CMSC 631, Spring 2009
Homework 5
Due Wednesday, April 8, in class
Problems 1–3 must be done individually.
Problem 4 may be done in pairs.
1. Write down a sequence of reduction steps reducing each of the following terms to normal form. For
this problem, reduction is allowed anywhere within a term, including under a λ.
(a) (λx.x(xy))(λu.u)
(b) (λxyz.z yx)aa(λpq.q)
(c) (λxyz.xz (yz))(λxy.x)(λxy.x)
Note: λxy.e is short for λx.λy.e. Remember also that the scope of λextends as far to the right as
possible, and that application associates to the left.
2. For each type, construct a simply-typed lambda calculus term (variables, functions, and function
application only) whose most general type is that type, or argue that no term has that type. (Hint:
You can double-check your answers in OCaml.)
(a) α→β→β
(b) (α→β→γ)→β→α→γ
(c) α→β
(d) α→α→α
3. Does the simply-typed lambda calculus with integers have a subject expansion property, meaning if
A`e:tand e0→e, does A`e0:t? Here →is reduction under call-by-value semantics. Either prove
that subject expansion holds, or give a counterexample showing that it does not hold.
4. Consider the simply typed lambda calculus with references added:
e::= x|n|λx :t.e |e e |ref e|!e|e:= e
t::= int |t→t|tref
Figure 1 gives an operational semantics for this language. In these rules, instead of just having an
expression reduce to another expression, we also track a store s, which is a (partial) mapping from
locations `to values. Locations are just our abstraction for pointers. In the S-Ref rule, we allocate
new memory by picking an `that is not already in the domain of sand then mapping `to vin the
store. Then we can apply S-Deref if we’re asked to dereference a value that is in the store. Similarly,
S-Assign handles assignment, which in these semantics not only updates the store, but also returns
the value of the right-hand side (so we don’t have a unit value, unlike OCaml).
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