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Problem 1
For each question, choose all correct answers.
(a) Which of the following are true of thread confinement?
A.all privatefinalfields are confined
B.all immutable values are confined
C.the values of all local variables are confined
D.a BlockingQueueused to implement message passing should be confined
E.the view tree in a Swing GUI should be confined
Solution. E.
(b) Select all the strings below where the entire string is
matched by the regular expression:
([a-z]+[1-9]+)*[1-9]
A.bb8b8b
B.r2d222
C.c3po000
D.8
E.r5d4
Solution. B, D.
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(c) Consider an immutable class ComplexNumfor representing
complex numbers:
private final double realPart;
private final double imaginaryPart;
// AF: represents (realPart + i * imaginaryPart)
// RI: true
// ... other creators, producers, observers ...
public double absoluteValue() {
return Math.sqrt(realPart*realPart +
imaginaryPart*imaginaryPart)
}
@Override public boolean equals(Object other) {
if (!(other instanceof ComplexNum)) { return false; }
ComplexNum that = (ComplexNum)other;
return this.absoluteValue() == that.absoluteValue();
}

This implementation of equals...
A.is reflexive
B.is symmetric
C.is transitive
D.returns true for ComplexNum pairs that should not be equal according to
the AF
E.returns false for ComplexNum pairs that should be equal according to the
AF
Solution. A, B, C, D. □
(d) Since ServerSocket.accept() is a blocking method, we can conclude
that...
A.accept()will not return until a client makes a connection
B.accept()will throw an exception if called when no clients are connecting
C.calling accept()could prevent the program from terminating
D.it is safe to call accept()simultaneously from multiple threads
E.while accept() is blocking, calling other methods on the same
ServerSocketinstance from other threads will block
Solution. A, C. □

Problem 2 (Map/Filter/Reduce)
Ben Bitdiddle is trying to select a new password for his social media accounts. Given the
following interface for representing passwords:
public interface Password { public String
plainText(); public int strength();
}
For each method below, fill in the blanks to implement the specification using map, filter,
and reduce. You must use exactly two operations, but you may use the same operation
twice. On each line, fill in the first blank with map, filter, or reduce, and fill in the second
blank with a lambda expression. Solutions will be graded for correctness and clarity.
(a) /**
* @param passwords the passwords to consider
* @param minStrength the minimum strength a password should have
* @return the plaintext strings of passwords that have strength at least minStrength
*/
public static List<String> strongEnough(List<Password> passwords, int minStrength) {
return passwords.stream()
. ( )
. ( )
.collect(toList());
}

(b) /**
* @param substring required substring
* @return the plaintext strings of passwords that contain substring
*/
public static List<String> containing(List<Password> passwords, String
substring) {
. ( )
. ( )
.collect(toList());
}
(c) /**
* @return the average strength of the passwords
*/
public static double averageStrength(List<Password> passwords) {
. ( )

. (0, )
/ (double)passwords.size();
}
Solution.
(a).filter(p -> p.strength() >= minStrength)
.map(p -> p.plainText())
(b).map(p -> p.plainText())
.filter(p -> p.contains(substring))
(c).map(p -> p.length())
.reduce(0, (a,b) -> a+b)
Problem 3 (Abstract Data Types)
A tree is a data structure that consists of a root node and zero or more
children which themselves are also trees. A binary tree is a tree in which
each node has at most two children, designated left and right. For example:

Simon Straightforward is an enthusiastic 6.005 student who just learned about ADTs
and wants to build a class to represent binary trees of positive integers. He comes up
with this internal representation:
/** A binary tree of positive integers. */
public class PosIntBinaryTree {
private final List<List<Integer>> nodes;
}
Simon wants each sub-list in nodes to represent a level in the binary tree, using 0 in
place of missing elements. For example, the binary tree shown above is represented by
the following list of lists:
[ [ 1 ]
[ 3, 5 ]
[ 0, 2, 6, 4 ] ]
(a) Silly Simon forgot to document his abstraction function and rep invariant!
Select statements to put into the AF and RI of PosIntBinaryTree by circling the letters of
all statements to include from the list below. Include all good statements that are
compatible with Simon’s design.
AF: A B C D E F G
RI: A B C D E F G
A.all 0 <= i < nodes.size(), nodes.get(i).get(0) > 0
B.if nodes.size() == 0, represents the empty tree

