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| [[Image:Coq plus comm screenshot.jpg|thumb|300px|A proof written as a functional program: the proof of commutativity of addition on natural numbers in the proof assistant [[Coq]]. ''nat_ind'' stands for [[mathematical induction]], ''eq_ind'' for substitution of equals and ''f_equal'' for taking the same function on both sides of the equality. Earlier theorems are referenced showing ''m'' = ''m'' + 0 and S (''m'' + ''y'') = ''m'' + S ''y''.]]
| | My name is Damaris and I am studying Athletics and Physical Education and Dramatic Literature and History at Kobenhavn K / Denmark.<br><br>Feel free to surf to my web blog; [http://Like-Boquete.com/?p=89 Fifa 15 Coin Generator] |
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| In [[programming language theory]] and [[proof theory]], the '''Curry–Howard correspondence''' (also known as the '''Curry–Howard isomorphism''' or '''equivalence''', or the '''proofs-as-programs''' and '''propositions-''' or '''formulae-as-types interpretation''') is the direct relationship between [[computer program]]s and [[mathematical proof]]s. It is a generalization of a syntactic [[analogy]] between systems of formal logic and computational calculi that was first discovered by the American [[mathematician]] [[Haskell Curry]] and [[logician]] [[William Alvin Howard]].{{Citation needed|date=August 2013}} It is the link between Logic and Computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by [[L. E. J. Brouwer]], [[Arend Heyting]] and [[Andrey Kolmogorov]].{{Citation needed|date =August 2013}}
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| == Origin, scope, and consequences == <!--This heading is used in other articles. Please update the links in the other articles if the heading name is changed. -->
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| At the very beginning, the '''Curry–Howard correspondence''' is
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| # the observation in 1934 by [[Haskell Curry|Curry]] that the [[typed lambda calculus|type]]s of the combinators could be seen as axiom-schemes for [[intuitionism|intuitionistic]] implicational logic,
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| # the observation in 1958 by Curry that a certain kind of [[proof calculus|proof system]], referred to as [[Hilbert-style deduction system]]s, coincides on some fragment to the typed fragment of a standard [[model of computation]] known as [[combinatory logic]],
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| # the observation in 1969 by [[William Alvin Howard|Howard]] that another, more "high-level" [[proof calculus|proof system]], referred to as [[natural deduction]], can be directly interpreted in its [[intuitionistic]] version as a typed variant of the [[model of computation]] known as [[lambda calculus]].
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| In other words, the Curry–Howard correspondence is the observation that two families of formalisms which had seemed unrelated—namely, the proof systems on one hand, and the models of computation on the other—were, in the two examples considered by Curry and Howard, in fact structurally the same kind of objects.
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| If one now abstracts on the peculiarities of this or that formalism, the immediate generalization is the following claim: ''a proof is a program, the formula it proves is a type for the program''. More informally, this can be seen as an [[analogy]] that states that the [[return type]] of a function (i.e., the type of values returned by a function) is analogous to a logical theorem, subject to hypotheses corresponding to the types of the argument values passed to the function; and that the program to compute that function is analogous to a proof of that theorem. This sets a form of [[logic programming]] on a rigorous foundation: ''proofs can be represented as programs, and especially as lambda terms'', or ''proofs can be '''run'''''.
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| The correspondence has been the starting point of a large spectrum of new research after its discovery, leading in particular to a new class of [[formal system]]s designed to act both as a [[proof calculus|proof system]] and as a typed [[functional programming language]]. This includes [[Martin-Löf]]'s [[intuitionistic type theory]] and [[Thierry Coquand|Coquand]]'s [[Calculus of Constructions]], two calculi in which proofs are regular objects of the discourse and in which one can state properties of proofs the same way as of any program. This field of research is usually referred to as modern [[type theory]].
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| Such [[typed lambda calculus|typed lambda calculi]] derived from the Curry–Howard paradigm led to software like [[Coq]] in which proofs seen as programs can be formalized, checked, and run.
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| A converse direction is to ''use a program to extract a proof'', given its [[program correctness|correctness]]— an area of research which is closely related to [[proof-carrying code]]. This is only feasible if the [[programming language]] the program is written for is very richly typed: the development of such type systems has been partly motivated by the wish to make the Curry–Howard correspondence practically relevant.
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| The Curry–Howard correspondence also raised new questions regarding the computational content of proof concepts which were not covered by the original works of Curry and Howard. In particular, [[classical logic]] has been shown to correspond to the ability to manipulate the [[continuation]] of programs and the symmetry of [[sequent calculus]] to express the duality between the two [[evaluation strategy|evaluation strategies]] known as call-by-name and call-by-value.
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| Speculatively, the Curry–Howard correspondence might be expected to lead to a substantial [[unifying theories in mathematics|unification]] between mathematical logic and foundational computer science:
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| Hilbert-style logic and natural deduction are but two kinds of proof systems among a large family of formalisms. Alternative syntaxes include [[sequent calculus]], [[proof net]]s, [[calculus of structures]], etc. If one admits the Curry–Howard correspondence as the general principle that any proof system hides a model of computation, a theory of the underlying untyped computational structure of these kinds of proof system should be possible. Then, a natural question is whether something mathematically interesting can be said about these underlying computational calculi.
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| Conversely, [[combinatory logic]] and [[simply typed lambda calculus]] are not the only [[models of computation]], either. Girard's [[linear logic]] was developed from the fine analysis of the use of resources in some models of lambda calculus; can we imagine a typed version of [[Turing machine|Turing's machine]] that would behave as a proof system? [[Typed assembly language]]s are such an instance of "low-level" models of computation that carry types.
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| Because of the possibility of writing non-terminating programs, [[Turing-complete]] models of computation (such as languages with arbitrary [[Recursion (computer science)|recursive functions]]) must be interpreted with care, as naive application of the correspondence leads to an inconsistent logic. The best way of dealing with arbitrary computation from a logical point of view is still an actively debated research question, but one popular approach is based on using [[Monad (functional programming)|monads]] to segregate provably terminating from potentially non-terminating code (an approach which also generalizes to much richer models of computation,<ref>{{Citation|first=Eugenio|last=Moggi|year=1991|title=Notions of Computation and Monads|journal=Information and Computation|volume=93|issue=1|url=http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf}}</ref> and is itself related to modal logic by a natural extension of the Curry–Howard isomorphism<ref name=PfenningDaviesJudgmental group=ext/>). A more radical approach, advocated by [[total functional programming]], is to eliminate unrestricted recursion (and forgo [[Turing completeness]], although still retaining high computational complexity), using more controlled [[corecursion]] where non-terminating behavior is actually desired.
