Church Encoding, Parametricity, and the Yoneda Lemma
The article explores the concepts of Church encoding, parametricity, and the Yoneda Lemma within functional programming. It discusses how natural numbers and lists can be represented as functions, revealing deeper connections between data types and category theory. The author emphasizes the importance of System F, which allows for polymorphic functions and the encoding of types from scratch.
- ▪Church encoding represents natural numbers as functions, where each number applies a successor function a specified number of times.
- ▪System F, the polymorphic lambda calculus, allows functions to take types as arguments, enabling greater flexibility in programming.
- ▪The Yoneda Lemma illustrates that the ways to consume a type fully determine the type itself, highlighting a fundamental principle in category theory.
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May 19, 202629 mins readTable of ContentsChurch Encoding, Parametricity, and the Yoneda LemmaSystem FParametricityAlgebraic Data Types and FunctorsF-AlgebraInitial AlgebraThe Yoneda LemmaThe Final Recap Church Encoding, Parametricity, and the Yoneda LemmaI still remember the shock I felt when I first encountered functional programming years ago. That was the moment I learned that natural numbers can be built within the language itself:data Nat = Zero | Succ NatI went on to learn that all computation can be expressed through functions (the lambda calculus), that recursion itself can be encoded as the mind-bending Y combinator:Y = λf. (λx. f (x x)) (λx. f (x x))And then there were Church numerals, where each number becomes a function:0 = λs. λz. z 1 = λs. λz. s z 2 = λs. λz.
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