Rob Recursive, an even more enthusiastic student, wants to represent binary trees of
any integers.
(b)What bad practice in Simon’s rep prevents Rob from easily extending it?
Solution. Magic number 0. □
(c)Rob decides to implement the binary trees recursively, using an interface
IntBinaryTree with two concrete variants, one of which represents the empty tree.
He also decides the type will be immutable.
/** An immutable binary tree of integers. */
public interface IntBinaryTree { ... }
Write a recursive datatype definition for IntBinaryTree:
Solution. IntBinaryTree = Empty() + Tree(value: int, left,right: IntBinaryTree) □
C.if nodes.size() > 0, nodes.get(0).get(0) is the root node
D.nodes.get(i).size() == 2^I
E.nodes.size() == 2^k for some nonnegative integer k
F.for node nodes.get(i).get(j),
its left child (if any) is nodes.get(i+1).get(j*2)
G.for node nodes.get(i).get(j),
its right child (if any) is nodes.get(i+1).get(j*2+1)
Solution. AF: B, C, F, and G. RI:
D.

(d) Start implementing the concrete variant that represents the empty tree by
writing its field declarations, abstraction function, and rep invariant. If parts of the
AF or RI would normally be assumed in 6.005, write them explicitly.
Fields:
AF:
RI:
Solution.
No fields
AF: represents the empty immutable
binary tree of integers RI: true
□
(e) Start implementing the other concrete variant by writing its field declarations,
abstraction function, and rep invariant. If parts of the AF or RI would normally be
assumed in 6.005, write them explicitly.
Fields:
AF:
RI:
Solution.
private final int value;
private final IntBinaryTree left, right;
AF: represents the immutable binary tree of integers with root node value, left
subtree left, and right subtree right

RI: left,right != null □
Problem 4 (Recursive Data Types) (20 points).
A Boolean formula is a propositional expression consisting of variables, ∧
(and) and ∨ (or) binary operators, and ¬ (not) unary operators.
For example, the following expression means “either P or Q is true, and
either P is false or R is true”:
(P ∨ Q) ∧ (¬P ∨ R)
A formula is in negation normal form if the negation operator (¬) is only
applied directly to variables. For example, P ∧ (¬Q ∨ R) is in negation
normal form, but P ∧ ¬(Q ∨ R) and ¬(P ∧ (¬Q ∨ R)) are not.
Here is a datatype definition for an ADT to represent negation normal form
Boolean formulas:
NegNormFormula = Variable(name: String, isNegated: boolean)
+ And(left: NegNormFormula, right:
NegNormFormula) + Or(left: NegNormFormula,
right: NegNormFormula)
Now consider this specification:
// returns a negation of the input formula
negate : NegNormFormula -> NegNormFormula
(a) Classify this operation according to our types of ADT operations:

□
(b) Implement the operation. Use De Morgan’s laws and the
rule for double negation:
¬(P ∧ Q) = ¬P ∨ ¬Q and
¬(P ∨ Q) = ¬P ∧ ¬Q and
public class Variable implements NegNormFormula {
private final String name;
private final boolean isNegated;
public Variable(String name, boolean isNegated) {
... } @Override public NegNormFormula negate() {
¬¬P = P
Solution.
return new Variable(name, ! isNegated);
□
}
}
public class And implements
NegNormFormula { private final
NegNormFormula left; private final
NegNormFormula right;
public And(NegNormFormula left, NegNormFormula
right) { ... } @Override public NegNormFormula negate() {
Soluti
on.
return new Or(left.negated(),
right.negated());
□

