CMPSCI 250: Introduction to Computation
Final Exam
David Mix Barrington
29 April 2016
Directions:
- Answer the problems on the exam pages.
- There are five problems, each with multiple parts, for 120 total
points plus 10 extra credit. Actual scale A = 110, C = 75.
- Some useful definitions precede the questions below.
- No books, notes, calculators, or collaboration.
- In case of a numerical answer, an arithmetic expression like
"2^{17} - 4" need not be reduced to a single integer.
Q1: 10 points
Q2: 20 points
Q3: 20+10 points
Q4: 30 points
Q5: 40 points
Total: 120+10 points
Question 3 deals with a set D of dogs, consisting of exactly the
five
dogs Arly (a), Cardie (c), Duncan (d), Knight (k), and Nico (n), and
a set of four activities A, consisting of "barking" (bar), "chasing"
(cha),
"swimming" (swi), and "wrestling" (wre). The predicate L(x, y, z)
means "dog x likes activity y with dog z". If y is any activity, the
binary relation L_{y} on D is defined by L_{y}(x, z)
↔ L(x, y, z).
Question 3 also refers to the following six statements, where the
variables are of type "dog" or type "activity":
- Statement I: For every activity y, the relation
L_{y}
is both reflexive and symmetric.
- Statement II: Cardie likes to wrestle with Duncan, but
she
does not like to wrestle with any other dog (except herself) and she
does not like any other activity with Duncan.
- Statement III: Cardie likes both swimming and chasing
with both Nico and Knight.
- Statement IV: ∀y: ∀z: L(a, y, z) ↔
((y = bar) ∨ (z = a))
- Statement V: ∃y: ∀z: L(c, y, z)
- Statement VI: ∀z: ∃y: L(c, y, z)
N is the set of naturals (non-negative integers),
{0, 1, 2, 3,...}.
Question 4 uses a recursive function from N to N
defined by the rules f(0) = 0 and f(n+1) = 2f(n) + n.
Questions 4 and 5 use the following DFA D, which has state set {p,
q, r}, start state p, final state set {p, q}, and transition function
δ given by δ(p, a) = p, δ(p, b) = q, δ(q, a) =
r,
δ(q, b) = q, and δ(r, a) = δ(r, b) = r.
We also use D to define three functions p(n), q(n), and r(n) from
N to N as follows. For any natural n, p(n) is the
number
of strings w of length n such that D (started in state p) finishes
reading
w in state p. (That is, such that δ^{*}(p, w) = p.)
Similarly, q(n) is the number of strings w of length n such that D
(again started in state p) finishes reading w in state q, and r(n) is
the number finishing in state r.
Question 5 also uses the following λ-NFA N, which has state set
{1, 2, 3, 4}, start state 1, final state set {4}, and transition
relation
Δ = {(1, λ, 2), (2, a, 2), (2, λ, 3), (3, b, 3),
(3, λ, 4)}.
The DFA D:
___ ___ ___
| | a | | b | | a, b
| V | V | V
>((p)) -----> ((q)) -----> (r)
b a
The lambda-NFA N ("L" = lambda):
___ ___
| | a | | b
| V | V
>(1) -----> (2) -----> (3) -----> ((4))
L L L
- Question 1 (10): Compare and contrast the two concepts
in each of the following pairs, giving enough detail to make it clear that
you are familiar with them (2 points each):
- (a, 2) A back edge and a forward edge in the
depth-first search tree of a directed graph.
- (b, 2) The prefix string and the postfix string
for
an arithmetic expression.
- (c, 2) The complement of a language and the one's
complement of a binary string.
- (d, 2) The empty string and the empty language.
- (e, 2) A Turing decidable language and a Turing
recognizable language.
- Question 2 (20):
The following are ten true/false questions, with no explanation needed
or wanted, no partial credit for wrong answers, and no penalty for
guessing. Each one counts two points.
- (a)
If Statement V above is true, then Statement VI must be true as well.
(Ignore
the other statements.)
- (b)
If we want to store the relation L above as a boolean array, without
assuming any of the statements, we will need exactly 100 boolean
variables.
- (c)
There is some natural number x such that for any natural number y, the
congruence 14xy ≡ y (mod 25) is true.
- (d)
The operation of concatenation on binary strings is associative but
not commutative, and has an identity element.
- (e)
If there is a path from node s to node t in a directed graph G,
then a depth-first search of G with start node s and goal node
t will always find a path from s to t with a minimum number of
edges.
- (f)
In any weighted graph with non-negative edge weights, the function h
that has h(x) = 0 for every node x is always an admissible and
consistent
heuristic for an A^{*} search.
- (g)
Let G be a finite two-player deterministic game of perfect
information, where the game ends with a real number payoff from
Black
to White. Assume that Black has a strategy that will ensure that
he pays at most 3. Then White must have a strategy that
guarantees her a payoff of at least 3.
- (h)
If a language X is not regular, then there exists a Turing machine M
such that X = L(M).
- (i)
Suppose that a single-tape Turing machine M has the transition
δ(p, ☐) = (p, a, R) where ☐ is the blank symbol.
Then if M reaches the configuration ☐ab☐p☐, it is
guaranteed never to halt.
- (j)
If the union of two languages X and Y is Turing decidable, then both
X and Y must themselves be Turing decidable.
- Question 3 (20+10):
This question deals with five statements about a set of dogs D, consisting
of exactly the five dogs Arly (a), Cardie (c), Duncan (d), Knight (k),
and Nico (n), and a set of four activities A, consisting of
"barking" (bar), "chasing" (cha), "swimming" (swi), and
"wrestling" (wre). The predicate L(x, y, z) means "dog x likes
activity y with dog z". If y is any activity, then the binary
relation
L_{y} is defined by L_{y} (x, z) ↔ L(x, y, z).
- Question 4 (30): This question uses the functions f, p,
q, and r from N to N, and the DFA D, defined above.
Recall
that for any natural n, p(n) is the number of strings w of length n
such that δ^{*}(p, w) = p, and that q(n) and r(n) are
defined similarly (with p still used as the start state).
- (a, 10) Recall that f is defined by the rules f(0) = 0 and, for
all naturals n, f(n+1) = 2f(n) + n. Prove by induction that for
every natural n, f(n) = 2^{n} - n - 1.
- (b, 5) Compute the value of p(n), q(n), and r(n) for all n
with 0 ≤ n ≤ 4.
- (c, 5) Explain why for any natural n, r(n+1) = 2r(n) + q(n).
Your explanation should refer specifically to the DFA D.
- (d, 10)
Let Z(n) be the statement "(p(n) = 1) ∧ (q(n) = n) ∧ (r(n) =
2^{n} - n - 1)". Prove that Z(n) is true for all naturals
n, by ordinary induction on n.
- Question 5 (40): This question is the usual one about
Kleene's Theorem constructions, using both the DFA D and the
λ-NFA N given above.
- (a, 10) Use the construction from lecture and the text to
obtain a regular expression denoting the language of D. Do not
simplify your expression -- I want the one from the construction.
- (b, 5) Use the construction from lecture and the
text to find a λ-NFA whose language is denoted by the
regular expression from part (a). (N is such a λ-NFA, but I
want the one from the construction.)
- (c, 10) Using the construction given in lecture and in the
text, find an ordinary NFA N' such that L(N') = L(N). (Use N
rather
than your answer from (b) == it will be easier.)
- (d, 10) Using the Subset Construction on N', find a DFA D'
such that L(D') = L(N).
- (e, 5) Either prove that your D' is minimal or prove that
it is not minimal. (If it is not minimal, you are not required to
minimize it.)
Last modified 14 May 2016