Examples of using Lambda calculus in English and their translations into Greek
{-}
-
Colloquial
-
Official
-
Medicine
-
Ecclesiastic
-
Financial
-
Official/political
-
Computer
The lambda calculus can be called the smallest universal programming language.
Many functional programming languages can be viewed as embellishments to the lambda calculus.
Lambda calculus can be used to define what is a computable function.
Many Functional Programming languages can thus be considered as elaborations on this lambda calculus.
The simply typed lambda calculus is the language of Cartesian closed categories(CCCs).
Lisp was originally created as a practical mathematicalnotation for computer programs, based on Alonzo Church's lambda calculus.
Turing machines μ-recursive functions Lambda calculus Post machines(Post- Turing machines and tag machines).
The lambda calculus provides a powerful and flexible syntax for defining and combining functions of several variables.
Stephen Cole Kleene, for his pioneering work with Alonzo Church in Lambda Calculus that first laid down the foundations of computation theory.
Combinatory logic and lambda calculus were both originally developed to achieve a clearer approach to the foundations of mathematics.
Originally created as a practical mathematical notation for computer programs,LISP is based on the notation of Alonzo Church's lambda calculus.
Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages.
The availability of predicates and the above definition of TRUE andFALSE make it convenient to write"if-then-else" expressions in lambda calculus.
Lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism.
Turing machines Finite state machines Recursive functions Lambda calculus Combinatory logic Cellular automaton Abstract rewriting systems.
Based on the lambda calculus, Lisp soon became the programming language of choice for AI applications after its publication in 1960.
Perhaps the first formal incarnation of operational semantics was the use of the lambda calculus to define the semantics of LISP by John McCarthy.
Instituting a simply typed lambda calculus over the type operators results in more than just a formalization of type constructors though.
Continuations are also used in models of computation including denotational semantics, the Actor model,process calculi, and lambda calculus.
For example the dependently typed lambda calculus with a type of all types(Type: Type) is not normalizing due to Girard's paradox.
As Peter Landin noted, the language Algol was the first language to combine seamlessly imperative effects with the(call-by-name) lambda calculus.
There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church numerals, which can be defined as follows.
The simply typed lambda calculus(formula_1), a formof type theory, is a typed interpretation of the lambda calculus with only one type constructor: formula_2 that builds function types.
The Montague grammar is based on formal logic,especially higher-order predicate logic and lambda calculus, and makes use of the notions of intensional logic, via Kripke models.
Alonzo Church, inventor of the lambda calculus, developed a higher-order logic commonly called"Church's Theory of Types", in order to avoid the Kleene-Rosser paradox afflicting the original pure lambda calculus. .
The class of computable functions can be defined in many equivalent models of computation,including Turing machines μ-recursive functions Lambda calculus Post machines(Post- Turing machines and tag machines).
(Church did this by inventing an entirely different system known as the lambda calculus, which would later encourage laptop languages like Lisp) The answer to the decision problem was adverse.
The simply typed lambda calculus( λ→{\displaystyle\lambda^{\to}}), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor:→{\displaystyle\to} that builds function types.
(Church did this by inventing a completely totally different system called the lambda calculus, which might later encourage laptop languages like Lisp) The answer to the choice problem was unfavourable.
Functional programming has its roots in lambda calculus, a formal system developed in the 1930s to investigate function definition, function application, and recursion.