Gadts in the lambda calculus and a proof of progress, preservation. The first language discussed is a variant of scott and plotkins pcf, which adds to the simplytyped lambda calculus products, fixed points of functions, and algebraic data types specified by a. The churchscott representation of inductive and coinductive data. Another result of the paper is an algebraic proof of consistency of the in nitary lambda calculus. Stratified type inference for generalized algebraic data types by francois pottier and yann regisgianas. Three interrelated topics at the heart of logic and computer science. A guarded recursive datatype is a datatype with local i.
Algebraic and topological methods in lambda calculus. The curryhoward isomorphism started life as an observation that the terms and typejudging rules of simplytyped lambda calculus correspond exactly to natural deduction as formulated by gentzen applied to intuitionistic propositional logic, with types taking the place of propositions, and terms taking the place of proofs, despite the two being independently inventeddiscovered. Comparative programming languages the lambda calculus haskell. The transformation is based on using the scottencoding for algebraic data types instead of the more common church encoding. Both 2 and 17 are concerned with a lambda calculus endowed with a structure of vector space. Singlestep reduction is defined as the union of the singlestep. To appear in mathematical structures in computer science, 20. We define variable substitution as usual and consider terms up to. Another, more recent take on pure lambda calculus as a certain kind of algebraic theory called a lambda theory can be found in. Semantics of a typed algebraic lambdacalculus arxiv.
It contains also a lot of links to other papers on gadts. Applications of generalized algebraic data types in ocaml. More than 40 million people use github to discover, fork, and contribute to over 100 million projects. Coq crash course examples in operational semantics pdf unit 2. In fact, sapl is equivalent to untyped lambda calculus with named. Algebraic data types and patternbased definitions need a special implemen tation in. Churchscott datatype as a kind of union of the two that supports primitive recursion.
Recall the definition of the simply typed lambda calculus, a small programming language. The language we consider is lambda calculus extended with a con structors and a form of type construct we call open case expressions. In computer programming, especially functional programming and type theory, an algebraic data type is a kind of composite type, i. Hindleymilner type inference and polymorphic types pdf 7. Pdf a typed lambda calculus with categorical type constructors. Thecalculus a pure calculus of functional abstraction and function application, with applications throughout logic and computer science.
A complete implementation of predicative system f with bidirectional type inference and algebraic data types. Data types and pattern matching by function application citeseerx. Nominal coalgebraic data types with applications to lambda calculus 3 alphastructural recursion pit05, pit06, pit11 is the induction principle that ensues from syntax as an initial algebra in the category nom of nominal sets. We will give reduction rules for this simply typed lambda calculus. Abstract data types 8 abstraction and types algebraic speci cation modules versus classes types as constraints additional keywords and phrases. Types the calculus is good tool for exploring type systems, invaluable both in the foundations of mathematics and for. The course is entitled lambda calculus and types and it appears in various forms. The lambdacalculus is nominal algebraic request pdf.
The meanings of the unusual terms are explained in the next section. Keywords and phrases lambda calculus, data types, coinductive types, iteration, primitive recursion. Add a description, image, and links to the lambdacalculus topic page so that developers can more easily learn about it. Gadts theory generalized algebraic data types were introduced by xi, chen and chen under the name guarded recursive datatype constructors 5. Request pdf on jan 1, 2008, mj gabbay and others published the lambdacalculus is nominal algebraic find, read and cite all the research you need on researchgate.