typed lambda calculus

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 ...

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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

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 ...

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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

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 ...

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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

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 ...

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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

Epigram uses a two-dimensional syntax, with a LaTeX version and an ASCII version. Here are some examples from The Epigram Tutorial: Epigram programming language - Examples. ...And in ASCII: ( ! ( ! ( n : Nat ! data !---------! where !----------! ; !-----------! ! Nat : * ) !zero : Nat) !suc n : Nat)

Epigram programming language, Epigram programming language - Syntax, Epigram programming language - Examples, Epigram programming language - Dependent Types in Epigram

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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 ...

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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

Programming languages can achieve some of the same algorithmic results as are obtained through higher-order functions by dynamically executing code (sometimes called "Eval" or "Execute" operations) in the scope of evaluation. Unfortunately, there are significant drawbacks to this approach: The argument code to be executed is usually not statically typed; these languages generally rely on dynamic typing to determine the well-formedness and safety of the code to be executed. The argument is usually provided as a string, t ...

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Higher-order function, Higher-order function - Alternatives

Read more here: » Higher-order function: Encyclopedia II - Higher-order function - Alternatives

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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Simply typed lambda calculus, Simply typed lambda calculus - Types, Simply typed lambda calculus - Terms, Simply typed lambda calculus - Important results

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

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - History

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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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

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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Simply typed lambda calculus, Simply typed lambda calculus - Types, Simply typed lambda calculus - Terms, Simply typed lambda calculus - Important results

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

Main article: Ordinal analysis Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Ordinal analysis

The three most well known proof calculi are: The Hilbert-style calculi The natural deduction calculus The sequent calculus To say these are proof calculi, rather than proof systems, is to say they are flexible frameworks for the study of many kinds of logical consequence relations. Each of these can formalise propositional or predicate logics of either the classical or intuitionistic flavour, or almost any modal logic studied, many substructural logics, such as relevance logic or lin ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Kinds of proof calculus

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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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

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 ...

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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

However, the proofs used in everyday mathematical practice are almost never like the formal proofs in proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, and given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code. Formal proofs are constructed, with the help of computers, in automated theorem proving. Significantly, these proofs can be checked automatically by com ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Formal and informal proof

Main article: Consistency proof As we have discussed, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable Pi-0-1 sentences) are finitarily true; once so grounded w ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Consistency proofs

Main article: Structural proof theory Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz; the definition is slightly more complex, we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Gira ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Structural proof theory

Main article: Tableau systems Tableau systems apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Tableau systems