In particular, linguists use typed λ-calculi to construct formal semantics for sentences. One surprising use for the λ-calculus outside of CS is in linguistics. Seeing proofs as programs really helped me come to terms (heh) with mathematical logic. I really like this notion because it gives me a concrete, first-class object to represent a proof. In some sense, it is a sort of "proof essence": the basic building block of what a proof is. Of course, as the accepted answer points out, the λ-calculus is also very useful for logic. Haskell uses a variant of System F which it actually calls "Core" to underscore how core this concept is :). Most of Haskell is like this: while the surface language has become rather complex, the majority of the features boil away rather transparently into typed lambda terms. If you can express a feature just in terms of the λ-calculus, you can implement it almost for free. It's also great as the core of a programming language. Fast iteration for programming language designers. It allows somebody designing a language feature or part of a type system to experiment with that feature in isolation. I don't know much about it, but it appears to provide a powerful set of tools for topology.The λ-calculus is basically the MVP of programming languages. It's one of the most fascinating new fields of study in math to emerge over the last 3 decades in my opinion.Ībstract Stone Duality is basically a lambda-calculus for topology. It also serves as a powerful new foundation of mathematics in its own right. It provides powerful tools for the study of homotopy theory and homotopy types. Homotopy Type Theory, and dependent type theory in general, is an extension of simply typed lambda calculus. The untyped lambda calculus is the language of PCAs and allows one to simply prove many results about them. Partial combinatory algebras allow one to develop all kinds of interesting models of set theory which have computational semantics. Understanding partial combinatory algebras.Researchers will develop variants of the lambda calculus to model the behaviour of real-world languages and then prove certain desirable properties hold about these versions of lambda calculus. There are also variants of the lambda calculus which model safe parallel programming, linear types, and many other intriguing concepts which are making their way into modern languages. When researchers set out to prove that Rust's safety guarantees actually meant that all code with no Unsafe blocks was safe, they developed a variant of the lambda calculus to do it. Modelling "unusual" kinds of computation.Simply typed lambda calculus is the "internal language" of these categories. This is the origin of the "propositions as types" Curry-Howard-Lambek correspondence. Other examples of a Cartesian Closed Category include any Heyting algebra, the algebras used to interpret (possibly non-classical) propositional logic. This includes important categories like the category of Sets and the category of directed graphs as well as other, more advanced examples like categories of sheaves on a site, which are highly useful in algebraic geometry and related fields. All Cartesian closed categories have models of the simply typed lambda calculus. Understanding Cartesian Closed CategoriesĬartesian closed categories are categories with a notion of "Cartesian product" and a notion of "exponential objects" (which are the analogue of the set $\$).Almost every modern language has support for defining a function using lambda. Functional languages tend to be particularly friendly to lambda.Įven languages which originally rejected lambda have now come around - even C++ and Java now include syntax inspired by that of the lambda calculus. Lambda calculus is used as the foundation for many of the most powerful and elegant programming languages, including Haskell and dialects of Lisp. Lambda calculus was the first formalism of "computability" in which what is now known as the "halting problem" was proved undecidable (by Alonzo Church, who mentored the more famous Alan Turing). Providing a foundational understanding of computation.Some applications of lambda calculus include: Contrast the definition of the untyped lambda calculus with the definition of a Turing machine. The only concept needed to understand what's going on is the concept of applying a function with a known definition to a value. The lambda calculus is far simpler than any other model of computation I know of. I highly disagree that the lambda calculus is too abstract and theoretical.
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