}
}
public class Or implements
NegNormFormula { private final
NegNormFormula left; private final
NegNormFormula right;
public Or(NegNormFormula left, NegNormFormula right)
{ ... } @Override public NegNormFormula negate() {
Solution.
return new And(left.negated(),
right.negated());
□
}
}
Problem 5 (Thread Safety)
It’s election season! With the tight race for president of Fictional Dystopia,
you’ve been asked to develop a secure online voting system.
In Fictional Dystopian elections, each voter has exactly two votes. Each
voter can vote twice for the same candidate, or split their vote between two
different candidates.
public class TwoVotesEachElection {
private final AtomicInteger[] voteCounts;
private final Set<String> voters;

// ... constructor initializes voteCounts to an array of AtomicIntegers,
// each with value zero, and initializes voters to a threadsafe Set ...
public int getNumberOfCandidates() { return voteCounts.length; }
// requires: 0 <= can1, can2 < getNumberOfCandidates()
// effects: if and only if the voter hasn’t already voted, casts both votes
public void vote(final String voterID, final int can1, final int can2) {
if (voters.contains(voterID)) { return; }
voters.add(voterID);
// use the atomic incrementAndGet operation
// (we don’t need the "get" part, but there is no plain "increment")
voteCounts[can1].incrementAndGet();
voteCounts[can2].incrementAndGet();
}
// ...
}
Suppose election is a TwoVotesEachElection with 4 candidates (so voteCounts.length
= 4).
(a) Unfortunately, even though TwoVotesEachElection uses threadsafe datatypes, the
vote() operation is not threadsafe. Explain clearly an interleaving that violates thread
safety for TwoVotesEachElection.

Solution. Two threads enter votewith the same voterIDnot yet in voters.
Both threads call voters.containsand obtain false, then both call voters.addsuccessfully.
At this point, both threads will increment the vote counts for their can1 and can2, giving this
voter 4 votes instead of 2, in violation of the postcondition. □
Louis Reasoner is convinced that using AtomicInteger is unnecessary, so he changes the
code to use primitive integers instead:
public class LouisTwoVotesEachElection {
private final int[] voteCounts;
private final Set<String> voters;
// ... constructor initializes voteCounts to an array of zeros,
// and initializes voters to a threadsafe Set ...
public int getNumberOfCandidates() { return voteCounts.length; }
// requires: 0 <= can1, can2 < getNumberOfCandidates()
// effects: if and only if the voter hasn’t already voted, casts both votes
public void vote(final String voterID, final int can1, final int can2) {
if (voters.contains(voterID)) { return; }

voters.add(voterID); voteCounts[can1]++;
voteCounts[can2]++;
}
// ...
}
Of course, this just makes the problem worse.
(b) Suppose 100 voters, each using a different thread, participate in our 4-candidate
election by concur- rently calling Louis’ vote(). Every voter uses a unique ID, and no one
casts both of their votes for the same candidate. The value of the sum
voteCounts[0] + voteCounts[1] + voteCounts[2] + voteCounts[3]
could be which of the following? Choose all that apply.
A.0
B.1
C.2
D.4
E. 100
F. 200
G. 400
Solution. C, D, E, F.
□

Solution. A, C, D.
Locking on different voterID strings doesn’t help. voteCounts[0] is a primitive in Louis’
code.
At least one thread will successfully increment the count for its first candidate; same
for some thread and its second candidate. With no races at all, 200 votes will be
counted.
(c) Louis wants to fix the thread safety problems by putting all the code inside vote()
in a synchronized block. Of the following objects in Louis’ version of the code,
which are suitable for us to synchronize on in order to ensure thread safety for
concurrent calls to vote()? Choose all that apply.
A.this
B.voterID
C.voters
D.voteCounts
E.voteCounts[0]

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