A Wisdom Archive on Simply Typed Lambda Calculus - Terms

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 ...

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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 ...

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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

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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

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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

In mathematics and computer science an algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will terminate in a corresponding recognizable end-state. Algorithms can be implemented by computer programs. Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). The concept of algorithms was formalized in 1936 by Alan Turing's Turing machines and Alonzo Church's lambda c ... Including:

• Algorithm - Formalization of algorithms
• Algorithm - Implementation
• Algorithm - Example
• Algorithm - History
• Algorithm - Classes
• Algorithm - Classification by implementation
• Algorithm - Classification by design paradigm
• Algorithm - Classification by field of study
• Algorithm - Classification by complexity
• Algorithm - Sources
• Algorithm - Legal issues

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• Turing machine - Definition
• Turing machine - Informal description
• Turing machine - Formal definition
• Turing machine - Example
• Turing machine - Deterministic and non-deterministic Turing machines
• Turing machine - Universal Turing machines
• Turing machine - Comparison with real machines
• Turing machine - Simulators

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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

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 ...

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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 ...

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It's clear that λ-expressions can have quite complicated types. One might ask whether there is a λ-expression with any given type. The problem of finding a λ-expression with a particular type is called the type inhabitation problem. The answer turns out to be remarkable: There is a closed λ-expression with a particular type only if the type corresponds to a theorem of logic, when the ...

Read more here: » Curry-Howard: Encyclopedia II - Curry-Howard - The type inhabitation problem