y combinator: Z = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) The Y combinator can be expressed in the SKI-calculus as Y = S (K (S I I)) (S (S (K S) K) (K (S I I))) The simplest fixed point combinator in the SK-calculus, found by John Tr ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Other fixed point combinators

simply typed lambda calculus: Encyclopedia II - Intuitionistic Type Theory - Categorical models of Type Theory

Using the language of category theory, Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of Type Theory. This has been refined by Hofmann and Dybjer to Categories with Families or Categories with Attributes based on earlier work by Cartmell. A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : C^op -> Fam(Set). Fam(Set) is the category in which the objects are pa ...

See also:

Intuitionistic Type Theory, Intuitionistic Type Theory - Connectives of Type Theory, Intuitionistic Type Theory - $Π$ -types, Intuitionistic Type Theory - $Σ$ -types, Intuitionistic Type Theory - Finite types, Intuitionistic Type Theory - Equality type, Intuitionistic Type Theory - Inductive types, Intuitionistic Type Theory - Universes, Intuitionistic Type Theory - Formalisation of Type Theory, Intuitionistic Type Theory - Categorical models of Type Theory, Intuitionistic Type Theory - Extensional versus intensional, Intuitionistic Type Theory - Implementations of Type Theory

Read more here: » Intuitionistic Type Theory: Encyclopedia II - Intuitionistic Type Theory - Categorical models of Type Theory

More material related to Simply Typed Lambda Calculus can be found here:

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