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| ==General formulation==
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| In its more general formulation, the Curry–Howard correspondence is a correspondence between formal [[proof calculus|proof calculi]] and [[type systems]] for [[model of computation|models of computation]]. In particular, it splits into two correspondences. One at the level of [[formula (mathematical logic)|formulas]] and [[data type|types]] that is independent of which particular proof system or model of computation is considered, and one at the level of [[mathematical proof|proofs]] and [[computer program|programs]] which, this time, is specific to the particular choice of proof system and model of computation considered.
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| At the level of formulas and types, the correspondence says that implication behaves the same as a function type, conjunction as a "product" type (this may be called a tuple, a struct, a list, or some other term depending on the language), disjunction as a sum type (this type may be called a union), the false formula as the empty type and the true formula as the singleton type (whose sole member is the null object). Quantifiers correspond to [[dependent type|dependent]] function space or products (as appropriate).
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| <!-- If quantification comes in two flavors, it is because there are two ways to think about the Curry–Howard correspondence. If one considers that every single proof step has to be computational, then [[universal quantification]] is interpreted as dependent product and [[existential quantification]] as a dependent sum (this corresponds to interpreting a proof as a term typed [[simply typed lambda calculus|à la Church]]). If otherwise one considers that some trivial computational step can be omitted, then universal quantification is interpreted as a dependent [[Type_system#Intersection_types|intersection type]] and existential quantification as a dependent [[Type_system#Union_types|union type]] (this corresponds to interpreting a proof as a term typed [[simply typed lambda calculus|à la Curry]], which was actually the approach taken by Howard). -->
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| This is summarized in the following table:
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:300px" | Logic side
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| ! style="width:300px" | Programming side
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| |-
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| | [[universal quantification]] || [[Dependent type#Formal definition|generalised function space]] (Π type)
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| |-
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| | [[existential quantification]] || [[Dependent type#Dependent pair type|generalised cartesian product]] (Σ type)
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| |-
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| | [[Logical implication|implication]] || [[function type]]
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| |-
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| | [[logical conjunction|conjunction]] || [[product type]]
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| |-
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| | [[logical disjunction|disjunction]] || [[sum type]]
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| |-
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| | true formula || [[unit type]]
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| |-
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| | false formula || [[bottom type]]
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| |-
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| |}
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| At the level of proof systems and models of computations, the correspondence mainly shows the identity of structure, first, between some particular formulations of systems known as [[Hilbert-style deduction system]] and [[combinatory logic]], and, secondly, between some particular formulations of systems known as [[natural deduction]] and [[lambda calculus]].
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:300px" | Logic side
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| ! style="width:300px" | Programming side
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| |-
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| | [[Hilbert-style deduction system]] || type system for [[combinatory logic]]
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| |-
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| | [[natural deduction]] || type system for [[lambda calculus]]
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| |}
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| Between the natural deduction system and the lambda calculus there are the following correspondences:
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:300px" | Logic side
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| ! style="width:300px" | Programming side
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| |-
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| | [[hypotheses]] || free variables
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| |-
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| | [[implication elimination]] (''modus ponens'') || [[apply|application]]
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| |-
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| | [[implication introduction]] || abstraction
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| |}
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| ==Correspondence between Hilbert-style deduction systems and combinatory logic==
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| It was at the beginning a simple remark in Curry and Feys's 1958 book on combinatory logic: the simplest types for the basic combinators K and S of [[combinatory logic]] surprisingly corresponded to the respective [[axiom scheme]]s α → (β → α) and (α → (β → γ)) → ((α → β) → (α → γ)) used in [[Hilbert-style deduction system]]s. For this reason, these schemes are now often called axioms K and S. Examples of programs seen as proofs in a Hilbert-style logic are given [[#Examples|below]].
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| If one restricts to the implicational intuitionistic fragment, a simple way to formalize logic in Hilbert's style is as follows. Let Γ be a finite collection of formulas, considered as hypotheses. We say that δ is derivable from Γ, and we write Γ {{Unicode|⊢}} δ, in the following cases:
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| * δ is an hypothesis, i.e. it is a formula of Γ,
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| * δ is an instance of an axiom scheme; i.e., under the most common axiom system:
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| ** δ has the form α → (β → α), or
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| ** δ has the form (α → (β → γ)) → ((α → β) → (α → γ)),
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| * δ follows by deduction, i.e., for some α, both α → δ and α are already derivable from Γ (this is the rule of [[modus ponens]])
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| This can be formalized using [[inference rules]], what we do in the left column of the following table.
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| We can formulate typed combinatory logic using a similar syntax: let Γ be a finite collection of variables, annotated with their types. A term T (also annotated with its type) will depend on these variables [Γ {{Unicode|⊢}} T:δ] when:
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| * T is one of the variables in Γ,
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| * T is a basic combinator; i.e., under the most common combinator basis:
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| ** T is K:α → (β → α) [where α and β denote the types of its arguments], or
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| ** T is S:(α → (β → γ)) → ((α → β) → (α → γ)),
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| * T is the composition of two subterms which depend on the variables in Γ.
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| The generation rules defined here are given in the right-column below. Curry's remark simply states that both columns are in one-to-one correspondence. The restriction of the correspondence to [[intuitionistic logic]] means that some [[Classical logic|classical]] [[tautology (logic)|tautologies]], such as [[Peirce's law]] ((α → β) → α) → α, are excluded from the correspondence.
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| {| style="text-align:center; margin: 0 auto;" class="wikitable"
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| ! width="200px"| Hilbert-style intuitionistic implicational logic
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| ! width="200px"| Simply typed combinatory logic
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| |- style="height:70px" valign=bottom
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| | <math>\frac{\alpha \in \Gamma}{\Gamma \vdash \alpha} \qquad\qquad\text{Assum}</math>
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| | <math>\frac{x:\alpha \in \Gamma}{\Gamma \vdash x:\alpha}</math>
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| |- style="height:70px" valign=bottom
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| | <math>\frac{}{\Gamma \vdash \alpha \rightarrow (\beta \rightarrow \alpha)} \qquad\text{Ax}_K</math>
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| | <math>\frac{}{\Gamma \vdash K: \alpha \rightarrow (\beta \rightarrow \alpha)}</math>
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| |- style="height:70px" valign=bottom
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| | <math>\frac{}{\Gamma \vdash (\alpha\!\rightarrow\!(\beta\!\rightarrow\!\gamma))\!\rightarrow\!((\alpha\!\rightarrow\!\beta)\!\rightarrow\!(\alpha\!\rightarrow\!\gamma))}\;\text{Ax}_S</math>
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| | <math>\frac{}{\Gamma \vdash S: (\alpha\!\rightarrow\!(\beta\!\rightarrow\!\gamma))\!\rightarrow\!((\alpha\!\rightarrow\!\beta)\!\rightarrow\!(\alpha\!\rightarrow\!\gamma))}</math>
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| |- style="height:70px" valign=bottom
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| | <math>\frac{\Gamma \vdash \alpha \rightarrow \beta \qquad \Gamma \vdash \alpha}{\Gamma \vdash \beta}\quad\text{Modus Ponens}</math>
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| | <math>\frac{\Gamma \vdash E_1:\alpha \rightarrow \beta \qquad \Gamma \vdash E_2:\alpha}{\Gamma \vdash E_1\;E_2:\beta} </math>
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| |}
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| Seen at a more abstract level, the correspondence can be restated as shown in the following table. Especially, the [[deduction theorem]] specific to Hilbert-style logic matches the process of [[Combinatory logic#Conversion of a lambda term to an equivalent combinatorial term|abstraction elimination]] of combinatory logic.
