A simply typed lambda calculus interpreter, written with GADTs.
This is an interpreter for the simply-typed lambda calculus. It is written making heavy use of generalized algebraic datatypes (GADTs), and is meant to serve as an example how how these GADTs can be useful. See the GitHub repo for more information about the syntax for the language and interpreter commands.
A Practical Introduction to Haskell GADTs
This repo and these notes are for a talk given at LambdaConf in Boulder, CO, USA, on 22 May, 2015.
See below for more information about glambda itself.
Setup
Do this first!
We will be working from my glambda project to learn about Generalized Algebraic Datatypes (GADTs). The glambda package has a non-trivial set of dependencies. While I'm introducing myself and GADTs, generally, it would be wise to download and compile all of the dependencies onto your laptop. Then, when we get to hands-on exercises, you'll be all ready to go.
Here's what to do
> git clone git://github.com/goldfirere/glambda.git
> cd glambda
> cabal sandbox init
> cabal install --only-dependencies --enable-tests -j<# of CPUs you have>
This should make your computer spin for a little while. In the meantime, enjoy the talk! (If you have trouble with cabal sandbox, possibly because of an old cabal, try the sequence of commands above without that step.)
Exercises 1 and 2 do not require those dependencies, so you can work on them while compiling dependencies.
Instructions for exercises:
More information about glambda appears below:
The Glamorous Glambda Interpreter
Glambda is a simply-typed lambda calculus interpreter. While it is intended to be easy-to-use and help users learn about the lambda calculus, its real strength is its implementation, which makes heavy use of GADTs, and is designed to serve as a showcase of writing a real-world program with extra compile-time guarantees.
This manual focuses only on the user experience. The structure of the code will be described in a series of GADT programming tutorials coming out Real Soon Now.
Example session
Saying cabal install glambda will produce an executable glam. glam is the lambda-calculus interpreter. It is GHCi-like, accepting commands beginning with a :. Here is an example session:
\\\\\\
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/-\ \\\\\\
| | \\\\\\
\-/| \\\\\\
| //\\\\\\
\-/ ////\\\\\\
//////\\\\\\
////// \\\\\\
////// \\\\\\
Welcome to the Glamorous Glambda interpreter, version 1.0.
λ> (\x:Int.x + 2) 5
7 : Int
λ> revapp = \x:Int.\y:Int->Int.y x
revapp = λ#. λ#. #0 #1 : Int -> (Int -> Int) -> Int
λ> not = \b:Bool.if b then false else true
not = λ#. if #0 then false else true : Bool -> Bool
λ> revapp (3 < 4) not
Bad function application.
Function type: Int -> (Int -> Int) -> Int
Argument type: Bool
in the expression 'revapp (3 < 4)'
λ> not (3 < 4)
false : Bool
λ> :type revapp (10 % 3)
(λ#. λ#. #0 #1) (10 % 3) : (Int -> Int) -> Int
λ> :step revapp (10 % 3) (\x:Int.x * 2)
(λ#. λ#. #0 #1) (10 % 3) (λ#. #0 * 2) : Int
--> (λ#. #0 (10 % 3)) (λ#. #0 * 2) : Int
--> (λ#. #0 * 2) (10 % 3) : Int
--> 10 % 3 * 2 : Int
--> 1 * 2 : Int
--> 2 : Int
2 : Int
λ> :quit
Good-bye.
As you can see, glambda uses de Bruijn indices to track variable binding. In the actual output (if your console supports it), the binders (#) and usage sites (#0, #1) are colored so that humans can easily tell which variable is used where.
You can also see above that the input to glambda must be fully annotated; glambda does not do type inference. However, note that types on binders do not appear in the output: once an input is type-checked, the type information is erased. Yet, because of the use of GADTs in the implementation, we can be sure that the reductions are type-safe.
The Language
The glambda language is an explicitly typed simply typed lambda calculus, with integers (Int) and booleans (Bool). The following operators are supported, with their usual meanings, associativity, and precedence:
+ - * / % < <= > >= ==
The only slightly unusual member of this list is %, which takes a modulus, like in C-inspired languages. The division operator / does integer division, naturally.
Glambda supports a ternary conditional operator, demonstrated in the snippet above, as if <boolean expression> then <exp> else <exp>.
Integer constants must be positive. Subtract from 0 to get a negative integer.
Boolean constants are spelled false and true.
Comments are exactly as in Haskell: -- starts a line comment, and {- ... -} is a block comment. Comments can be nested.
Variable names are as in Haskell: names must start with a letter or underscore (although case is immaterial) and then may have letters, numbers, and underscores.
The language is not whitespace-aware.
Most of what we have seen are expressions. Glambda also supports statements. A statement is either an expression or has the form <name> = <expression>. This latter form assigns a global variable to the expression. These global variables are expanded during type-checking: they are more like macros than proper cells holding information. Statements can be separated by ;.
The Interface
When you type an expression into the glambda interpreter, it is evaluated fully, and the value is printed, along with its type.
When you type a global variable assignment, that variable is assigned, and its (unevaluated) contents are printed, along with its type.
You can also run commands, as described below. Commands all start with a leading :, and that : must be the first character on the input line. Arguments to a command are given after the command itself. Commands can be abbreviated by typing an unambiguous prefix to a command. For example, :t can be used to get an expression's type, because no other command begins with t.
Commands
:quit quits the glambda interpreter.
:lex lexes the given text and pretty-prints the result.
:parse parses the given text and pretty-prints the result.
:eval type-checks and evaluates the given expression. This is the default behavior at the command line.
:step runs the given expression through the single-step semantics. This shows you every step of the way from your expression down to a value. This uses a different evaluation strategy than :eval does, but the result should always be the same.
:type gives you the type of an expression.
:all runs both :eval and :step on an expression.