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Topic: Curry-Howard isomorphism

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Isomorphism Isomorphism class An isomorphism class is a collection of mathematical objects isomorphic with a certain mathematical ob... Group isomorphism In group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective fun... Isomorphism theorem In universal algebra, stating the existence of certain natural isomorphisms. www.brainyencyclopedia.com /topics/isomorphism.html

Curry-Howard - Wikipedia, the free encyclopedia A second aspect of the Curry-Howard isomorphism is that a program whose type corresponds to a logical formula is itself analogous to a proof of that formula. Therefore one concrete realisation of the Curry-Howard isomorphism is to study in detail how proofs from intuitionistic logic should be written into lambda terms. Secondly, and more formally, it specifies a formal isomorphism between two mathematical domains, that of natural deduction formalised in a particular manner, and of the simply-typed lambda calculus, where there is a bijection between, in the first place proofs and terms, and in the second, proof reduction steps and beta reductions. en.wikipedia.org /wiki/Curry-Howard

curry Curry -Howard isomorphism In theoretical computer science, the Curry -Howard isomorphism is an important underlying principle connecting the... Curry powder Curry powder is a mixture of spices of widely varying composition developed by the British during... Curry Curry is a distinctively spiced dish which is common in Indian cuisine but is found in the... www.wikisearch.net /curry

15-399, 80317/617 Constructive Logic We devolop the isomorphism between proofs and programs, and discuss the notion of reduction, which is the basis of an operational semantics for the small programming language. www.andrew.cmu.edu /course/80-317/lectures/lecture10.html

Fall 99, CSE 520: Lectures The Curry-Howard isomorphism is a guide to the design of types and constructs in programming languages. There is an interesting correspondence between the symply typed lambda calculus and the intuitionistic propositional logic, known as Curry-Howard isomorphism. In fact, given a term M with type T, the proof of M:T is isomorphic to the parse tree of M. www.lix.polytechnique.fr /~catuscia/teaching/cg520/99Fall/lecture_notes/L11and12.html

A New Basic Set or Proof Transformations Thus, because of the Curry-Howard isomorphism, we have that every LND derivation converts to a normal form (normalization theorem) and it is unique (strong normalization theorem) [deOl95,deOldeQ95]. A New Basic Set of Transformations between Proofs (Abstract) (Submitted for presentation at Logic Colloquium '95, August 9-17, Haifa, Israel) Deductive systems based on the so-called Curry-Howard isomorphism [How80] have an interesting feature: normalization and strong normalization (Church-Rosser property) theorems can be proved by reductions on the terms of the functional calculus. In J.R. Seldin and J.R. Hindley (editors), To H. Curry: Essays on Combinatory Logic Lambda Calculus and Formalism. www.seas.upenn.edu /~sweirich/types/archive/1995/msg00086.html

NASSLLI 2002 Proof polynomials considerably extend the Curry-Howard Isomorphism and lead to a joint calculus of propositions and proofs which unifies modal and epistemic logic with combinatory logic and typed lambda-calculus. This is the Curry-Howard Isomorphism between typed lambda-terms and intuitionistic deductions. A discovery of a natural system of self-referential proof terms, which we call "proof polynomials", was essential in recent solution to the famous open problem of Goedel (1933) concerning formalization of provability. www.stanford.edu /group/nasslli/courses/Proof.htm

DIKU Graduate/Ph.D. course: The Curry-Howard isomorphism The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene's realizability interpretation---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The isomorphism has many aspects, even at the syntactic level: formulae correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. Moreover, there are many other aspects of the isomorphism. www.diku.dk /topps/activities/type-theory

Lambda calculus - Wikipedia, the free encyclopedia A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see currying). For instance, the function f ( x, y) = x - y would be written as λ x. en.wikipedia.org /wiki/Lambda_calculus

[Haskell] Correct interpretation of the curry-howard isomorphism Regarding Curry-Howard isomorphism: it has two parts: - a term is an encoding of a proof of a formula that is its type - proof normalization corresponds to term normalization (Thanks to Ken Shan for explaining this). Let us concentrate on the first part (which we would call Curry-Howard correspondence). > So, either the interpretation of the isomorphism is wrong, or Haskell > type syste m is in fact unsound. www.haskell.org /pipermail/haskell/2004-April/013993.html