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:200px" | Logic side
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| ! style="width:200px" | Programming side
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| |-
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| | assumption || variable
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| |-
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| | axioms || combinators
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| |-
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| | modus ponens || application
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| |-
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| | [[deduction theorem]] || [[Combinatory logic#Conversion of a lambda term to an equivalent combinatorial term|abstraction elimination]]
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| |}
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| Thanks to the correspondence, results from combinatory logic can be transferred to Hilbert-style logic and vice-versa. For instance, the notion of [[combinatory logic#Reduction in combinatory logic|reduction]] of terms in combinatory logic can be transferred to Hilbert-style logic and it provides a way to canonically transform proofs into other proofs of the same statement. One can also transfer the notion of normal terms to a notion of normal proofs, expressing that the hypotheses of the axioms never need to be all detached (since otherwise a simplification can happen).
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| Conversely, the non provability in intuitionistic logic of [[Peirce's law]] can be transferred back to combinatory logic: there is no typed term of combinatory logic that is typable with type ((α → β) → α) → α.
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| Results on the completeness of some sets of combinators or axioms can also be transferred. For instance, the fact that the combinator '''X''' constitutes a [[Combinatory logic#One-point basis|one-point basis]] of (extensional) combinatory logic implies that the single axiom scheme
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| :(((α → (β → γ)) → ((α → β) → (α → γ))) → ((δ → (ε → δ)) → ζ)) → ζ,
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| which is the [[principal type]] of '''X''', is an adequate replacement to the combination of the axiom schemes
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| :α → (β → α) and
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| :(α → (β → γ)) → ((α → β) → (α → γ)).
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| ==Correspondence between natural deduction and lambda calculus==
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| After [[Haskell Curry|Curry]] emphasized the syntactic correspondence between [[Hilbert-style deduction system|Hilbert-style deduction]] and [[combinatory logic]], [[William Alvin Howard|Howard]] made explicit in 1969 a syntactic analogy between the programs of [[simply typed lambda calculus]] and the proofs of [[natural deduction]]. Below, the left-hand side formalizes intuitionistic implicational natural deduction as a calculus of [[sequent]]s (the use of sequents is standard in discussions of the Curry–Howard isomorphism as it allows the deduction rules to be stated more cleanly) with implicit weakening and the right-hand side shows the typing rules of [[lambda calculus]]. In the left-hand side, Γ, Γ<sub>1</sub> and Γ<sub>2</sub> denote ordered sequences of formulas while in the right-hand side, they denote sequences of named (i.e., typed) formulas with all names different.
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| <br> | |
| {| class="wikitable" style="text-align:center; margin: 0 auto;" cellpadding="10"
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| ! width="200px"| Intuitionistic implicational natural deduction
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| ! width="200px"| Lambda calculus type assignment rules
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| |- style="height:70px" valign=bottom
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| | <math>\frac{}{\Gamma_1, \alpha, \Gamma_2 \vdash \alpha} \text{Ax}</math>
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| | <math>\frac{}{\Gamma_1, x:\alpha, \Gamma_2 \vdash x:\alpha}</math>
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| |- style="height:70px" valign=bottom
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| | <math>\frac{\Gamma, \alpha \vdash \beta}{\Gamma \vdash \alpha \rightarrow \beta} \rightarrow I</math>
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| | <math>\frac{\Gamma, x:\alpha \vdash t:\beta}{\Gamma \vdash \lambda x.t : \alpha \rightarrow \beta}</math>
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| |- style="height:70px" valign=bottom
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| | <math>\frac{\Gamma \vdash \alpha \rightarrow \beta \qquad \Gamma \vdash \alpha}{\Gamma \vdash \beta} \rightarrow E</math>
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| | <math>\frac{\Gamma \vdash t:\alpha \rightarrow \beta \qquad \Gamma \vdash u:\alpha}{\Gamma \vdash t\;u:\beta} </math>
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| |}
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| To paraphrase the correspondence, proving Γ {{Unicode|⊢}} α means having a program that, given values with the types listed in Γ, manufactures an object of type α. An axiom corresponds to the introduction of a new variable with a new, unconstrained type, the → I rule corresponds to function abstraction and the → E rule corresponds to function application. Observe that the correspondence is not exact if the context Γ is taken to be a set of formulas as, e.g., the λ-terms λx.λy.x and λx.λy.y of type α → α → α would not be distinguished in the correspondence. Examples are given [[#Examples|below]].
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| Howard showed that the correspondence extends to other connectives of the logic and other constructions of simply typed lambda calculus. Seen at an abstract level, the correspondence can then be summarized as shown in the following table. Especially, it also shows that the notion of normal forms in [[lambda calculus]] matches [[Dag Prawitz|Prawitz]]'s notion of normal deduction in [[natural deduction]], from what we deduce, among others, that the algorithms for the [[type inhabitation problem]] can be turned into algorithms for deciding [[intuitionistic]] provability.
