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(a)Which of thefollowing mustbe trueof anunderdeterminedfunction specification?(choose all that apply)
A. An underdeterminedspecmeanstheimplementationisunwritten.
B. An underdeterminedspecmeanstheimplementationisnondeterministic.
C. An underdeterminedspecallows multiple valid outputsfor some input.
D.An underdeterminedspecallows multiple valid inputsthatgive some output.
(b) After thefollowing code is executed, whatis thevalue of arrayarr? (choose one answer)
final String[] arr = new String[2]; String s = "6.005";
arr[0] = s;
s = "is";
arr[1] = s; String t = arr[0]; t = "fun";
A. [ "is", "is" ]
B. [ "6.005", "is" ]
C. [ "fun", "is" ]
D. none of the above
(c) The line of Javacode String t = arr[0]; involves... (choose all that apply)
A. assignment
B. equivalence relation
C. mutation
D.static typing

(d)Alyssa P.Hackeris designing animmutabletype torepresentusersinhercomputersystem. The User’s login
nameis stored as:
private final String kerberos;
She definestwo User objects asequalif theirlogin namesarethesame,ignoring case:
@Override public boolean equals(Object other) { if (!(other instanceof
User)) { return false; } User that = (User)other;
return this.kerberos.equalsIgnoreCase(that.kerberos);
}
@Override public int hashCode() { /* TODO */ }
Which of thefollowing implementations of hashCode() would bevalid, satisfyingtheObject contract? (choose
all that apply)
A.return 31;
B.return this.kerberos.hashCode();
C.return this.kerberos.toLowerCase().hashCode();
D.return this.kerberos.toUpperCase().hashCode();
(e)If thecode inanswerchoice (A) aboveappearedinyour 6.005code review assignment,which of the following
commentswould beappropriatecriticisms? (choose all that apply)
A. The code isn’t DRY.
B. The code usesmagic numbers.
C. The code exposes User’s representation.
D.The code is unnecessary,we don’t needtooverridehashCode if we only returna constant.

Problem 2 (Specifications)
A tetromino is a shape made out of four adjacent squares. These shapes are most famous from the game
Tetris where the player must rotate and translate falling tetrominoes in order to fit them together. There are
sevenpossible tetrominoesthatlie onthe2D plane,andeachis identifiedby aletter it lookslike:
I O T J L S Z
Define atetromino shape letter asoneof thesevenletters IO T JL S Z, eitherupper-orlowercase.
Tetrominoesmayberotatedby 0,90,180, or270 degrees.
A diagram of all seven tetrominoes in all four orientations is on the last page of this quiz.
Let’s defineanabstractdatatype torepresenttetrominoes:
public class Tetromino {
// determine if a character is a valid tetromino shape letter
public static boolean isValidShape(char shape) { ... }
// make a new tetromino
public Tetromino(char shapeLetter, int rotationDegrees) { ... }
// rotate this tetromino
public void rotateClockwise() { ... }
// get the shape of this tetromino
public char shape() { ... }
}
(a) Fill inthistable with informationabouttheoperationsof Tetromino. The
answersfor isValidShape arealready given.

operation type signature
classify thetypeof
ADT operation
Java implementation
strategy
isValidShape char→ boolean
not applicable static method
Tetromino
rotateClockwise
shape

(b) Consider thesedifferent specificationsfor theisValidShapefunction:
public static boolean isValidShape(char shape)
/**
* @param shape any character
* @return true iff shape is a lowercase tetromino shape letter
*/
/**
* @param shape an English alphabetic character
*/
/**
* @param shape a lowercase tetromino shape letter
* @return true
*/
SpecA
SpecB
SpecC
Comparethesespecifications.Foreachpairbelow, circle one correct option andwrite write a brief
explanation tocompletethesentence.
A vs. B:
equivalent to
weakerthan
strongerthan
incomparable to
SpecA is specB because
B vs. C:
equivalent to
weakerthan
strongerthan
incomparable to
SpecB is
specC because