Logic and Computation For instance, it is an old idea (due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene's realizability interpretation) that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism states a correspondence between systems of formal logic and computational calculi. In this seminar we will study the Curry-Howard isomorphism and cover relevant parts of proof theory and related aspects of typed lambda-calculi. cs-people.bu.edu /metayer/logic.html

Articles - Calculus of constructions 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 can be considered an extension of the Curry-Howard isomorphism. The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions"). www.kamero.net /articles/Calculus_of_constructions?mySession=7368ff828e5ff0fd9f0cdae6bd7456fb

The Curry-Howard Isomorphism However, this brings up an important point about the Curry-Howard Isomorphism, which basically says that types are statements in first-order logic, and the corresponding values are proofs of those statements. This equivalence can be "proven" using the function: swap :: (a,b) -> (b,a) swap (x,y) = (y, x) Which, using the Curry-Howard isomorphism, is also a proof that /\ is commutative. In short, just because a type "a" implies type "b", and type "b" implies type "a", doesn't mean they are interchangeable as statements are in logic. www.haskell.org /pipermail/haskell-cafe/2002-August/003300.html

Diary for chalst The labels are needed for the isomorphism to work, but before the work of Curry and Howard, no-one working with natural deduction thought to keep track of these; instead the assumption was simply that all assumptions matching the implies elimination rule are discharged. There's something artificial about Howard's isomorphism: while the simply-typed lambda calculus is a calculus of an obviously fundamental nature, there's something forced about the logical end, namely the idea of labelling assumptions and implies elimination rules. I've got a long past with this topic, so much so that I probably can't be objective about certain details: my doctorate was on the so-called classical Curry-Howard correspondence and begins with a rant about the misleading way the topic is made use of in many works in the field. www.advogato.org /person/chalst/diary.html?start=131

Project-Team-CALLIGRAMME Although the Curry-Howard isomorphism owes its existence to the functional character of intuitionistic logic, it can be extended to fragments of classical logic. This extension of the Curry-Howard isomorphism to classical logic and its applications has a perennial place as research field in the project. It turns out that some constructions that one meets in functional progamming languages, such as control operators, can presently only be explained by the use of deduction rules that are related to proof by contradiction [42] www.inria.fr /rapportsactivite/RA2004/calligramme2004/uid10.html

Curry-Howard Isomorphism and Intuitionistic Linear Logic - Roversi (ResearchIndex) Abstract: The notion of Curry-Howard Isomorphism (CHI) was originally introduced for formalizing to which extent the computational behavior of the typed -calculus (fi t) is joined at the semantics of the natural deduction for Intuitionistic Logic (IL). , used to encode Intuitionistic Linear Logic [11] by Curry Howard Isomorphism [13] refine the simply typed typable... @misc{ roversi96curryhoward, author = "L. Roversi", title = "Curry-Howard isomorphism and intuitionistic linear logic", text = "L. Roversi. citeseer.ist.psu.edu /106731.html

LtU Classic Archives This is a very good introduction to an important idea: The Curry-Howard Isomorphism between systems of formal logic and computational type systems. The isomorphism has many aspects, even at the syntactic level: formuals correspond to types, proofs correspond to terms, provability corresponds inhabitation, proof normalization corresponds to term reduction etc. (from the preface) The text is fairly self contained, and starts from an explanation of the Lambda Calculus ( LC) and the typed lambda calculus. lambda-the-ultimate.org /classic/message2352.html

Cahiers du centre de logique, volume 8 This volume is devoted to the Formulae-as-Types correspondence, also widely known as the Curry-Howard isomorphism. DE GROOTE (ed.), The Curry-Howard Isomorphism, volume 8 of the Cahiers du Centre de logique, Academia, Louvain-la-Neuve (Belgium), 1995, 364 pages, ISBN: 2-87209-363-X. This Cahier can be ordered from the publisher: H.B. CURRY and R. The basic theory of functionality. www.lofs.ucl.ac.be:16080 /cnrl/Cahiers/Cahier8.html

Proof-Term Synthesis on Dependent-Type Systems via Explicit Substitutions - Storming Media This is possible since the Curry-Howard isomorphism relates proof trees with typed lambda-terms. Abstract: Typed lambda-terms are used as a compact and linear representation of proofs in intuitionistic logic. The proofs-as-terms principle can be used to check a proof by type checking the lambda-term extracted from the complete proof tree. www.stormingmedia.us /07/0781/A078173.html

Summary: logic texts for computer scientists The first part of the notes covers natural deduction, sequent calculus, and the simply typed lambda-calculus in the context of the Curry-Howard isomorphism. Intuitionistic logic and types (This chapter includes a discussion of the Curry-Howard isomorphism.) 10. ---------------------------------------------------------------------- From: Andrew Ian Schein I am not sure this is as broad a text as you would prefer, but Philip Wadler's article: "A taste of linear logic" served me as a straightforward introduction to intuitionistic logic, linear logic, and the Curry-Howard Isomorphism. www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg00127.html

tr-2002-117-abs.html The Curry-Howard isomorphism permits the representation of intuitionistic logic as a constructive type thoery. In this paper, we present a constructive logical system for reasoning about imperative programs to which the Curry-Howard isomosphism may be adapted. This allows us to take advantage of the isomorphism for theorem proving implementation, and for the synthesis of correct imperative programs, following the proofs-as-programs paradigm. www.csse.monash.edu.au /publications/2002/tr-2002-117-abs.html

An Introduction to the Curry-Howard Correspondence Disscussion of the Curry-Howard Isomorphism or Propositions-as-Types Analogy as an amazingly beautiful, powerful and robust mathematical idea. research.microsoft.com /research/pubs/view.aspx?pubid=704

CTO : Programming Languages Coq - An higher-order proof system based on the Curry-Howard isomorphism between propositions and type s, proof s and terms in a pure functional programming language : a functional term is a proof of its type's realizability Logic - The term for a paradigm of programming that originates in the Curry-Howard Isomorphism, which relates type s and proof s to terms and expressions Curry - A functional logic programming language with many implementations cliki.tunes.org /Programming%20Languages

Heyting algebra Details, Meaning Heyting algebra Article and Explanation Guide See Curry-Howard isomorphism for the general context, of what this implies in type theory. From what has just been said, this does show that it cannot be derived. www.e-paranoids.com /h/he/heyting_algebra.html

Ftp-able papers Errata for Lecture Notes on the Curry-Howard Isomorphism --- with M.H. Sorensen Lecture Notes on the Curry-Howard Isomorphism --- with M.H. Sorensen Lecture notes for School of Logic and Computation, Edinburgh, April 1999 slides (corrected) zls.mimuw.edu.pl /~urzy/ftp.html

Technical report: Intellektik 91-21 The Curry-Howard Isomorphism is used to represent proof constructions in a term-functional language and to specify analogies by transformation rules on these terms. Building Proofs by Analogy Using NuPRL's Built-in Curry-Howard Isomorphism The method has the advantage to admit complete formalization and to make use of well-known techniques like higher-order unification. www.intellektik.informatik.tu-darmstadt.de /TR/1991/91-21.html

CTO : Logic The term for a paradigm of programming that originates in the Curry-Howard Isomorphism, which relates type s and proof s to terms and expressions. SPIKE was written in Caml Light, a functional language of the ML family. In most cases, logical deduction or inference is equated with term evaluation or state update. cliki.tunes.org /Logic

milScript.sml (* =====================================================================*) (* FILE : milScript.sml *) (* DESCRIPTION : Defines a proof system for minimal intuitionistic *) (* logic, and proves the Curry-Howard isomorphism for *) (* this and typed combinatory logic. www.comp.nus.edu.sg /~hugh/chlproject/CODE/milScript.sml

Thirty Five years of Automath For example, de Bruijn indices still play an important role in the implementation of programming languages and theorem provers, de Bruijn's new type systems were influential in the discovery of new powerful type systems, and de Bruijn re-invented the Curry-Howard isomorphism (which should be referred to as the Curry-Howard-de Bruijn isomorphism). During his work on Automath, de Bruijn discovered many concepts that still remain of great relevance to the theory and practice of computation. The Landau book on the foundations of analysis remains the only fully encoded and checked mathematical book in any theorem prover. www.cee.hw.ac.uk /~fairouz/automath2002

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