Homework 6

Untyped lambda calculus

Please submit homeworks via the new submission page.

In this homework, you’ll implement an interpreter for the untyped lambda calculus.

Your submission should be a zipfile including:

  • a Makefile, with a default target that builds an executable named interp;
  • your source files;
  • a file README.md, which lists the collaborators on this assignment (not more than three); and
  • a file fact.lc, which computes the factorial of 5 using a recursive factorial function.

If you’re going to use a language other than Haskell, please tell me about your choices in README.md—which version I should use, which libraries I need, and why you chose the given language.

Please submit a clean zipfile, i.e., you should include just your Makefile, source, README.md, and fact.lc.

I will grade your homework by unzipping the zipfile and running ‘make’. If that doesn’t work and the cause is something that is not my fault (like needing to install library you used), you will get a zero.

Do not:

  • have your files inside a directory
  • include an existing interp executable
  • include any .hi or .o files
  • include weirdness like __MACOSX or .DS_Store files
  • include version control information like .git directories

To avoid getting a zero, test out your zipfile by unzipping it in a new directory and running make. Does it build your interpreter correctly?

Two part submissions

You’ll submit this homework twice. The first checkpoint is Thursday, March 29th; the second (and final!) checkpoint is Thursday, April 5th.

I’m going to run the same grader on your code twice: once at the first checkpoint, once at the second. Your grade on this assignment will be the better of the two.

How should you do this assignment? I’d recommend the following three-part plan:

  1. Implement the parser and the -c checking-mode flag. Leave the interpreter stubbed out, but write tests for the parser and the checker. Write the command-line interface/argument parser.
  2. Implement the interpreter. Write tests for the interpreter. Extend the argument parser.
  3. Implement the -n Church-numeral conversion flag. Write tests for the Church-numeral conversion and extend the argument parser.

At a minimum, plan on having part (1) done by the first checkpoint. You might be able to get part (2) done as well. Once you have feedback from the first checkpoint, you can fix bugs in your parser and checker. Then you can implement the interpreter and the Church numeral conversion.

To be clear: my implementation plan means you’ll miss out on more than half the credit at checkpoint 1. A different strategy—that tries to do everything up front—might get a few more points the first go round. But trying to do everything at once will be much more likely to introduce hard-to-find bugs and structural infelicities. Slow down: this assignment lasts two weeks for a reason.

Please read the entire specification before starting on the homework. If you have a question, please double check that it’s not answered in this document before posting to Piazza.

The language

You will fundamentally be implementing the lambda calculus. A lambda calculus expression is defined by the following grammar:

e ::= x | e1 e2 | lambda x. e

Your concrete syntax should allow arbitrary whitespace between tokens (like lambda or .) and parenthesization for disambiguation (lambda x. x (x x) is different from lambda x. x x x).

Variable names should begin with an alphabetical character followed by zero or more alphanumeric or single-quote (') characters, i.e., x and foo3'5bar are valid variable names, but 12 and 'quoted' are not. As for the While language, please be careful to ensure that the identifiers and keywords are kept distinct.

Application should be left associative, i.e., x y z should parse like (x y) z.

Lambda expressions should be allowed to have more than one argument, i.e., lambda s z. s z should parse like lambda s. (lambda z. (s z)).

Even though they’re not in the grammar, you should support let expressions let x = e1 in e2.

A program in our language is just a single expression. For example, the following program computes the Church numeral representing the number two:

let zero = lambda s z. z in
let succ = lambda n. lambda s z. s (n s z) in
succ (succ zero)

The interpreter

Your task is to implement an interpreter that reads in and evaluates programs. You’ll need to define syntax, a parser, a pretty printer, and an interpreter. Write a Makefile that compiles your code into a program called interp.

Your interpreter should use a call-by-value (CBV) semantics, i.e., you only apply a beta rule when the argument has fully reduced to a lambda. Be certain to test this! Since Haskell is lazy, it’s very easy to think you’ve implemented CBV but in fact implement CBN.

After your interpreter evaluates all of the program, it should, by default, print out the final result of the expression/program; for example, running interp on:

let zero = lambda s z. z in
let succ = lambda n. lambda s z. s (n s z) in
succ (succ zero)

should produce:

lambda s z. s ((lambda s z. s ((lambda s z. z) s z)) s z)

and running interp on:

let zero = lambda s z. z in 
let succ = lambda n. lambda s z. s (n s z) in
succ (succ (succ zero))

should produce:

lambda s z. s ((lambda s z. s ((lambda s z. s ((lambda s z. z) s z)) s z)) s z)

It’s critical (for grading) that your interpreter output parseable lambda expressions. For example:

let id = lambda x. x in
lambda y. id

should yield:

lambda y x. x


lambda y. lambda x. x

or something similar. If it yields something like λy x→x, you will lose points, even though Unicode is fun.

Your interpreter should signal errors appropriately. The precise content of your error message isn’t the most important thing—though the more detailed they can be, the better!—but it is critical that interp exits with a non-zero exit code when there is an error. For example, when I run my version of interp on lambda. lambda lambda, I get the following error message:

Parse error: "parse.lc" (line 1, column 7):
unexpected keyword in place of variable (lambda)
expecting letter or digit, space or "'"

There are other errors that can occur, like unbound variables. For example, the program (lambda x. y) (lambda x. x) produces Error: unbound variable y when run. Note that (lambda a b. a) (lambda x. x) (lambda y. z) should not produce an error.

If you don’t encounter an error, your program should exit with a zero exit code.

Command-line arguments

Your interpreter should, when run without arguments, read all of the input from standard input, parse the input as a program, and then evaluate the program, pretty printing the final result of the expression.

When given an argument, your interpreter should read the file as input (and then proceed to parse, evaluate, and print as above). If the file is specified as -, then you should follow UNIX convention and read from standard input. (Pro tip: never name a file -.) I don’t care what your program does when given more than one argument; mine uses the rightmost file given.

You should also implement two flags to your interpreter: -c for checking that all variables are well scoped, and -n for converting the final results from Church numerals to Arabic numerals.

The -c flag should turn on a “checking mode”, wherein you should check that a program is well scoped before running it. If a program is not well scoped, you should display an error message and exit with a non-zero exit code. For example, running:

$ echo lambda x. y >bad.lc
$ ./interp bad.lc
lambda x. y
$ ./interp -c bad.lc
Unbound variables: y
$ echo $? # displays the exit code of the last command

Note that the checker should run before your program does. For example:

$ echo '((lambda x. x x) (lambda x. x x))' z >bad2.lc
$ ./interp bad2.lc # runs forever
$ ./interp -c bad2.lc
Unbound variables: z

The -n flag should turn on a “Church numeral conversion mode”. After you finish evaluating the program in -n mode, you should convert the final result from a Church numeral to an Arabic numeral. For example, suppose three.lc contains the text:

let zero = lambda s z. z in
let succ = lambda n. lambda s z. s (n s z) in
succ (succ (succ zero))

We should then have:

$ ./interp three.lc
lambda s z. s ((lambda s z. s ((lambda s z. s ((lambda s z. z) s z)) s z)) s z)
$ ./interp -n three.lc
$ echo three.lc | ./interp -n -

Implementing the -n flag will require some creativity in extending your language. You’ll have to make some interesting internal changes—but please make no externally visible changes beyond adding the -n flag. I expect you to be able to convert any Church numeral—note that the two lambda terms we get out of two-three.lc aren’t the “standard” way of writing two and three, which would be lambda s z. s (s z) and lambda s z. s (s (s z)), respectively.

If you’re evaluating in -n mode and you get a term that can’t be interpreted as a Church numeral, you should issue an error message and exit with a non-zero exit code. For example:

$ echo lambda s z. s | ./interp -n
Couldn't extract a number from (alleged) Church numeral lambda s z. s

Note that -c and -n are orthogonal: I expect your code to work with every possible combination.

Finally, you’ll be a good citizen if you implement a usage message, as in:

$ ./interp --help
interp [OPTIONS] FILE (defaults to -, for stdin)
  lambda calculus interpreter

  -c --check    Check scope
  -n --numeral  Convert final Church numeral to a number
  -? --help     Display help message

I don’t require that you write a usage message, but it’s a good habit to get into.


You are free to write whatever error messages you like, though please be sure (a) that they go to stderr (not stdout) and (b) that you exit with a non-zero exit code.

Do not output unnecessary text when a program succeeds. While such debugging information may be valuable when you’re programming it’s (a) not part of the specification in this document and (b) will confuse my grader. If you’re not sure what output to give, please ask on Piazza.


I strongly recommend that you build a test suite. Include short programs and long programs; programs that fail and programs that succeed. In class we’ve been building up the booleans, the naturals, etc.—use these to test your code!

At a minimum, I expect you turn in a file fact.lc which builds up enough of the naturals to compute the factorial of 5 using a recursively defined factorial function. We should have:

$ ./interp -cn fact.lc

A note on efficiency

While this class is explicitly not about efficiency, I expect your code to run in a reasonable amount of time. For example, here’s how long my code takes to calculate factorial of 5 (using Church numerals and the Y combinator):

$ time ./interp -cn fact.lc

real	0m0.019s
user	0m0.007s
sys	0m0.007s

Your code doesn’t need to be this fast, but if it takes more then five seconds, I think there’s cause for concern.

A note on Makefiles

Here’s the Makefile from my solutions:

interp : Main.hs Eval.hs Check.hs Test.hs Syntax.hs
	stack ghc -- -o $@ $^

test : interp tests/*
	for t in tests/*; do ./interp $$t; done

clean :
    rm interp
    rm *.hi *.o

.PHONY: test clean

You can read up in the manual, but the general format is as follows. A Makefile should be named exactly that, including capitalization. To run it, you run make, optionally followed by one or more targets. If you don’t list a target, it’ll choose the first. In the Makefile above, there are three targets: interp, which builds the executable; test, which runs my tests; and clean, which removes the generated files.

Each target is followed by a colon and any dependencies it has. When you run make interp, the commands below will be run if the target is older than the dependencies. For the interp target, make will check to see if any of the specified Haskell files have been updated since the interp file was last updated; if they have, it will decide to rebuild. Note that each of the commands after the target must be indented with tabs! Weird, right?

Concretely, interp will run the command ghc -o interp Main.hs Eval.hs Check.hs Test.hs Syntax.hs. Why? The make program interprets the $@ variable as the current target and the $^ variable as all of the dependencies.

My test and clean targets don’t correspond to real files. I’ve marked them at the bottom as “phony” targets. (If I didn’t do that and someone created a file named test or clean, things would get confused… so be careful!)

The language of Makefiles is big and complicated and powerful—and a handy thing to know!