Problem 3 (AFs & RIs)
Hereis animplementationof theTetrominoADT.
/** A rotatable tetromino. */
// ... static method isValidShape ...
private final char shape;
private int rotation;
// Abstraction function
// TODO
// Representation invariant
// TODO
// Safety from rep exposure
// TODO
/**
* Make a new Tetromino with the given shape and rotation.
* @param shapeLetter uppercase tetromino shape letter
* @param rotationDegrees clockwise rotation in degrees,
* must be 0, 90, 180, or 270

*/
public Tetromino(char shapeLetter, int rotationDegrees) {
this.shape = shapeLetter;
this.rotation = rotationDegrees / 90;
}
/**
* TODO
*/
public void rotateClockwise() { rotation = (rotation + 1) %
4;
}
/**
* @return shape of this tetromino: ’I’ ’O’ ’T’ ’J’ ’L’ ’S’ or ’Z’
*/
public char shape() {
return shape;
}
@Override
public String toString() {
return shape + "-shape@" + rotation * 90 + "deg";
}
}

(a) For each of the statements below, say whether it should be included in the internal documentation of
Tetromino by writing:
AF if the statement belongs in the abstraction function
RI ... the rep invariant
EXP ... the argument that type has no rep exposure
NONE if it should not be included in any of those
You should include in the AF, RI, or EXP all good statements that are compatible with the code and specs on
the previous page.
Do not include statements that are not compatible with the code and specs.
shape is private and an immutable value
rotation is private and an immutable value
this Tetromino is never returned to clients
shape is an uppercase tetromino shape letter
shapeLetter is an uppercase tetromino shape letter
the tetromino has the shape given by shape
rotation = 0
the tetromino is rotated clockwise by rotation

the tetromino is rotated clockwise rotation times 90 degrees
0 <= rotation < 4
0 <= rotation < 360
rotation is one of { 0, 90, 180, 270 }
Consider thesedifferent specificationsfor therotateClockwisemethod:
public void rotateClockwise()

Foreachpossiblespecificationbelow, write a one-sentence code review comment that identifies themost
serious problem with the spec.
(b) /**
* Update this tetromino’s rotation number to add 1, mod 4.
*/
Comment:
(c) /**
* @return this tetromino rotated by 90 degrees clockwise
*/
Comment:
(d) /**
* Rotate this tetromino (does not otherwise modify the tetromino).
*/
Comment:

Problem 4
Let’s consider the problem of packing tetrominoes into a rectangle by rotating and translating them in the 2D
plane. For example, here are tetrominoes I, L, & Z packed into a 5 × 4 rectangle, and a 7 × 2 rectangle:
For those three tetrominoes, the minimum area they can be packed into is 7 × 2 = 14 squares.
(a) /**
* @param shapes string of tetromino shape letters
* @return a 2-element list representing a minimum-area rectangle into which
* the tetrominoes given by shapes can be packed
* (for example, pack("ILZ") might return the list [ 2, 7 ])
*/
public static List pack(String shapes) { ... }
Start writing a black box testing strategy for pack(..) by giving one good partitioning for input shapes:
For each of the test cases below, in the first box write YES or NO in the first box to say whether the
test valid or not. If the test is not valid, write a one-sentence reason why not. We’ll use Python’s
syntax to represent lists for brevity

Problem 5 (Rep Exposure) . Let’s define a mutable abstract data type TetrominoGrid to represent tetrominoes arranged
on a fixed-size grid, where every tetromino fits on the grid and none of the tetrominoes overlap. For example:

On the next page is an implementation of TetrominoGrid. In its rep, TetrominoGrid stores the location of
each tetromino using a map from integers to tetrominoes. The integer keys are square numbers: Each square
in the grid is numbered starting from 0 in the upper-left corner.
On a grid of width width, square at row row and column col is numbered row × width + col. For example,
here’s how our type would represent the example above:
width = 5, height = 4
tetrominoLocations = {
0: (Tetromino: I-shape rotated 0 degrees),
3: (Tetromino: L-shape rotated 0 degrees),
6: (Tetromino: Z-shape rotated 90 degrees)
}
The TetrominoGrid ADT is implemented using the same mutable Tetromino ADT from previous questions.
(a) Identify all instances of rep exposure in TetrominoGrid on the next page. For each one, write:
1. the line number most directly responsible for the problem,
2. at most one-sentence description of the rep exposure, and
3. at most one-sentence description of how to fix the problem, or a single corrected line
of code.
There may be more boxes than you need.

