Overview

This assignment will give you practice with the lambda calculus:

• Using lambda calculus to write simple functions
• Implementing lambda calculus using substitution, reduction, and alpha-conversion

Substitution, reduction, and alpha-conversion are ubiquitous ideas in programming-language semantics.

Setup

You will build on an existing interpreter for lambda calculus, which you will get by cloning the following git repository:

git clone linux.cs.tufts.edu:/comp/105/git/lambda

Cloning should give you a directory lambda with files linterp.sml, Lhelp.ui, Lhelp.uo, Makefile, and predefined.lam.

Learning about the lambda calculus

There is no book chapter on the lambda calculus. Instead, we refer you to these resources:

1. Raúl Rojas’s “A Tutorial Introduction to the Lambda Calculus” is short, easy to read, and covers the same points that are covered in lecture:

   • Syntax
   • Free and bound variables
   • Capture-avoiding substitution
   • Addition and multiplication with Church numerals
   • Church encoding of Booleans and conditions
   • The predecessor function on Church numerals
   • Recursion using the Y combinator

   Rojas doesn’t provide many details, but he covers everything you need to know in 8 pages, with no distracting theorems or proofs. When you want a short, easy overview to help you solidify your understanding, Rojas’s tutorial is the best source.

2. Wikipedia offers two useful pages:¹

   • The Lambda Calculus page covers everything you’ll find in Rojas and much more besides. In particular, it discusses reduction strategies.

   • The Church Encoding page goes into more detail about how to represent ordinary data as terms in the lambda calculus. The primary benefit relative to Rojas is that Wikipedia describes more kinds of arithmetic and other functions on Church numerals.

   You need to know that the list encoding used on Wikipedia is not the list encoding used in COMP 105. In order to complete all the homework problems successfully, you must use the list encoding described in the lecture notes.

   When you want a quick, easily searchable reference to some particular point, Wikipedia is your best source. Wikipedia is particularly useful for explaining the difference between normal-order reduction and applicative-order reduction, both of which you will implement.

3. Prakash Panangaden’s “Notes on the Lambda-Calculus” cover the same material but with more precision and detail. Prakash is particularly good on capture-avoiding substitution and change of bound variables, which you will implement.

   Prakash also discusses more theoretical ideas, such as how you might prove inequality (or inequivalence) of lambda-terms. And instead of just presenting the Y combinator, Prakash goes deep into the ideas of fixed points and solving recursion equations—which is how you achieve recursion in lambda calculus.

   When you are getting ready to implement substitution and reduction strategies, Prakash’s notes are your best source.

Introduction to the lambda interpreter

You will implement the key components of a small, interactive interpreter for the lambda calculus. This section explains how to use the interpreter and the syntax it expects. A reference implementation of the interpreter is available in /comp/105/bin/linterp-nr.

Syntax

The syntax of definitions

Like the interpreters in the book, the lambda interpreter processes a sequence of definitions.
The concrete syntax is very different from the book languages. Every definition must be terminated with a semicolon. Comments are line comments in C++ style, starting with the string // and ending at the next newline.

The interpreter supports four forms of definition: a binding, a term, the extended definition “use”, and an extended definition “check-equiv”.

Bindings

A binding has the form

-> noreduce name = term;

or

-> name = term;

In both forms, every free variable in the term must be bound in the environment—if a right-hand side contains an unbound free variable, the result is a checked run-time error. The first step of computation is to substitute for each of the free variables: each occurrence of each free variable is replaced by that variable’s definition.

In the first form, where noreduce appears, no further computation takes place. The substituted right-hand side is simply associated with the name on the left, and this binding is added to the environment.

The noreduce form is intended only for terms that cannot be normalized, such as

noreduce bot = (\x.x x)(\x.x x);
noreduce Y   = \f.(\x.f(x x))(\x.f(x x));

In the second form, after the free variables are replaced, the term on the right is reduced until there are no more beta-redexes or eta-redexes. (You will implement the two reduction strategies presented in class.) If reduction doesn’t terminate, the interpreter might loop.

Loading files with use

The use extended definition loads a file into the interpreter as if it had been typed in directly. It takes the form

-> use filename;

Comparing normal forms with check-equiv

The check-equiv form immediately reduces two terms to normal form and compares them for equivalence. It has the form

-> check-equiv term = term;

And here are some examples:

-> check-equiv x = x;
The test passed
-> check-equiv \x.x = \y.y;
The test passed
-> check-equiv \x.x = \y.x;
The test failed: terms \x.x and \y.x do not have equivalent normal forms
-> check-equiv (\x.x)(\y.y) = \z.z;
The test passed

Unlike the check-expect in the other interpreters, check-equiv is not “saved for later”—the given terms are normalized right away.

Terms as definitions

As in the book, a term can be entered at the read-eval-print loop, just as if it were a definition. Every free variable in the term is checked to see if it is bound in the environment; if so, each free occurrence is replaced by its binding. Free variables that are not bound in the environment are permissible; they are left alone.² The term is reduced to normal form (if possible) and the result is printed.

