A Wisdom Archive on Simply Typed Lambda Calculus - Types

The types of the simply typed lambda calculus are constructed from base types (or type variables) and given types σ,τ we can construct . Church used only two base types o for the type of propositions and ι for the type of individuals. Frequently the calculus with only one base type, usually o, is considered. associates to the right: we read as . To each type σ we assign a ...

Read more here: » Simply typed lambda calculus: Encyclopedia II - Simply typed lambda calculus - Types

To define the set of well typed lambda terms of a given type, we introduce typing contexts which are sequences of typing assumptions of the form x:σ where x is a variable. We introduce the judgment which means that t is a term of type σ in context Γ which is given by the following typing rules: Examples of closed terms are: (I), (K), and (S). These are the typed lambda calculus represen ...

Read more here: » Simply typed lambda calculus: Encyclopedia II - Simply typed lambda calculus - Terms

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 ...

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

FALSE is an esoteric programming language designed by Wouter van Oortmerssen in 1993, named after his favourite boolean value. It is a small Forth-like stack-oriented language, with syntax designed to make the code inherently obfuscated, confusing, and unreadable. It is also noteworthy for having a compiler of only 1024 bytes (written in 68000 assembly). According to van Oortmerssen, FALSE provided the inspiration for various well known esoteric ... Including:

• FALSE - The stack
• FALSE - Data types
• FALSE - Basic operators
• FALSE - Variables
• FALSE - Subroutines
• FALSE - Control flow
• FALSE - If
• FALSE - While
• FALSE - Strings
• FALSE - Input / Output
• FALSE - Comments
• FALSE - Code examples

Read more here: » FALSE: Encyclopedia - FALSE

Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. Combinatory logic - Combinatory logic in mathematics. Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for ... Including:

• Combinatory logic - Combinatory logic in mathematics
• Combinatory logic - Combinatory logic in computing
• Combinatory logic - Summary of the lambda calculus
• Combinatory logic - Combinatory calculi
• Combinatory logic - Combinatory terms
• Combinatory logic - Examples of combinators
• Combinatory logic - Completeness of the S-K basis
• Combinatory logic - Simplifications of the transformation
• Combinatory logic - Reverse conversion
• Combinatory logic - Undecidability of combinatorial calculus
• Combinatory logic - Combinatory terms as graphs
• Combinatory logic - Applications
• Combinatory logic - Compilation of functional languages
• Combinatory logic - Logic

Read more here: » Combinatory logic: Encyclopedia - Combinatory logic

This article is about type polymorphism. For another kind of polymorphism in computer science, related only in name to type polymorphism, see polymorphic code. In computer science, polymorphism means allowing a single definition to be used with different types of data (specifically, different classes of objects). For instance, a polymorphic function definition can replace several type-specific ones, and a single polymorphic operator can act in expressions of various ... Including:

• Polymorphism computer science - Parametric polymorphism
• Polymorphism computer science - Predicative and impredicative polymorphism
• Polymorphism computer science - Subtyping polymorphism
• Polymorphism computer science - Ad-hoc polymorphism
• Polymorphism computer science - Overloading
• Polymorphism computer science - Coercion
• Polymorphism computer science - Example
• Polymorphism computer science - Overloading
• Polymorphism computer science - Coercion
• Polymorphism computer science - Parametric polymorphism

Read more here: » Polymorphism computer science: Encyclopedia - Polymorphism computer science

In computer science, the Actor model, first published in 1973 (Hewitt et al. 1973), is a mathematical model of concurrent computation. The Actor model treats “Actors” as the universal primitives of concurrent digital computation: in response to a message that it receives, an Actor can make local decisions, create more Actors, send more messages, and determine how to respond to the next message received. The Actor model has been used both as a framework within which to develop a theor ... Including:

• Actor model - History
• Actor model - Fundamental concepts
• Actor model - Formal systems
• Actor model - Applications
• Actor model - Models prior to the Actor model
• Actor model - Lambda calculus
• Actor model - Simula
• Actor model - Smalltalk
• Actor model - Petri nets
• Actor model - Message Passing Semantics
• Actor model - The unbounded nondeterminism controversy
• Actor model - Direct communication and asynchrony
• Actor model - Actor creation plus addresses in messages means variable topology
• Actor model - Inherently concurrent
• Actor model - No requirement on order of message arrival
• Actor model - Not sequentiality not buffering not synchrony and not fixed topology
• Actor model - Locality
• Actor model - Compositionality
• Actor model - Behaviors
• Actor model - Relationship to mathematical logic
• Actor model - Migration
• Actor model - Security
• Actor model - Synthesizing addresses of Actors
• Actor model - Why is the Actor model important now?
• Actor model - Actor researchers

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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 ...

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

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 ...

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

In cartesian closed categories, a "function of two variables" (a morphism f:X×Y → Z) can always be represented as a "function of one variable" (the morphism λf:X → ZY). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any cartesian closed category. Certain cartesian closed categories, the topoi, have been proposed as a general s ...

Read more here: » Cartesian closed category: Encyclopedia II - Cartesian closed category - Applications

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 ...

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

Following the lambda calculus, we will use λx.E to denote the function with formal parameter x and body E. When applied to an argument, say a, this function yields E, with every free appearance of x replaced with a. Valid λ-calculus expressions have one of these forms: v (where v is a variable) λv.E (where v is a variable and E is a λ-calculus expression) (E F) (where E and F

Read more here: » Curry-Howard: Encyclopedia II - Curry-Howard - Types

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 ...

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

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 ...

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