Project 2: Untyped Lambda Calculus

Hand in: 18 Mar 2022, 23:59 (Hongkong)

Project template: 2-untyped.zip

In this exercise, you’ll be studying untyped λ-calculus. This classic language is defined in Chapter 5 of the TAPL book.

t ::= ident              terms
    | "\" ident "." t
    | t t
    | "(" t ")"

v ::= "\" ident "." t    values

We use standard syntax to express λ-terms, with lambdas replaced by backslashes. As a reminder, note that the bodies of abstractions are taken to extend as far to the right as possible, so that, for example, λx. λy. x y x stands for the same tree as λx. (λy. ((x y) x)) (see TAPL, p54).

Evaluation rules

The evaluation rules for full beta-reduction are as follows.

   t1 → t1'
--------------
t1 t2 → t1' t2

   t2 → t2'
--------------
t1 t2 → t1 t2'

    t1 → t1'
----------------
λx. t1 → λx. t1'

(λx. t1) t2 → [x → t2] t1

The last rule uses substitution, whose definition we reproduce here (see TAPL, p71):

The part marked with an (*) doesn’t handle the case where y ∈ FV(s). So what shall we do then? We first use of alpha-conversion for consistently renaming a bound variable in a term - actually a lambda abstraction - and then continue to apply the substitution rules. To rename a bound variable in a lambda abstraction λx. t1, one chooses a fresh name x1 for bound variable x, and consistently renames all free occurrences of x in the body t1. Note: you will need to define a function that “freshens” variable names. You can rely on variable names in the input term containing only characters, never any numbers.

We use the following rules to test if a variable is free in some term:

Evaluation strategies

TAPL p56 presents several evaluation strategies for the λ-calculus:

• Under full beta-reduction any redex may be reduced at any time. This is the strategy described by the evaluation rules listed above, but this is too unrestricted in practice: we need a deterministic way to choose a certain redex, when more than one is available.

• Under normal order strategy, the leftmost, outermost redex is always reduced first. This strategy allows to reduce inside unapplied lambda terms.

• The call-by-name strategy is yet more restrictive, allowing no reductions inside lambda abstractions. Arguments are not reduced before being substituted in the body of lambda terms when applied.

  Haskell uses an optimized version known as call-by-need that, instead of re-evaluating an argument each time it is used, overwrites all occurrences of the argument with its value the first time it is evaluated, avoiding the need for subsequent re-evaluation.

• Most languages use a call-by-value strategy, in which only outermost redexes are reduced and where a redex is reduced only when its right-hand side has already been reduced to a value (a function).

  The call-by-value strategy is strict, in the sense that the arguments to functions are always evaluated, whether or not they are used by the body of the function. In contrast, lazy strategies such a call-by-name and call-by-need evaluate only the arguments that are actually used.

What you have to do

1. Implement the reduceNormalOrder method that uses the normal-order strategy, which applies alpha-conversion and term substitution following the above reduction rules. If none of the rules apply it should throw a NoReductionPossible exception containing the corresponding irreducible term.

2. Implement the reduceCallByValue method that uses the call-by-value evaluation strategy. For that you need to define a new set of evaluation rules, similar to the ones given above. Again, if none of the rules apply it should throw NoReductionPossible exception containing the corresponding irreducible term.

   Since we reason about values, we need to define what a value is. We will follow the book in saying that the only values are lambda abstractions. Does it simplify the substitution operation? What would happen if we added variables to the set of values (do not implement, these are self-check questions)? Compare the output of the two reducers, and try to understand why it is different.

Development & debugging

We provide a simple parser and runner for your application which let you quickly debug your current implementation. When you implement reduceNormalOrder and reduceCallByValue, it should produce results like:

input:

\y. ((\x. x) y)

output:

normal order:
Abs(y,App(Abs(x,Var(x)),Var(y)))
Abs(y,Var(y))
call-by-value:
Abs(y,App(Abs(x,Var(x)),Var(y)))