Question 1 (15):
Using the definition of boolean straight-line program (SLP) from Lecture
1, prove that any fixed finite boolean function can be computed by an
SLP. That is, let k be a number and let f(x1,...,xk)
be any boolean-valued function on k boolean inputs. Prove that there is an
SLP whose output is f.
Question 2 (20):
Describe two different SLP's that input a string of n boolean values and
output 1 iff the input has exactly two ones. Your first SLP should separately
consider each possible pair of locations for the two ones, and have length
O(n3). The other should have length O(n) -- it is probably easiest
to simulate the action of a finite-state machine.
Question 3 (10): The depth of an SLP instruction is one
greater than the maximum of the depth of its inputs, where input variables have
depth zero. The depth of an SLP is the depth of its return instruction.
Determine the depth of your SLP's in Question 2, in big-O terms. Can you
get depth O(log n)? Depth O(1)?
Question 4 (15): Using Question 1 and quoting Kleene's Theorem,
prove that any regular language can be decided by an SLP with length O(n).
Question 5 (10): Using the definition in Lecture 1, show that
the set of numbers that have exactly two ones in their binary expansion is
Bloop-definable and hence primitive recursive. (Hint: First define a
function isPowerOfTwo(x) that returns 1 if its input x is a
power of two and returns zero otherwise.)
Question 6 (10): Let X be the regular language
1(01+10)*0. Find all the Myhill-Nerode equivalence classes for
X (that is, the equivalence classes of the relation ~X defined
so that u ~X v iff for all strings w, uw is in X iff vw is in X)
and give a regular expression for each.
Question 7 (10): Write a first-order sentence, with input
predicate I(x), that is true for an input string w iff w is in the regular
Question 8 (10): Describe a PDA whose language is given by
the grammar with rules: