# Bruijn A purely academic programming language based on lambda calculus and De Bruijn indices written in Haskell. [Jump to examples](#Examples) ## Features - De Bruijn indices\[0\] eliminate the complexity of α-equivalence and α-conversion - Unique bracket-style representation for lambda abstractions enables improved human-readability and faster syntactic perception - Balanced ternary allows negative numbers while having a reasonably compact representation, operator and time complexity (in comparison to unary/binary church numerals)\[1\] - Highly space-efficient compilation to binary lambda calculus (BLC)\[2\]\[3\] additionally to normal interpretation and REPL - Recursion can be implemented using combinators such as Y or ω - Included standard library featuring many useful functions (`std.bruijn`) ## Basics ### De Bruijn indices De Bruijn indices\[0\] replace the concept of variables in lambda calculus. The index basically represents the abstraction layer you want to reference beginning at 0 with the innermost layer. For example, λx.x becomes λ0 because x referenced the first abstraction layer. Furthermore, λx.λy.xy becomes λλ10 and so forth. You can read more about De Bruijn indices on [Wikipedia](https://en.wikipedia.org/wiki/De_Bruijn_index). ### Syntax In general the syntax of bruijn is pretty similar to the previously presented normal lambda calculus syntax with De Bruijn indices. You can use any function that you’ve previously defined. You can also overwrite previously defined functions. The environment gets interpreted from bottom to top (starting at `main`). The following are the main syntax specifications in the (minorly extended) [Backus-Naur form](https://en.wikipedia.org/wiki/Backus%E2%80%93Naur_form). Additional spaces are optional but allowed. ::= [a-ω,A-Ω,_][a-ω,A-Ω,0-9,?,!,',-]* ::= "[" "]" ::= ("+" | "-")[0-9]+ ::= [0-9] ::= | | | "(" ")" | ::= ::= | ::= ":test " = ::= ":import " ::= "# " * The following are the differences in syntax between REPL and file: **For files**: The execution of a file begins at the `main` function. Its existence is mandatory. ::= ":print " ::= ::= | | | | | "\n" **For REPL**: ::= = ::= | | | | | "\n" ### Numerals Numbers in bruijn always have a sign in front of them or else they will be mistaken for De Bruijn indices. Generally the decimal representation is only syntactic sugar for its internal balanced ternary representation. We use balanced ternary because it’s a great compromise between performance and size (according to \[1\]). You don’t have to care about the internal side too much though, if you use the included operations from the standard library. The REPL even tries its best at displaying expressions that look like ternary numbers as decimal numbers in paranthesis next to it. ### Standard library You may want to use the included standard library to reach your program’s full potential. It includes many common combinators as well as functions for numerical, boolean and IO operations and much more. You can import it from `std.bruijn` using `:import std`. ### Examples You can try these by experimenting in the REPL or by running them as a file. Note, however, that you need an equal sign between the function name and its definition if you’re using the REPL. Plain execution: # this is a comment # we now define a function returning a ternary 1 get-one +1 # we can use the function in all functions below its definition get-one2 get-one :test get-one2 = +1 # equivalent of λx.x id [0] # equivalent of (λx.x) (λx.λy.x) = λx.λy.x :test id [[1]] = [[1]] # multiple arguments set-of-three [[[[0 1 2 3]]]] number-set set-of-three +1 +2 +3 access-first [0 [[[0]]]] :test access-first number-set = +1 # endless loop using omega combinator om [0 0] nom om main om nom # you may have realized you can't easily operate with numbers # or anything else really without writing crazy functions # -> luckily the standard library defines many standard operations! Using standard library: :import std # pairs with some values me [[[1]]] you [[[2]]] love pair me you :test fst love = me :test snd love = you # numerical operations five pred (sub (add +8 -4) -2) not-five? [if (eq? 0 +5) false true] :test not-five? five = false # boolean main not (or (and false true) true) :test main = false # read the std.bruijn file for an overview of all functions ### Compilation to BLC You can compile bruijn to John Tromp’s BLC\[2\]\[3\]. Only the used functions actually get compiled in order to achieve a minimal binary size. BLC uses the following encoding: | term | lambda | bruijn | BLC | |:-------------|:-------|:-------|:-----------------| | abstraction | λM | \[M\] | 00M | | application | MN | MN | 01MN | | bruijn index | i | i | 1i+10 | ## Installation You first need to install Haskell and Haskell Stack using the guidelines of your operating system. Using Haskell Stack, run `stack run -- [args]` to play around and use `stack install` to install bruijn into your path. ## REPL config You can configure the REPL by editing the `config` file. `stack install` or `stack run` will move the file into a data directory. More options can be found [here](https://github.com/judah/haskeline/wiki/UserPreferences). ## Usage Please read the usage information in the executable by using the `-h` argument. ## References - \[0\] De Bruijn, Nicolaas Govert. “Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem.” Indagationes Mathematicae (Proceedings). Vol. 75. No. 5. North-Holland, 1972. - \[1\] Mogensen, Torben. “An investigation of compact and efficient number representations in the pure lambda calculus.” International Andrei Ershov Memorial Conference on Perspectives of System Informatics. Springer, Berlin, Heidelberg, 2001. - \[2\] Tromp, John. “Binary lambda calculus and combinatory logic.” Randomness and Complexity, from Leibniz to Chaitin. 2007. 237-260. - \[3\] Tromp, John. “Functional Bits: Lambda Calculus based Algorithmic Information Theory.” (2022).