Line #: Explanation:
You may detach this page from your quiz, but you must write your name above and turn in all pages.
/**
* Mutable type representing a fixed-size grid with a valid arrangement of
* tetrominoes: every tetromino fits on the grid without overlapping.
*/
public class TetrominoGrid {
1) private final int width;
2) private final int height;
3) public final Map<Integer, Tetromino> tetrominoLocations;

// create a new grid and try to add some tetrominoes
public TetrominoGrid(int width, int height, List<Tetromino> initial) {
this.width = width;
this.height = height; this.tetrominoLocations = new
HashMap<>(); for (Tetromino tetromino : initial) {
this.add(tetromino.shape());
}
}
4)
5)
6)
7)
// get the tetromino whose upper-left corner is the given square
public Tetromino getTetrimino(int row, int col) {
return tetrominoLocations.get(row * width + col);
}
8)
// return a list of all tetriminos currenty on the grid
public List<Tetromino> getTetriminosOnBoard() {
List<Tetromino> tetrominoes = new ArrayList<>();
for (Tetromino tetromino : tetrominoLocations.values()) {
tetrominoes.add(tetromino);
}
return Collections.unmodifiableList(tetrominoes);
}
9)
10)
11)
// try to add a tetromino to the grid, if it can fit without overlap
public boolean add(char shape) {
boolean canBePlaced = false; // can we fit the tetromino anywhere?
int topLeft = -1;
}
12)
13)
14)
15)

int rotation = 0;
// ... code to check whether the new tetromino fits and doesn’t
overlap ...
// ... updates the values of the local variables accordingly ...
if (canBePlaced) {
tetrominoLocations.put(topLeft, new Tetromino(shape,
rotation));
}
return canBePlaced;
16)
// ... other operations ...
}

Problem 1 (Multiple Choice)
(a)Which of the following must be true of an underdetermined function specification? (choose all that apply)
A.An underdetermined spec means the implementation is unwritten.
B. An underdetermined spec means the implementation is nondeterministic.
C. An underdetermined spec allows multiple valid outputs for some input.
D.An underdetermined spec allows multiple valid inputs that give some output.
Solution. C. None of the other options must be true. □
(b) After the following code is executed, what is the value of array arr? (choose one answer)
final String[] arr = new String[2]; String s = "6.005";
arr[0] = s;
s = "is";
arr[1] = s; String t = arr[0]; t =
"fun";
A. [ "is", "is" ]
B. [ "6.005", "is" ]
C. [ "fun", "is" ]
D. none of the above
Solution. B. The reassignment of sto "is"or tto "fun"do not affect arr[0].
□

(c) The line of Java code String t = arr[0]; involves... (choose all that apply)
A.assignment
B. equivalence relation
C.mutation
D.static typing
Solution. A, D. This line assigns to t, which has static type String. □
(d) Alyssa P. Hacker is designing an immutable type to represent users in her computer system. The User’s
login name is stored as:
private final String kerberos;
She defines two Userobjects as equal if their login names are the same, ignoring case:
@Override public boolean equals(Object other) { if (!(other
instanceof User)) { return false; } User that = (User)other;
return this.kerberos.equalsIgnoreCase(that.kerberos);
}