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:200px" | Logic side
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| ! style="width:200px" | Programming side
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| |-
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| | axiom || variable
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| |-
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| | introduction rule || constructor
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| |-
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| | elimination rule || destructor
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| |-
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| | normal deduction || normal form
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| |-
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| | normalisation of deductions || [[Normalization property (lambda-calculus)|weak normalisation]]
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| |-
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| | provability || [[type inhabitation problem]]
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| |-
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| | intuitionistic tautology || inhabited type
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| |}
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| Howard's correspondence naturally extends to other extensions of [[natural deduction]] and [[simply typed lambda calculus]]. Here is a non exhaustive list:
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| * Girard-Reynolds [[System F]] as a common language for both second-order propositional logic and polymorphic lambda calculus,
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| * [[higher-order logic]] and Girard's [[System F|System F<sub>ω</sub>]]
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| * inductive types as [[algebraic data type]]
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| * necessity <math>\Box</math> in [[modal logic]] and staged computation<ref name=DaviesPfenningStaged group=ext>{{Citation|doi=10.1145/382780.382785|first=Rowan|last=Davies|first2=Frank|last2=Pfenning|year=2001|title=A Modal Analysis of Staged Computation|journal=Journal of the ACM |volume=48|issue=3|pages=555–604|url=http://www.cs.cmu.edu/~fp/papers/jacm00.pdf}}</ref>
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| * possibility <math>\Diamond</math> in [[modal logic]] and monadic types for effects<ref name=PfenningDaviesJudgmental group=ext/>
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| * The λ<sub>I</sub> calculus corresponds to [[relevant logic]].<ref>{{Citation|first=Morten|last=Sørenson|first2=Paweł|last2=Urzyczyn|title=Lectures on the Curry-Howard Isomorphism|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7385}}</ref>
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| * The local truth (∇) modality in [[Grothendieck topology]] or the equivalent "lax" modality (∘)<!-- should be a large circle --> of Benton, Bierman, and de Paiva (1998) correspond to CL-logic describing "computation types".<ref>{{Citation|last=Goldblatt|title=Mathematical Modal Logic: A Model of its Evolution|chapter=7.6 Grothendieck Topology as Intuitionistic Modality|pages=76–81|url=http://homepages.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf}}. The "lax" modality referred to is from {{Citation|last=Benton|last2=Bierman|last3=de Paiva|title=Computational types from a logical perspective|journal=Journal of Functional Programming|volume=8|pages=177–193|year=1998}}</ref>
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| ==Correspondence between classical logic and control operators==
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| At the time of Curry, and also at the time of Howard, the proofs-as-programs correspondence concerned only [[intuitionistic logic]], i.e. a logic in which, in particular, [[Peirce's law]] was ''not'' deducible. The extension of the correspondence to Peirce's law and hence to [[classical logic]] became clear from the work of Griffin on typing operators that capture the evaluation context of a given program execution so that this evaluation context can be later on reinstalled. The basic Curry–Howard-style correspondence for classical logic is given below. Note the correspondence between the [[double-negation translation]] used to map classical proofs to intuitionistic logic and the [[continuation-passing style|continuation-passing-style]] translation used to map lambda terms involving control to pure lambda terms. More particularly, call-by-name continuation-passing-style translations relates to [[Kolmogorov]]'s double negation translation and call-by-value continuation-passing-style translations relates to a kind of double-negation translation due to Kuroda.
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! width="200px"| Logic side
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| ! width="200px"| Programming side
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| |-
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| | [[Peirce's law]]: ((α → β) → α) → α || [[call-with-current-continuation]]
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| |-
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| | double-negation translation || [[continuation-passing style|continuation-passing-style translation]]
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| |}
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| A finer Curry–Howard correspondence exists for classical logic if one defines classical logic not by adding an axiom such as [[Peirce's law]], but by allowing several conclusions in sequents. In the case of classical natural deduction, there exists a proofs-as-programs correspondence with the typed programs of Parigot's [[lambda-mu calculus|λμ-calculus]].
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| ==Sequent calculus==
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| A proofs-as-programs correspondence can be settled for the formalism known as [[Gentzen]]'s [[sequent calculus]] but it is not a correspondence with a well-defined pre-existing model of computation as it was for Hilbert-style and natural deductions.
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| Sequent calculus is characterized by the presence of left introduction rules, right introduction rule and a cut rule that can be eliminated. The structure of sequent calculus relates to a calculus whose structure is close to the one of some [[abstract machine]]s. The informal correspondence is as follows:
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| {| class="wikitable" style="margin: 0 auto; text-align: center;"
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| ! style="width:200px" | Logic side
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| ! style="width:200px" | Programming side
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| |-
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| | cut elimination || reduction in a form of abstract machine
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| |-
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| | right introduction rules || constructors of code
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| |-
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| | left introduction rules || constructors of evaluation stacks
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| |-
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| | priority to right-hand side in cut-elimination
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| | [[call-by-name]] reduction
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| |-
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| | priority to left-hand side in cut-elimination
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| | [[call-by-value]] reduction
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| |}
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| ==Related proofs-as-programs correspondences==
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| ===The role of de Bruijn===
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| [[Nicolaas Govert de Bruijn|N. G. de Bruijn]] used the lambda notation for representing proofs of the theorem checker [[Automath]], and represented propositions as "categories" of their proofs. It was in the late 1960s at the same period of time Howard wrote his manuscript; de Bruijn was likely unaware of Howard's work, and stated the correspondence independently (Sørensen & Urzyczyn [1998] 2006, pp 98–99). Some researchers tend to use the term Curry–Howard–de Bruijn correspondence in place of Curry–Howard correspondence.
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| ===BHK interpretation===
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| The [[BHK interpretation]] interprets intuitionistic proofs as functions but it does not specify the class of functions relevant for the interpretation. If one takes lambda calculus for this class of function, then the [[BHK interpretation]] tells the same as Howard's correspondence between natural deduction and lambda calculus.
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| ===Realizability===
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| [[Stephen Cole Kleene|Kleene]]'s recursive [[realizability]] splits proofs of intuitionistic arithmetic into the pair of a recursive function and of
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| a proof of a formula expressing that the recursive function "realizes", i.e. correctly instantiates the disjunctions and existential quantifiers of the initial formula so that the formula gets true.
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| [[Georg Kreisel|Kreisel]]'s modified realizability applies to intuitionistic higher-order predicate logic and shows that the [[simply typed lambda calculus|simply typed lambda term]] inductively extracted from the proof realizes the initial formula. In the case of propositional logic, it coincides with Howard's statement: the extracted lambda term is the proof itself (seen as an untyped lambda term) and the realizability statement is a paraphrase of the fact that the extracted lambda term has the type that the formula means (seen as a type).
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| [[Kurt Gödel|Gödel]]'s [[dialectica interpretation]] realizes (an extension of) intuitionistic arithmetic with computable functions. The connection with lambda calculus is unclear, even in the case of natural deduction.
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| ===Curry–Howard–Lambek correspondence===
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| [[Joachim Lambek]] showed in the early 1970s that the proofs of intuitionistic propositional logic and the combinators of typed [[combinatory logic]] share a common equational theory which is the one of [[cartesian closed categories]]. The expression Curry–Howard–Lambek correspondence is now used by some people to refer to the three way isomorphism between intuitionistic logic, typed lambda calculus and cartesian closed categories, with objects being interpreted as types or propositions and morphisms as terms or proofs. The correspondence works at the equational level and is not the expression of a syntactic identity of structures as it is the case for each of Curry's and Howard's correspondences: i.e. the structure of a well-defined morphism in a cartesian-closed category is not comparable to the structure of a proof of the corresponding judgment in either Hilbert-style logic or natural deduction. To clarify this distinction, the underlying syntactic structure of cartesian closed categories is rephrased below.
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| Objects (types) are defined by
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| * <math>\top</math> is an object
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| * if <math>\alpha</math> and <math>\beta</math> are objects then <math>\alpha \times \beta</math> and <math>\alpha \rightarrow \beta</math> are objects.
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| Morphisms (terms) are defined by
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| * <math>id</math>, <math>\star</math>, <math>\operatorname{eval}</math>, <math>\pi_1</math> and <math>\pi_2</math> are morphisms
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| * if <math>t</math> is a morphism, <math>\lambda t</math> is a morphism
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| * if <math>t</math> and <math>u</math> are morphisms, <math>(t, u)</math> and <math>u \circ t</math> are morphisms.
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| Well-defined morphisms (typed terms) are defined by the following [[type rule|typing rules]] (in which the usual categorical morphism notation <math>f : \alpha \to \beta</math> is replaced with [[sequent calculus]] notation <math>f :\!\!-~~ \alpha ~\vdash~ \beta</math>).
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| Identity:
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| :<math>\frac{}{id:\!\!-~~ \alpha ~\vdash~ \alpha}</math>
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| Composition:
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| :<math>\frac{t:\!\!-~~ \alpha ~\vdash~ \beta\qquad u:\!\!-~~ \beta ~\vdash~ \gamma}{u \circ t:\!\!- ~\alpha ~\vdash~ \gamma}</math>
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| [[Unit type]] ([[terminal object]]):
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| :<math>\frac{}{\star:\!\!-~~\alpha ~\vdash~ \top}</math>
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| Cartesian product:
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| :<math>\frac{t:\!\!-~~\alpha ~\vdash~ \beta\qquad u:\!\!-~~\alpha ~\vdash~ \gamma}{(t,u):\!\!-~~ \alpha ~\vdash~ \beta \times \gamma}</math>
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| Left and right projection:
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| :<math>\frac{}{\pi_1:\!\!-~~ \alpha \times \beta ~\vdash~ \alpha}\qquad\frac{}{\pi_2:\!\!-~~ \alpha \times \beta ~\vdash~ \beta}</math>
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| [[Currying]]:
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| :<math>\frac{t:\!\!-~~ \alpha \times \beta ~\vdash~ \gamma}{\lambda t:\!\!-~~ \alpha ~\vdash~ \beta \rightarrow \gamma}</math>
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| [[Apply|Application]]:
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| :<math>\frac{}{eval:\!\!-~~ (\alpha \rightarrow \beta) \times \alpha ~\vdash~ \beta}</math>
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| Finally, the equations of the category are
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| * <math>id \circ t = t</math>, <math>t \circ id = t</math>, <math>(v \circ u) \circ t = v \circ (u \circ t)</math>
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| * <math>\star \circ t = \star</math>
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| * <math>\pi_1 \circ (t, u) = t, \pi_2 \circ (t,u) = u, (\pi_1 \circ t, \pi_2 \circ t) = t</math>
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| * <math>eval \circ (\lambda t \circ \pi_1, \pi_2) = t, \lambda eval = id</math>
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| Now, there exists <math>t</math> such that <math>t:\!\!-~ \alpha_1 \times \ldots \times \alpha_n \vdash \beta</math> iff <math>\alpha_1, \ldots, \alpha_n \vdash \beta </math> is provable in implicational intuitionistic logic,.
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| ==Examples==
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| Thanks to the Curry–Howard correspondence, a typed expression whose type corresponds to a logical formula is analogous to a proof of that formula. Here are examples.
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| ===The identity combinator seen as a proof of <tt>α → α</tt> in Hilbert-style logic===
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| As a simple example, we construct a proof of the theorem <tt>α → α</tt>. In [[lambda calculus]], this is the type of the identity function '''I''' = λ''x''.''x'' and in combinatory logic, the identity function is obtained by applying ''S'' twice to '''K'''. That is, we have '''I''' = (('''S''' '''K''') '''K'''). As a description of a proof, this says that to prove <tt>α → α</tt>, we can proceed as follows:
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| * instantiate the second axiom scheme with the formulas <tt>α</tt>, <tt>β → α</tt> and <tt>α</tt>, so that to obtain a proof of <tt>(α → ((β → α) → α)) → ((α → (β → α)) → (α → α))</tt>,
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| * instantiate the first axiom scheme once with <tt>α</tt> and <tt>β → α</tt>, so that to obtain a proof of <tt>α → ((β → α) → α)</tt>,
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| * instantiate the first axiom scheme a second time with <tt>α</tt> and <tt>β</tt>, so that to obtain a proof of <tt>α → (β → α)</tt>,
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| * apply modus ponens twice so that to obtain a proof of <tt>α → α</tt>
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| In general, the procedure is that whenever the program contains an application of the form (''P'' ''Q''), we should first prove theorems corresponding to the types of ''P'' and ''Q''. Since ''P'' is being applied to ''Q'', the type of ''P'' must have the form α → β and the type of ''Q'' must have the form α for some <tt>α</tt> and <tt>β</tt>. We can then detach the conclusion, <tt>β</tt>, via the modus ponens rule.
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| ===The composition combinator seen as a proof of <tt>(β → α) → (γ → β) → γ → α</tt> in Hilbert-style logic===
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| As a more complicated example, let's look at the theorem that corresponds to the '''B''' function. The type of '''B''' is (β → α) → (γ → β) → γ → α. '''B''' is equivalent to ('''S''' ('''K''' '''S''') '''K'''). This is our roadmap for the proof of the theorem (β → α) → (γ → β) → γ → α.
