Assignment 4

Due on Oct 23 before lecture.

Please bring a hardcopy of your solutions to lecture. Or submit a hardcopy of your solutions to Paul Darga before lecture.

Problem 1 (6 Points)

Consider the language of boolean and arithmetic expressions discussed in class (Figures 3-1 and 3-2 in the Pierce book). Suppose we add a new syntactic form:

double t

Suppose we also add new evaluation rules:

t1 ---> t1'
-------------------------
double t1 ---> double t1'
double 0 ---> 0
double (succ nv1) ---> succ (succ (double nv1))

Which of the theorems discussed in class (Theorems 3.5.4, 3.5.7, 3.5.8, 3.5.11, 3.5.12 in the Pierce book) remain valid?

If a theorem is no longer valid, then present a counter example. If a theorem is still valid but its proof is different, then present the new proof. (Note that the Pierce book contains proofs of these theorems for the original language. You do not have to reproduce these proofs. Just show the portions of the proofs that are different now.)

Problem 2 (3 Points)

Consider the language of boolean and arithmetic expressions discussed in class (Figures 3-1 and 3-2 in the Pierce book). Suppose we add a new evaluation rule:

succ (pred nv1) ---> nv1

Problem 3 (7 Points)

Consider the language of boolean expressions discussed in class (Figure 3-1 in the Pierce book). Suppose we add a new syntactic form:

and t t

Suppose we also add new evaluation rules:

and true true ---> true
and true false ---> false
and false true ---> false
and false false ---> false

t1 ---> t1'
-------------------------
and t1 t2 ---> and t1' t2

t2 ---> t2'
-------------------------
and t1 t2 ---> and t1 t2'