@Override public int hashCode() { /* TODO */ }
Which of the following implementations of hashCode() would be valid, satisfying the Object contract?
(choose all that apply)
A.return 31;
B.return this.kerberos.hashCode();
C.return this.kerberos.toLowerCase().hashCode();
D.return this.kerberos.toUpperCase().hashCode();
Solution. A, C, D.
A is correct, although not necessarily a good hash function.
C and D are correct because Strings that are equal ignoring case will have the same value for both
toLowerCase()and toUpperCase().
B is incorrect because two Users with with login names that differ only in case will have different hash
codes even though the Userobjects are equal. □
(e)If the code in answer choice (A) above appeared in your 6.005 code review assignment, which of the
following comments would be appropriate criticisms? (choose all that apply)
A.The code isn’t DRY.
B.The code uses magic numbers.
C.The code exposes User’s representation.
D.The code is unnecessary, we don’t need to override hashCodeif we only return a constant.
Solution. B. 31 is a magic number.
□
Problem 2 (Specifications)
A tetromino is a shape made out of four adjacent squares. These shapes are most famous from the game
Tetris where the player must rotate and translate falling tetrominoes in order to fit them together. There are
seven possible tetrominoes that lie on the 2D plane, and each is identified by a letter it looks like:

I O T J L S Z
Define a tetromino shape letter as one of the seven letters I O T J L S Z, either upper- or lowercase.
Tetrominoes may be rotated by 0, 90, 180, or 270 degrees.
A diagram of all seven tetrominoes in all four orientations is on the last page of this quiz.
Let’s define an abstract data type to represent tetrominoes:
// determine if a character is a valid tetromino shape letter
public static boolean isValidShape(char shape) { ... }
// make a new tetromino
public Tetromino(char shapeLetter, int rotationDegrees) { ... }
// rotate this tetromino
public void rotateClockwise() { ... }
// get the shape of this tetromino
public char shape() { ... }
}
(a) Fill in this table with information about the operations of Tetromino. The
answers for isValidShapeare already given.

operation type signature
classify the type of
ADT operation
Java implementation
strategy
isValidShape char→ boolean
not applicable static method
Tetromino char×int→Tetromino
creator constructor
rotateClockwise Tetromino→ void
mutator instance method
shape Tetromino→ char observer instance method
(b) Consider these different specifications for the isValidShapefunction:
public static boolean isValidShape(char shape)
/**
* @param shape any character
*/
/**
* @param shape an English alphabetic character
*/
SpecA
Spec B

Compare these specifications. For each pair below, circle one correct option and write write a brief explanation to
complete the sentence.
Solution. Spec A is stronger than spec B because its precondition is weaker and its postcondition is the same.
Solution. Spec B is stronger than spec C because it has a weaker precondition, and it satisfies C’s postcondition for any
input that satisfies C’s precondition.
Problem 3 (AFs & RIs)
Here is an implementation of the Tetromino ADT.
/** A rotatable tetromino. */

// ... static method isValidShape ...
private final char shape;
private int rotation;
// Abstraction function
// TODO // Representation invariant
// TODO // Safety from rep exposure
// TODO
public Tetromino(char shapeLetter, int rotationDegrees) {
this.shape = shapeLetter;
this.rotation = rotationDegrees / 90;

(a) For each of the statements below, say whether it should be included in the internal documentation of Tetromino by
writing:
AF if the statement belongs in the abstraction function
RI ... the rep invariant
EXP ... the argument that type has no rep exposure
NONE if it should not be included in any of those
You should include in the AF, RI, or EXP all good statements that are compatible with the code and specs on the
previous page.
Do not include statements that are not compatible with the code and specs.
shape is private and an immutable value
rotation is private and an immutable value
this Tetromino is never returned to clients
shape is an uppercase tetromino shape letter
shapeLetter is an uppercase tetromino shape letter
the tetromino has the shape given by shape
rotation = 0
the tetromino is rotated clockwise by rotation
the tetromino is rotated clockwise rotation times 90 degrees
0 <= rotation < 4
0 <= rotation < 360
rotation is one of { 0, 90, 180, 270 }

Solution.
EXP EXP
NONE (doesn’t make sense)
RI
NONE (part of the precondition of the constructor)
AF NONE NONE
AF (see e.g. toString)
RI (established by constructor, maintained by rotateClockwise)
NONE NONE
□
Consider these different specifications for the rotateClockwisemethod:
public void rotateClockwise()
For each possible specification below, write a one-sentence code review comment that identifies the most
serious problem with the spec.
(b) /**
* Update this tetromino’s rotation number to add 1, mod 4.
*/
Comment:
Solution. The specification must be abstract, it should not talk about the representation.
The specification should be declarative, not operational.
□