-> term;

The syntax of terms

A lambda term can be either a variable, a lambda abstraction, an application, or a parenthesized lambda term. Precedence is as in ML.

A lambda abstraction abstracts over exactly one variable; it is written as follows:

\name.term

Application of one term to another is written:

term1 term2

The lambda interpreter is very liberal about names of variables. A name is any string of characters that contains neither whitespace, nor control characters, nor any of the following characters: \ ( ) . = /. Also, the string use is reserved and is therefore not a name. But a name made up entirely of digits is OK; the lambda calculus has no numbers, and names like 105 have no special status.

As examples, all the following definitions are legal:

1    = \f.\x.f x;
True = \x.\y.x;
one  = True 1;

A short example transcript

A healthy lambda interpreter should be capable of something like the following transcript:

-> true  = \x.\y.x;
-> false = \x.\y.y;
-> pair  = \x.\y.\f.f x y;
-> fst = \p.p (\x.\y.x);
-> snd = \p.p (\x.\y.y);
-> true;
\x.\y.x
-> fst (pair true false);
\x.\y.x
-> snd (pair true false);
\x.\y.y
-> if = \x.\y.\z.x y z;
if
-> (if true fst snd) (pair false true);
\x.\y.y
-> (if false fst snd) (pair true false);
\x.\y.y

For more example definitions, see the predefined.lam file distributed with the assignment.

Software provided for you

Both capture-avoiding substitution and normal-order reduction can be tricky to implement.³ So that you may have a solid foundation on which to write your lambda code, I provide an interpreter linterp-nr. Running use comp105 should give you access to that interpreter.

Even with a correct interpreter, lambda code can be hard to debug. So I also provide an interpreter called lamstep, which shows every reduction step. Some reductions require a lot of steps and produce very big intermediate terms. Don’t be alarmed.

All questions and problems

• There are four problems on programming with Church numerals, which you’ll do on your own.

• There are four problems on implementing the lambda calculus, which you can do with a partner. Modify the linterp.sml file from the git repository described under the heading Setup, above.

Reading comprehension

These problems will help guide you through the reading. We recommend that you complete them before starting the other problems below. You can download the questions.

Programming in the lambda calculus (individual problems)

These problems give you a little practice programming in the lambda calculus. All functions must terminate in linear time, and you must do these exercises by yourself. You can use the reference interpreter linterp-nr.

Place your solutions in file church.lam.

Not counting code copied from the lecture notes, my solutions to all four problems total less than fifteen lines of code. And all four problems rely on the same related reading.

0    = \f.\x.x;
succ = \n.\f.\x.f (n f x);
+    = \n.\m.n succ m;
*    = \n.\m.n (+ m) 0;

true  = \x.\y.x;
false = \x.\y.y;

-> binary 0;
\f.\x.x
-> binary 1;
\f.f
-> binary 2;
\f.\x.f (f (f (f (f (f (f (f (f (f x))))))))) // f applied 10 times
-> binary 3;
\f.\x.f (f (f (f (f (f (f (f (f (f (f x)))))))))) // f applied 11 times

  noreduce binary = ... ;

-> binary-sym O l 0;      
O
-> binary-sym O l 1;
l
-> binary-sym O l 2;
l O
-> binary-sym O l (+ 2 4);
l l O
-> binary-sym Zero One (+ 2 4);  
One One Zero
-> binary-sym O l (+ 1 (* 4 (+ 1 2)));
l l O l

-> 0;
\f.\x.x
-> 2;
\f.\x.f (f x)
-> nth 0 (cons Alpha (cons Bravo (cons Charlie nil)));
Alpha
-> nth 2 (cons Alpha (cons Bravo (cons Charlie nil)));
Charlie

Implementing the lambda calculus (possibly with a partner)

For problems 5 to 7 below, you may work on your own or with a partner. These problems help you learn about substitution and reduction, the fundamental operations of the lambda calculus. The problems also give you a little more practice in continuation passing, which is an essential technique in lambda-land.

For each problem, define appropriate types and functions in linterp.sml. When you are done, you will have a working lambda interpreter. Some of the code we give you (Lhelp.ui and Lhelp.uo) is object code only, so you will have to build the interpreter using Moscow ML. Typing make should do it.

