aspects of type theory relevant for the Curry-Howard isomorphism. Outline . (D IK U). Roughly one chapter was presented at each lecture, sometimes. CiteSeerX – Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The Curry-Howard isomorphism states an amazing correspondence between. Lectures on the. Curry-Howard Isomorphism. Morten Heine B. Sørensen. University of Copenhagen. Pawe l Urzyczyn. University of Warsaw.
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Log in lecttures sign up in seconds. The internal language of these categories is the linear type system corresponding to linear logicwhich generalizes simply-typed lambda calculus as the internal language of cartesian closed categories.
The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems e. These notes give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. Well-defined morphisms typed terms are defined by the following typing rules in which the usual categorical morphism notation f: Abstract 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.
Juan Bono marked it as to-read Apr 19, I will give it a go. Views Read Edit View history. Here is a non-exhaustive list:. In general, the procedure is that whenever the program contains an application of the form P Qthese steps should be followed:.
Lambda-calculus, types and models isomorpnism, Ellis Horwood, 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. Account Options Fazer login.
Course 6 17th Oct. That is, the Curry—Howard correspondence that proofs are elements of inhabited types is generalized to the notion homotopic equivalence of proofs as paths in space, the identity type or equality type of type theory being interpreted as a path.
Jun 19, Saeede Kermani rated it really liked it. Bob Coecke, Lecture Notes in Physics vol. For instance, it is an old ideadue to Brouwer, Kolmogorov, and Heytingthat 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.
Joachim Lambek showed in the early s 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. Quantifiers correspond to dependent function space or products as appropriate. Chapter 13 Dependent types. Advanced Search Include Citations. There’s more that could be said but that’s a taste.
curry-howatd Can you recommend few books for self-learner? One at the level of formulas and types that is independent of which particular proof system or model of computation is considered, and one at the level of proofs and programs which, this time, is specific to the particular choice of proof system and model of computation considered.
Think of a type in your program. For instance, the fact that the combinator X constitutes a one-point basis of extensional combinatory logic implies that the single axiom scheme. 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.
Durant carried the game with 42 pts. Please help to improve this article by introducing more precise citations.
reference request – Book on Curry-Howard Isomorphisms – Mathematics Stack Exchange
Nick marked it as to-read Apr 26, Post as a guest Name. Stanley marked it as to-read Jun 13, This page was last edited on 19 Augustat Girard’s linear logic was developed from the fine analysis of the use of resources in some models of lambda calculus; is there typed version of Turing’s machine that would behave as a proof system?
Sequent Calculus for Predicate Logic: Sequent calculus is characterized by the presence of left introduction rules, right introduction rule and a cut rule that can be eliminated. Thanks to the Curry—Howard correspondence, a typed expression whose type corresponds to a logical formula is analogous to a proof of that formula.
Typing rules and their correspondence with Natural Deduction Rules summary in these [slides] Course 4 3rd Oct. Chapter 12 Secondorder arithmetic. Sandro Mancuso marked it as to-read Apr 21, Examples of programs seen as proofs in a Hilbert-style logic are given below. 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.
Ben Jeffrey marked it as to-read Jun 15, Lectures on the Curry-Howard Isomorphism.
Course notes of Laurent, O. Alyssa Picard added it Dec 29, For those topics, please consider one of the subreddits in the sidebar instead.
The correspondences listed here go much farther and deeper. CreateInstance any type, or have a reference to any type that is actually null, correspond to trivial logics in which everything is true.
Hardcoverpages. An extended set of equivalences is also explored in homotopy type theorywhich became a isomorphsim active area of research around and as of [update] still is.