Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages
Control Flow Analysis and Logic Programming
Section and Practice Problems
Monday April 27, 2015
1 Control Flow Analysis
Consider the following lambda calculus program.
(λf. (f76) + (f77)) (λa. a)
(a) Add labels to the program. That is, make it an expression in the labeled lambda calculus of Lecture
22, where every label is unique.
(b) Let ebe your labeled lambda calculus program. Write out C[[e]]e, i.e., the set of constraints for the
program e. (Hint, you should have 20 constraints in total. In particular, for each of the 3 applications,
you should have 4 constraints, 2 for each of the lambda terms in the program.)
(c) Find C∗and r∗, the smallest functions that satisfy the constraints you generated in the question above.
(d) Check that your functions C∗and r∗make sense. That is, if an expression labeled lcan evaluate to an
expression labeled l0, do you have l0∈C∗(l)?
(e) Consider adding the expression (let x=e1in e2)lto the language. Define C[[(let x=e1in e2)l]]e. Try
rewriting the program above using one or more let expressions, and make sure that the constraints
you generate for the modified program produce the same solution C∗and r∗.
2 Logic Programming
(a) Consider the following Prolog program (where [] is a constant representing the empty list, [t]is short-
hand for cons(t, []) and [t1, t2|t3]is shorthand for cons(t1,cons(t2, t3)).
foo([],[]).
foo([X],[X]).
foo([X, Y |S],[Y, X|T]) :– foo(S, T )
For each of the following queries, compute the substitutions that Prolog will generate, if any. (Note
that there is a difference between an empty substitution, and no substitution.) If the query evaluation
will not terminate, explain why.
•foo([a,b], X ).
•foo([a,b,c], X ).
•foo([a,b],[a,b])
•foo(X, [a])
•foo(X, Y ).