 -> thing = \x.\y.y x;
 thing
 -> thing;
 (\thing.thing) \x.\y.y x

exception MissingTest of string
val N : term = app (app (var "fst") (var "x")) (var "y")
val test_a = subst ("x", N) (var "x") = N
val test_b = raise MissingTest "(b)"
val test_c = raise MissingTest "(c)"
val test_d = raise MissingTest "(d)"
val test_e = raise MissingTest "(e)"
val test_renaming = raise MissingTest "renaming"

val _ = ListPair.app
        (fn (t, name) => if t then () else print ("BAD TEST (" ^ name ^ ")\n"))
        ( [test_a, test_b, test_c, test_d, test_e, test_renaming]
        , ["a", "b", "c", "d", "e", "renaming"]
        )

fun tick show n f =  (* show info about term every n reductions *)
  let val count = ref 0
      fun apply arg =
        let val _ = if !count = 0 then
                      ( List.app print ["[", Int.toString (show arg), "] "]
                      ; TextIO.flushOut TextIO.stdOut
                      ; count := n - 1)
                    else
                      count := !count - 1
        in  f arg
        end
  in  apply
  end

val reduceN_debug = tick size 1 reduceN  (* show size after every reduction *)

More Extra Credit

Solutions to any of the extra-credit problems below should be placed in your README file. Some may be accompanied by code in your linterp.sml file.

What and how to submit: Individual work

Using script submit105-lambda-solo, submit

• A README file containing
  • The names of the people with whom you collaborated
  • Any extra credit you may have earned
• File cqs.lambda.txt, containing your answers to the reading-comprehension questions
• File church.lam containing your solutions to the Church-numeral

As soon as you have the files listed above, run submit105-lambda-solo to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

What and how to submit: Pair work

Using script submit105-lambda-pair, submit

• File linterp.sml, containing your modified lambda-calculus interpreter and your tests of substitution

• File string-tests.sml, containing your tests of var, app, lam, and toString

As soon as you have the files listed above, run submit105-lambda-pair to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

Avoid common mistakes

Common mistakes with Church numerals

Here are some common mistakes to avoid when programming with Church numerals:

• Don’t forget the question mark in the name of even?.

• When using a fixed-point combinator, don’t forget noreduce.

• Don’t use the list representation or primitives from Wikipedia. We will test your code using the representation and primitives described in class, which you will also find in the file predefined.lam.

• Don’t include any use directives in church.lam.

• Don’t copy predefined terms from predefined.lam. We will load the predefined terms before running your code.

  To make sure your code is well formed, load it using

  cat predefined.lam church.lam | ./linterp

  or

  cat predefined.lam church.lam | linterp-nr

  If you want to build a test suite, put your tests in file test.lam and run

  cat predefined.lam church.lam test.lam | ./linterp

Common mistakes with the lambda interpreter

Here are some common mistakes to avoid in implementing the interpreter:

• Don’t forget the eta rule:

     \x.M x --> M   provided x is not free in M

  Here is a reduction in two eta steps:

    \x.\y.cons x y --> \x.cons x --> cons

  Your interpreters must eta-reduce when possible.

• Don’t forget to reduce under lambdas.

• Don’t forget to use normal-order reduction in the code you submit.

• Don’t forget that in an application M₁ M₂, just because M₁ is in normal form doesn’t mean the whole thing is in normal form. In particular, if reducing M₁ raises the NormalForm exception, you must catch the exception and try to reduce M₂.

• Don’t clone and modify your code for reduction strategies; people who do this wind up with wrong answers. The code should not be that long; use a clausal definition with nested patterns, and write every case from scratch.

  Do make sure to use normal-order reduction, so that you don’t reduce a divergent term unnecessarily.

• Don’t try to be clever about a divergent term; just reduce it. (It’s a common mistake to try to detect the possibility of an infinite loop. Mr. Turing proved that you can’t detect an infinite loop, so please don’t try.)

• When implementing freshVar, don’t try to repurpose function freshTyvar from Section 7.6. That function isn’t smart enough for your needs.

How your work will be evaluated

Your ML code will be judged by the usual criteria, emphasizing

• Correct implementation of the lambda calculus
• Good form
• Names and contracts for helper functions
• Structure that exploits standard basis functions, especially higher-order functions, and that avoids redundant case analysis

Your lambda code will be judged on correctness, form, naming, and documentation, but not so much on structure. In particular, because the lambda calculus is such a low-level language, we will especially emphasize names and contracts for helper functions.

• This is low-level programming, and if you don’t get your code exactly right, the only way we can recognize and reward your learning is by reading the code. It’s your job to make it clear to us that even if your code isn’t perfect, you understand what you’re doing.

• Try to write your contracts in terms of higher-level data structures and operations. For example, even though the following function does some fancy manipulation on terms, it doesn’t need much in the way of a contract:

     double = \n.\f.\x. n (\y.f (f y)) x;  // double a Church numeral

  Documenting lambda calculus is like documenting assembly code: it’s often sufficient to say what’s happening at a higher level of abstraction.

• Although it is seldom ideal, it can be OK to use higher-level code to document your lambda code. In particular, if you want to use Scheme or ML to explain what your lambda code is doing, this can work only because Scheme and ML operate at much higher levels of abstraction. Don’t fall into the trap of writing the same code twice—if you are going to use code in a contract, it must operate at a significantly higher level of abstraction than the code it is trying to document.

In more detail, here are our criteria for names:

And here are our criteria for contracts: