LambdaCalculus 1.0.0-lambda

This is a prerelease version of LambdaCalculus.
Install-Package LambdaCalculus -Version 1.0.0-lambda
dotnet add package LambdaCalculus --version 1.0.0-lambda
<PackageReference Include="LambdaCalculus" Version="1.0.0-lambda" />
For projects that support PackageReference, copy this XML node into the project file to reference the package.
paket add LambdaCalculus --version 1.0.0-lambda
The NuGet Team does not provide support for this client. Please contact its maintainers for support.
#r "nuget: LambdaCalculus, 1.0.0-lambda"
#r directive can be used in F# Interactive, C# scripting and .NET Interactive. Copy this into the interactive tool or source code of the script to reference the package.
// Install LambdaCalculus as a Cake Addin
#addin nuget:?package=LambdaCalculus&version=1.0.0-lambda&prerelease

// Install LambdaCalculus as a Cake Tool
#tool nuget:?package=LambdaCalculus&version=1.0.0-lambda&prerelease
The NuGet Team does not provide support for this client. Please contact its maintainers for support.

λ calculus in F#

Fun library for lambda calculus made purely in F#. It has a web app made for demonstration.

Syntax of λ-calculus


  • Variables are single-character lower-case Latin letters.
  • Lambda starts with \, then is followed by 1 or more variable, then a dot (.), then expression. Until wrapped with parentheses, the body of a lambda has a lower priority than () (e. g. \x.(x)(y) is the same as (\x.x)(y)).
  • Associativity is left-sided: xyz is (xy)z.
  • A nested lambda can be shortcuted: \x.\y.\ is the same as \
  • Symbol λ can be used instead of \, but is not essential.


  • x - an expression of one free variable x
  • xy - y applied to x
  • \x.x - an identity
  • \x.xx - a lambda which returns its only parameter applied to itself
  • (\x.x)y - y applied to identity (will return y after beta reduction)
  • (\x.xx)(\x.xx) - simple example of beta-irreducible expression

Library API

Expression is defined as following:

type Expression =
    | Variable of char
    | Lambda of Head : Variable * Body : Expression
    | Applied of Lambda : Expression * Argument : Expression

in LambdaCalculus.Atoms.

In the same module LambdaCalculus.Atoms there are

  • betaReduce or βReduce: Expression -> Expression performs beta-reduction, that is, simplification of an expression (starting from bottom, replaces parameters of lambdas in their bodies with the applied expressions)
  • alphaEqual or αEqual: Expression -> Expression -> Expression compares two expressions up to lambdas parameter names
  • substitute: char -> Expression -> Expression -> Expression replaces the given variable with a given expression. In case of conflicts between it and a local variable in a lambda, it will alpha-convert the lambda's parameter to a new name.

In the module LambdaCalculus.Parsing there is parse which takes a string as an argument, and returns a Result of Expression and string.

Fun facts

The library is made in pure functional programming. Even the parser has no exceptions, early returns, for-loops. The logic is implemented with recursion, pattern matching, and computation expressions.

NuGet packages

This package is not used by any NuGet packages.

GitHub repositories

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Version Downloads Last updated
1.0.0-lambda 46 8/14/2021

First release