In C#, they can be viewed as t => f => t and t => f => f, which are curried from (t, f) => t and (t, f) => f. In this tutorial, for consistency and intuition, function definition with multiple variables is always represented in the latter curried form. True function simply output the first parameter, and False function output the second parameter: True := λtf.tĪs fore mentioned, λtf.E is just the abbreviation of λt.λf.E, so these definitions actually are: True := λt.λf.t Church Booleanīoolean values True and False can be both represented by anonymous function with 2 parameters.
This part discusses Church Boolean - modeling Boolean values and logic operators with functions. Church encoding is named after Alonzo Church, who first discovered this approach. data and operation can be modeled by higher-order anonymous functions and their application. Anonymous function is actually very powerful. Lambda calculus is a formal system for function definition and function application, so in lambda calculus, the only primitive is anonymous function. NET Lambda Calculus Functional Programming Church Encoding Church Booleans Let us know what you think about this lambda-calculus tutorial in the comments below….LINQ via C# C#. Let’s check that conjunction respects the truth table: (disp ((disjunction T) T) This is exactly the definition of the disjunction in logic.
#Lambda calculus boolean code
Let’s check that conjunction respects the truth table: (disp ((conjunction T) T)īasically, this code says: if x is T return T else return y. This is exactly the definition of the conjunction in logic. We will answer that in another article, when we present the Y-combinator.īack to our boolean operations… Conjunction (defn conjunction īasically, this code says: if x is T return y else return F. You might wondering how we are going to implement recursions in a language with no names? It works exactly the same as negation: (dispįrom now on, we will use names to make our code readable and we will keep in the back on our mind that in real lambda calculus there are no names. In real lambda calculus, we cannot name things, so the real implementation of negation is: (defn negation-lambda Let’s check that negation respects the truth table: (disp This is exactly what the negation is supposed to do. And if x is F then return the second argument - which is T. Keep in mind that in boolean algebra, the operations are defined by their truth tables: xīasically, what this code says is: if x is T then return the first argument - which is F. Now, we are going to build the basic boolean operations, using the definition of T and F: In a nutshell, T returns the first argument and F returns the second argument. They are invokable as functions: ((T T) F) Now, the lambda booleans are viewed properly: Now, let’s redefine T and F using Lambda type: (def T (Lambda. (Check here for an explanation about the code of the Lambda type.) (-pr-writer (-write writer (str (view-bool f)))) Like this (defn view-bool Īnd it works as expected: T returns "T" (view-bool T)Īs we did in Numbers and Arithmetics with functions only: lambda calculus live tutorial., we are going to define the Lambda type to make the visualisation and comparison of the lambda booleans more convenient. You can see T as a 2-arity function that returns the first argument and F as a 2-arity function that returns the second argument.Īs we did with numerals, we can view a lambda boolean by passing to it T and F. Here is how we define T ( true) and F ( false) in lambda calculus: (defn T In order to load code from a github repository, we use data-external-libs attribute on the code snippet as explained here. More about the disp macro in this article… The code snippets are powered by the Klipse plugin.įirst, let’s create a namespace for our journey, and refer the disp macro, from the gadjett repository on github:
#Lambda calculus boolean free
We are using clojure for the code snippets - as it belongs to the LISP family, and LISP is founded on top of lambda calculus.Īll the code snippets on this article are live and interactive: feel free to modify the code and it will evaluate instantaneously! And to show the code of the basic boolean operations: negation, conjunction and disjunction a.k.a not, and and or. The purpose of this article is to show how we represent boolean values in lambda calculus. In our previous article, we showed how the numbers are represented in lambda calculus.
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