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| First we need to construct ('''K''' '''S'''). We make the antecedent of the '''K''' axiom look like the '''S''' axiom by setting α equal to <tt>(α → β → γ) → (α → β) → α → γ</tt>, and <tt>β</tt> equal to <tt>δ</tt> (to avoid variable collisions):
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| : <tt>'''K''' : α → β → α</tt>
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| : <tt>'''K'''[α = (α → β → γ) → (α → β) → α → γ, β=δ] : ((α → β → γ) → (α → β) → α → γ) → δ → (α → β → γ) → (α → β) → α → γ</tt>
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| Since the antecedent here is just '''S''', we can detach the consequent using Modus Ponens:
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| : <tt>'''K S''' : δ → (α → β → γ) → (α → β) → α → γ</tt>
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| This is the theorem that corresponds to the type of ('''K''' '''S'''). We now apply '''S''' to this expression. Taking '''S'''
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| : <tt>'''S''' : (α → β → γ) → (α → β) → α → γ</tt>
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| we put <tt>α</tt> = <tt>δ</tt>, <tt>β</tt> = <tt>α → β → γ</tt>, and <tt>γ</tt> = <tt>(α → β) → α → γ</tt>, yielding
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| : <tt>'''S'''[α = δ, β = α → β → γ, γ = (α → β) → α → γ] : (δ → (α → β → γ) → (α → β) → α → γ) → (δ → (α → β → γ)) → δ → (α → β) → α → γ</tt>
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| and we then detach the consequent:
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| : <tt>'''S (K S)''' : (δ → α → β → γ) → δ → (α → β) → α → γ</tt>
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| This is the formula for the type of ('''S''' ('''K''' '''S''')). A special
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| case of this theorem has <tt>δ</tt> = <tt>(β → γ)</tt>:
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| : <tt>'''S (K S)'''[δ = β → γ] : ((β → γ) → α → β → γ) → (β → γ) → (α → β) → α → γ</tt>
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| We need to apply this last formula to '''K'''. Again, we specialize '''K''', this time by replacing <tt>α</tt> with <tt>(β → γ)</tt> and <tt>β</tt> with <tt>α</tt>:
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| : <tt>'''K''' : α → β → α</tt>
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| : <tt>'''K'''[α = β → γ, β = α] : (β → γ) → α → (β → γ)</tt>
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| This is the same as the antecedent of the prior formula, so we detach the consequent:
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| : <tt>'''S (K S) K''' : (β → γ) → (α → β) → α → γ</tt>
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| Switching the names of the variables <tt>α</tt> and <tt>γ</tt> gives us
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| : <tt>(β → α) → (γ → β) → γ → α</tt>
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| which was what we had to prove.
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| ===The normal proof of <tt>(β → α) → (γ → β) → γ → α</tt> in natural deduction seen as a λ-term===
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| We give below a proof of <tt>(β → α) → (γ → β) → γ → α</tt> in natural deduction and show how it can be interpreted as the λ-expression λ ''a''. λ''b''. λ ''g''.(''a'' (''b'' ''g'')) of type <tt>(β → α) → (γ → β) → γ → α</tt>.
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| a:β → α, b:γ → β, g:γ {{Unicode|⊢}} b : γ → β a:β → α, b:γ → β, g:γ {{Unicode|⊢}} g : γ
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| ——————————————————————————————————— ————————————————————————————————————————————————————————————————————
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| a:β → α, b:γ → β, g:γ {{Unicode|⊢}} a : β → α a:β → α, b:γ → β, g:γ {{Unicode|⊢}} b g : β
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| ————————————————————————————————————————————————————————————————————————
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| a:β → α, b:γ → β, g:γ {{Unicode|⊢}} a (b g) : α
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| ————————————————————————————————————
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| a:β → α, b:γ → β {{Unicode|⊢}} λ g. a (b g) : γ → α
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| ————————————————————————————————————————
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| a:β → α {{Unicode|⊢}} λ b. λ g. a (b g) : (γ → β) -> γ → α
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| ————————————————————————————————————
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| {{Unicode|⊢}} λ a. λ b. λ g. a (b g) : (β → α) -> (γ → β) -> γ → α
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| ==Other applications==
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| Recently, the isomorphism has been proposed as a way to define search space partition in [[Genetic programming]].<ref>F. Binard and A. Felty, "Genetic programming with polymorphic types and higher-order functions." In ''Proceedings of the 10th annual conference on Genetic and evolutionary computation,'' pages 1187 1194, 2008.[http://www.site.uottawa.ca/~afelty/dist/gecco08.pdf]
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| </ref> The method indexes sets of genotypes (the program trees evolved by the GP system) by their Curry–Howard isomorphic proof (referred to as a species).
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| ==Generalizations==
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| The correspondences listed here go much farther and deeper. For example, cartesian closed categories are generalized by [[closed monoidal category|closed monoidal categories]]. The [[internal language]] of these categories is the [[linear type system]] (corresponding to [[linear logic]]), which generalizes simply-typed lambda calculus as the internal language of cartesian closed categories. What's more, these can be shown to correspond to [[cobordism]]s,<ref>John c. Baez and Mike Stay, "[http://math.ucr.edu/home/baez/rosetta/rose3.pdf Physics, Topology, Logic and Computation: A Rosetta Stone]", (2009) [http://arxiv.org/abs/0903.0340/ ArXiv 0903.0340] in ''New Structures for Physics'', ed. Bob Coecke, ''Lecture Notes in Physics'' vol. '''813''', Springer, Berlin, 2011, pp. 95-174.</ref> which play a vital role in [[string theory]].
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| An extended set of equivalences is also explored in [[homotopy type theory]], which is a very active area of research at this time (2013). Here, [[type theory]] is extended by the [[univalence axiom]], ('equivalence is equivalent to equality') which permits homotopy type theory to be used as a foundation for all of mathematics (including [[set theory]] and classical logic, providing new ways to discuss the [[axiom of choice]] and many other things). That is, the Curry–Howard correspondence that proofs are elements of inhabited types is generalized to the notion [[homotopy|homotopic equivalence]] of proofs (as paths in space, the [[identity type]] or [[Intuitionistic type theory#Connectives of type theory|equality type]] of type theory being interpreted as a path).<ref>[http://homotopytypetheory.org/book/ ''Homotopy Type Theory: Univalent Foundations of Mathematics'']. (2013) The Univalent Foundations Program. [[Institute for Advanced Study]].</ref>
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| ==References==
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| {{More footnotes|date=April 2010}}
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| {{Reflist}}
| |
| | |
| === Seminal references ===
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| * {{Citation | last1=Curry | first1=Haskell | title=Proceedings of the National Academy of Sciences | chapter=Functionality in Combinatory Logic | year=1934 | volume=20 | pages=584–590}}.
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| * <cite id="CurryFeys_paragraph9E">{{Citation | last1=Curry | first1=Haskell B. | last2=Feys | first2=Robert | editor1-last=Craig | editor1-first=William | title=Combinatory Logic Vol. I | publisher=North-Holland | location=Amsterdam | year=1958}}, with 2 sections by William Craig, see paragraph 9E</cite>. <!-- How to explain to the citing mechanism to tell that William Craig wrote two sections of the book? -->
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| * De Bruijn, Nicolaas (1968), ''Automath, a language for mathematics'', Department of Mathematics, Eindhoven University of Technology, TH-report 68-WSK-05. Reprinted in revised form, with two pages commentary, in: ''Automation and Reasoning, vol 2, Classical papers on computational logic 1967–1970'', Springer Verlag, 1983, pp. 159–200.