(c) /**
* @return this tetromino rotated by 90 degrees clockwise
*/
Comment:
Solution. The specification, which says a Tetrominois returned, does not agree with return type void.
□
(d) /**
* Rotate this tetromino (does not otherwise modify the tetromino).
*/
Comment:

Solution. This specification is too weak, it does not specify the amount of rotation. □
Problem 4
Let’s consider the problem of packing tetrominoes into a rectangle by rotating and translating
them in the 2D plane. For example, here are tetrominoes I, L, & Z packed into a 5 × 4 rectangle,
and a 7 × 2 rectangle:
Start writing a black box testing strategy for pack(..) by giving one good partitioning for input shapes: For each of
the test cases below, in the first box write YES or NO in the first box to say whether the test valid or not. If the test
is not valid, write a one-sentence reason why not. We’ll use Python’s syntax to represent lists for brevity.

(b) shapes = "X"
rectangle = [ 0, 0 ]
Valid? Reason if invalid:
Solution. Invalid: shape X violates the precondition
(c) shapes = "I"
Solution. Invalid: rectangle [ 4, 1 ] is also allowed by the
postcondition.
d) shapes = "LO“
Solution. Valid.
Problem 5
Let’s define a mutable abstract data type TetrominoGrid to represent tetrominoes arranged on a fixed-size
grid, where every tetromino fits on the grid and none of the tetrominoes overlap. For example:

On the next page is an implementation of TetrominoGrid. In its rep, TetrominoGrid stores the location of
each tetromino using a map from integers to tetrominoes. The integer keys are square numbers:
Each square in the grid is numbered starting from 0 in the upper-left corner
On a grid of width width, square at row row and column col is numbered row × width + col.
For example, here’s how our type would represent the example above:
The TetrominoGrid ADT is implemented using the same mutable Tetromino ADT from previous
questions.
(a) Identify all instances of rep exposure in TetrominoGrid on the next page. For each one, write:
1. the line number most directly responsible for the problem,
2. at most one-sentence description of the rep exposure, and
3. at most one-sentence description of how to fix the problem, or a single corrected line of code.
There may be more boxes than you need.

Solution.
Line 3: tetrominoLocations can be accessed and mutated by clients. It should be private.
Line 8: returns a Tetromino that is in the rep, clients could mutate it and break the rep invariant. Should
return a defensive copy.
Line 11 (or 10): returns an unmodifiable list, but the Tetromino objects are still mutable
parts of the rep. Should return defensive copies.
public class TetrominoGrid {
1) private final int width;
2) private final int height;
3) public final Map<Integer, Tetromino> tetrominoLocations;
// create a new grid and try to add some tetrominoes
public TetrominoGrid(int width, int height, List<Tetromino> initial) {
this.width = width;
this.height = height; this.tetrominoLocations = new
HashMap<>(); for (Tetromino tetromino : initial) {
this.add(tetromino.shape());
}
}
4)
5)
6)
7)

// get the tetromino whose upper-left corner is the given square
public Tetromino getTetrimino(int row, int col) {
return tetrominoLocations.get(row * width + col);
}
8)
// return a list of all tetriminos currenty on the grid
public List<Tetromino> getTetriminosOnBoard() {
List<Tetromino> tetrominoes = new ArrayList<>();
for (Tetromino tetromino : tetrominoLocations.values()) {
tetrominoes.add(tetromino);
}
return Collections.unmodifiableList(tetrominoes);
}
9)
10)
11)
// try to add a tetromino to the grid, if it can fit without overlap
public boolean add(char shape) {
boolean canBePlaced = false; // can we fit the tetromino anywhere?
int topLeft = -1;
int rotation = 0;
// ... code to check whether the new tetromino fits and doesn’t overlap ...
// ... updates the values of the local variables accordingly ...
if (canBePlaced) {
tetrominoLocations.put(topLeft, new Tetromino(shape, rotation));
}
return canBePlaced;
}
12)
13)
14)
16)
// ... other operations ...
}
15)

0 degrees 90 degrees 180 degrees 270 degrees
I
O
T
J
L
S
Z

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