12/18/2023 0 Comments Type lambda calculus![]() $\hspace(a)$, we shouldĬhoose a fresh variable name for $x$ and perform $\alpha$-conversion on theĪbstraction before proceeding. What do we mean by free occurrences of a variable? Consider Performing substitution is a bit tricky, so before we give a type-levelĪlgorithm to do so, we should talk about some subtle cases. $\beta$-reductions can be applied are called $\beta$-redexes. ( reducible expressions) in particular, expressions to which Where the substitution $b$ replaces free occurrences of the variable $x$ in $b$ with $a$.Įxpressions to which reductions can be applied are called redexes The untyped lambda calculus is a language consisting of three forms: The latter and an introduction to computation with the TS type system, see my However, assume familiarity with TypeScript's syntax and type system. Nevertheless, I will not assume existing knowledge of the lambda calculus. Perhaps only the type-level implementation in TypeScript will be interesting. Our construction of the lambdaĬalculus will not be anything new, so if you are already familiar with it, ![]() I mention these consequences not because they're particularly important to know,īut to say that applications of the lambda calculus across mathematics andĬomputer science are well known. Logic, and category theory are equivalent. That says that computation in the model of typed lambda calculi, mathematical Known as the Curry-Howard-Lambek correspondence The simply typed lambda calculus is part of a beautiful trilogy With types, known as the simply typed lambda calculus. Almost all functional languages, type systems,Īnd type-based theorem provers are built on an extension of the lambda calculus The lambda calculus is one of the most well-studied topics in computer science,įundamental in computational theory. Similarly to typical functional implementations of the lambda calculus. Perhaps this because it is straightforward (as we will see), or Turing machineįurthermore, our construction will not be esoteric in fact, it will read However, IĪm not aware of any prior implementations of the lambda calculus in the type system. Type system, we will have a proof that the type system is Turing-complete It has been well-knownįor several years that TypeScript's type system is Turing-complete. So, if we can emulate the lambda calculus in the TypeScript In other words, the lambda calculus can simulate anyĬomputer algorithm. But there are programs for whichĪ Turing machine does not halt. Out that all well-typed terms in the typed lambda calculus halt. Terms that are "well-typed", is not computationally equivalent to the Turing machine. Turing machine model, which means that the lambda calculus can compute anythingĪ Turing machine can The typed lambda calculus, which only admits The (untyped) lambda calculus is a model of computation equivalent to the This time, we'll solve the computationally-harder problem of emulating the We weren't trying to do everything a Turing machine can do. ![]() Those problems were relatively simple as far as computational complexity goes. It's been half year since I last wrote about exploiting the TypeScript type ![]() About cc visual Emulating the Lambda Calculus in TypeScript's Type System ![]()
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