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| * {{Citation | last1=Howard | first1=William A.
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| | chapter=The formulae-as-types notion of construction
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| | pages=479–490
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| | editor1-last=Seldin | editor1-first=Jonathan P.
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| | editor1-link=Jonathan P. Seldin
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| | editor2-last=Hindley | editor2-first=J. Roger
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| | editor2-link=J. Roger Hindley
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| | title=To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism
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| | origyear=original paper manuscript from 1969
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| | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-349050-6 | month=09 | year=1980}}. <!-- How to explain to the citing mechanism to tell that the paper had remained unpublished from 1969 to 1980? -->
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| | |
| === Extensions of the correspondence ===
| |
| | |
| <references group=ext>refs=<ref name=PfenningDaviesJudgmental group=ext>{{Citation|first=Frank|last=Pfenning|first2=Rowan|last2=Davies|title=A Judgmental Reconstruction of Modal Logic|journal=Mathematical Structures in Computer Science|volume=11|pages=511–540|year=2001|url=http://www.cs.cmu.edu/~fp/papers/mscs00.pdf|doi=10.1017/S0960129501003322}}</ref> </references>
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| * {{Citation | last1=Griffin | first1=Timothy G. | title=Conf. Record 17th Annual ACM Symp. on Principles of Programming Languages, POPL '90, San Francisco, CA, USA, 17–19 Jan 1990 | year=1990 | chapter=The Formulae-as-Types Notion of Control | pages=47–57}}.
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| * {{Citation | last1=Parigot | first1=Michel | title=Logic Programming and Automated Reasoning: International Conference LPAR '92 Proceedings, St. Petersburg, Russia | publisher=[[Springer-Verlag]] | location=Berlin; New York | series=Lecture Notes in Computer Science | isbn=978-3-540-55727-2 | year=1992 | volume=624 | chapter=Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction | pages=190–201}}.
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| * {{Citation | last1=Herbelin | first1=Hugo | editor1-last=Pacholski | editor1-first=Leszek | editor2-last=Tiuryn| editor2-first=Jerzy | title=Computer Science Logic, 8th International Workshop, CSL '94, Kazimierz, Poland, September 25–30, 1994, Selected Papers | publisher=[[Springer-Verlag]] | location=Berlin; New York | series=Lecture Notes in Computer Science | isbn= 978-3-540-60017-6 | year=1995 | volume=933 | chapter=A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure | pages=61–75}}.
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| * {{Citation | last1=Gabbay | first1=Dov | last2=de Queiroz | first2=Ruy | title=Journal of Symbolic Logic | chapter=Extending the Curry–Howard interpretation to linear, relevant and other resource logics | year=1992 | volume=57 | pages=1319–1365}}. (Full version of the paper presented at ''Logic Colloquium '90'', Helsinki. Abstract in ''JSL'' 56(3):1139–1140, 1991.)
| |
| * {{Citation | last1=de Queiroz | first1=Ruy | last2=Gabbay | first2=Dov | editor1-last=Dekker | editor1-first=Paul | editor2-last=Stokhof | editor2-first=Martin | title=Proceedings of the Ninth Amsterdam Colloquium | chapter=Equality in Labelled Deductive Systems and the Functional Interpretation of Propositional Equality | year=1994|publisher=ILLC/Department of Philosophy, University of Amsterdam | isbn=90-74795-07-2 | pages=547–565}}.
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| * {{Citation | last1=de Queiroz | first1=Ruy | last2=Gabbay | first2=Dov | title=Bulletin of the Interest Group in Pure and Applied Logics | chapter=The Functional Interpretation of the Existential Quantifier | year=1995 | volume=3(2–3) | pages=243–290}}. (Full version of a paper presented at ''Logic Colloquium '91'', Uppsala. Abstract in ''JSL'' 58(2):753–754, 1993.)
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| * {{Citation | last1=de Queiroz | first1=Ruy | last2=Gabbay | first2=Dov | editor1-last=de Rijke | editor1-first=Maarten | title=Advances in Intensional Logic | chapter=The Functional Interpretation of Modal Necessity | isbn=978-0-7923-4711-8 | year=1997|publisher=[[Springer-Verlag]] |series=Applied Logic Series | volume=7 | pages=61–91}}.
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| * {{Citation | last1=de Queiroz | first1=Ruy | last2=Gabbay | first2=Dov | editor1-last=Ohlbach | editor1-first=Hans-Juergen | editor2-last=Reyle | editor2-first=Uwe | title=Logic, Language and Reasoning. Essays in Honor of Dov Gabbay | url=http://www.springer.com/philosophy/logic/book/978-0-7923-5687-5 | publisher=[[Kluwer Acad. Pub.]] | series=Trends in Logic | isbn=978-0-7923-5687-5 | year=1999 | volume=7 | chapter=Labelled Natural Deduction | pages=173–250}}.
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| * {{Citation | last1=de Oliveira | first1=Anjolina | last2=de Queiroz | first2=Ruy | title=Logic Journal of the Interest Group in Pure and Applied Logics | chapter=A Normalization Procedure for the Equational Fragment of Labelled Natural Deduction | publisher=[[Oxford Univ Press]] | year=1999 | volume=7 | pages=173–215 | issue=2}}. (Full version of a paper presented at ''2nd WoLLIC'95'', Recife. Abstract in ''Journal of the Interest Group in Pure and Applied Logics'' 4(2):330–332, 1996.)
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| * {{citation | last1=Poernomo | first1=Iman | last2=Crossley | first2=John | last3=Wirsing | last4=Martin | title=Adapting Proofs-as-Programs: The Curry–Howard Protocol | publisher=[[Springer Science+Business Media|Springer]] | year=2005 | origyear=2005 | isbn=978-0-387-23759-6 | series=Monographs in Computer Science}}, concerns the adaptation of proofs-as-programs program synthesis to coarse-grain and imperative program development problems, via a method the authors call the Curry–Howard protocol. Includes a discussion of the Curry–Howard correspondence from a Computer Science perspective.
| |
| * {{Citation | last1=de Queiroz | first1=Ruy J.G.B. | last2=de Oliveira | first2=Anjolina | title=Electronic Notes in Theoretical Computer Science | chapter=The Functional Interpretation of Direct Computations | doi=10.1016/j.entcs.2011.03.003 | publisher=[[Elsevier]] | year=2011 | volume=269 | pages=19–40}}. (Full version of a paper presented at ''LSFA 2010'', Natal, Brazil.)
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| | |
| === Philosophical interpretations ===
| |
| * {{Citation | last1=de Queiroz | first1=Ruy J.G.B. | title=Dialectica | chapter=Normalisation and language-games | chapter-url=http://www3.interscience.wiley.com/journal/119262585/abstract?CRETRY=1&SRETRY=0 | year=1994 | volume=48 | pages=83–123 | issue=2}}. (Early version presented at ''Logic Colloquium '88'', Padova. Abstract in ''JSL'' 55:425, 1990.)
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| * {{Citation | last1=de Queiroz | first1=Ruy J.G.B. | title=Logic Journal of the Interest Group in Pure and Applied Logics | chapter=Meaning, function, purpose, usefulness, consequences – interconnected concepts | chapter-url=http://jigpal.oxfordjournals.org/cgi/content/abstract/9/5/693 | year=2001 | volume=9 | pages=693–734 | issue=5}}. (Early version presented at ''Fourteenth International Wittgenstein Symposium (Centenary Celebration)'' held in Kirchberg/Wechsel, August 13–20, 1989.)
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| * {{Citation | last1=de Queiroz | first1=Ruy J.G.B. | title=Studia Logica | chapter=On reduction rules, meaning-as-use and proof-theoretic semantics | chapter-url=http://www.springerlink.com/content/27nk266126k817gq/ | year=2008 | volume=90 | pages=211–247 | issue=2}}.
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| | |
| === Synthetic papers ===
| |
| * {{Citation | last1=De Bruijn | first1=Nicolaas Govert | editor1-last=Groote | editor1-first=Philippe de | title=The Curry–Howard isomorphism | publisher=Academia-Bruyland | series=Cahiers du Centre de logique (Université catholique de Louvain) | isbn=978-2-87209-363-2 | volume=8 | chapter=On the roles of types in mathematics | chapterurl=http://alexandria.tue.nl/repository/freearticles/597627.pdf | pages=27–54 | year=1995}}, the contribution of de Bruijn by himself.
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| *{{Citation | last1=Geuvers | first1=Herman | editor1-last=Groote | editor1-first=Philippe de | title=The Curry–Howard isomorphism | publisher=Academia-Bruyland | series=Cahiers du Centre de logique (Université catholique de Louvain) | isbn=978-2-87209-363-2 | volume=8 | chapter=The Calculus of Constructions and Higher Order Logic | pages=139–191 | year=1995}}, contains a synthetic introduction to the Curry–Howard correspondence.
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| * {{Citation | last1=Gallier | first1=Jean H. | editor1-last=Groote | editor1-first=Philippe de | title=The Curry–Howard isomorphism | publisher=Academia-Bruyland | series=Cahiers du Centre de logique (Université catholique de Louvain) | isbn=978-2-87209-363-2 | volume=8 | chapter=On the Correspondence between Proofs and Lambda-Terms | chapterurl=ftp://ftp.cis.upenn.edu/pub/papers/gallier/cahiers.pdf | pages=55–138 | year=1995}}, contains a synthetic introduction to the Curry–Howard correspondence.
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| | |
| === Books ===
| |
| *{{Citation | editor1-last=De Groote | editor1-first=Philippe | title=The Curry–Howard Isomorphism | publisher=Academia-Bruylant | series=Cahiers du Centre de Logique (Université catholique de Louvain) | isbn=978-2-87209-363-2 | year=1995 | volume=8 | author=edited by Ph. de Groote.}}, reproduces the seminal papers of Curry-Feys and Howard, a paper by de Bruijn and a few other papers.
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| * {{citation | last1=Sørensen | first1=Morten Heine | last2=Urzyczyn | first2=Paweł | title=Lectures on the Curry–Howard isomorphism | publisher=[[Elsevier Science]] | year=2006 | origyear=1998 | isbn=978-0-444-52077-7 | volume=149 | series=Studies in Logic and the Foundations of Mathematics | url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7385}}, notes on proof theory and type theory, that includes a presentation of the Curry–Howard correspondence, with a focus on the formulae-as-types correspondence
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| * Girard, Jean-Yves (1987–90). ''[http://www.monad.me.uk/stable/Proofs+Types.html Proof and Types]''. Translated by and with appendices by Lafont, Yves and Taylor, Paul. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7), ISBN 0-521-37181-3, notes on proof theory with a presentation of the Curry–Howard correspondence.
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| * Thompson, Simon (1991). ''[http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ Type Theory and Functional Programming]'' Addison–Wesley. ISBN 0-201-41667-0.
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| * {{citation | last1=Poernomo | first1=Iman | last2=Crossley | first2=John | last3=Wirsing | last4=Martin | title=Adapting Proofs-as-Programs: The Curry–Howard Protocol | publisher=[[Springer Science+Business Media|Springer]] | year=2005 | origyear=2005 | isbn=978-0-387-23759-6 | series=Monographs in Computer Science}}, concerns the adaptation of proofs-as-programs program synthesis to coarse-grain and imperative program development problems, via a method the authors call the Curry–Howard protocol. Includes a discussion of the Curry–Howard correspondence from a Computer Science perspective.
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| * F. Binard and A. Felty, "Genetic programming with polymorphic types and higher-order functions." In ''Proceedings of the 10th annual conference on Genetic and evolutionary computation,'' pages 1187 1194, 2008.[http://www.site.uottawa.ca/~afelty/dist/gecco08.pdf]
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| * {{citation | last1=de Queiroz | first1=Ruy J.G.B. | last2=de Oliveira | first2=Anjolina G. | last3=Gabbay | first3=Dov M. | title=''[http://www.worldscibooks.com/compsci/8215.html The Functional Interpretation of Logical Deduction]'' | publisher=[[Imperial College Press / World Scientific]] | year=2011 | origyear=2011 | isbn=978-981-4360-95-1 | volume=5 | series=Advances in Logic }}.
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| == Further reading ==
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| * [[P.T. Johnstone]], 2002, ''Sketches of an Elephant'', section D4.2 (vol 2) gives a [[categorical logic|categorical]] view of "what happens" in the Curry–Howard correspondence.
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| ==External links==
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| {{Wikibooks|Haskell|The Curry–Howard isomorphism}}
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| *[http://www.thenewsh.com/~newsham/formal/curryhoward/ The Curry–Howard Correspondence in Haskell]
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| *[http://www.haskell.org/wikiupload/1/14/TMR-Issue6.pdf The Monad Reader 6: Adventures in Classical-Land]: Curry–Howard in Haskell, Pierce's law.
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| {{DEFAULTSORT:Curry-Howard Correspondence}}
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| [[Category:1934 in computer science]]
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| [[Category:1958 in computer science]]
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| [[Category:1969 in computer science]]
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| [[Category:Dependently typed programming]]
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| [[Category:Proof theory]]
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| [[Category:Logic in computer science]]
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| [[Category:Type theory]]
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| [[Category:Philosophy of computer science]]
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