50.530 Exercise Solutions

2014 Fall Term

Week 2: Exercise 1public static Boolean repOK(Stack mystack) {if (mystack.capacity() < 0) {return false;}

if (mystack.empty() != mystack.elements().hasMoreElements()) {return false;}

if (mystack.isEmpty() && mystack.size() != 0) {return false;}

//...

return true;}

Week 2: Exercise 2

• Valid test case:1. OrdSet newobj = new OrdSet(5);2. OrdSet newobj = new OrdSet(5); newobj. add(3);

• Redundant test case:OrdSet newobj = new OrdSet(5); newobj.toString();

• Illegal test case: OrdSet newobj = new OrdSet(-1);

Week 3: Exercise 1

DD+({1..8}, {}, 2)

DD+({1..8}, {}, 4)

try again

DD+({1..4,7,8}, {5,6}, 8)

test({1..4}) =? = test({5...8})

test({5,6}) = PASS

try again

test({5,6,8}) = FAIL

DD+({8}, {5,6}, 2)

found in {8}

return {8};

Week 3: Exercise 2

Use DDmin algorithm to minimize the input• Cut the input in a half and test on the first half• It fails and thus apply DDmin again and repeat.Once the minimum input is reached, identify the bug by tracing through the code based on the minimum test case.

Week 5: Exercise 1

• Using Set Union: bug is at line 7 (since only line 7 is unique to the failed test case).

• Using Set Intersection: the result is {} and thus no predication.

Week 5: Exercise 2

• Using Set Union: the result is {} and thus no predication.

• Using Set Intersection: the result is {} and thus no predication.

Week 5: Exercise 3

• The nearest neighbor would be the second last test case and the predicate would be: line 7.

Week 5: Exercise 4

• The nearest neighbor would be the first test case and the predicate would be: none.

Week 5: Exercise 5

Week 5: Exercise 6

• To apply the method A/B/C/D, you need to instruct the program to monitor its execution (one for each block of code without any branches). • None of the methods would locate the bug (for instance, method E

would not work as the buggy line is not updating any variable) except perhaps method D (which depends on the test cases you generate).

Week 6: Exercise 1

• The slice of the assertion includes all the statements.

Week 8: Exercise 1

• Answer 1: No, there is no race condition according to the happens-before approach.• Answer 2: Yes, there is a race condition on variable count (the first

and last line).

Week 8: Exercise 2

• The lockset algorithm finds the race condition at line 1 since count is a shared variable and it is accessed without any lock held (so that the intersection is {}).

Week 8: Exercise 3

• For variable count, after the first count++, it moves from state Virgin to state Exclusive• After the second count++, it moves from state Exclusive to Shared-

Modified state since it is accessed by a new thread with write privilege. • The checking then happens and since there is no lock held, a race

condition is reported.

Week 8: Exercise 4

• The report would be there is no race.

Week 8: Exercise 5

• Lockset-based algorithm• Race on x: line 10 and 1• Race on z: line 7 and 5

• Happens-before filtering• No race on Z since there is happens-before from unlock(L) to lock(L)

Week 8: Exercise 6

• Assume that thread 1 is scheduled first, after executing line 1, it is postponed; • Thread 2 is scheduled to run line 7 and then is blocked at line 8.• Thread 1 moves on and thus no race is reported.

Week 9: Exercise 1

(3*y + z)*y - z > 0

Week 9: Exercise 2

5 * y - z > 0

Week 9: Exercise 3

We get the following Hoare triple first.{K>0&&M>0&&k=K&&m=M}while(k!=m) {

if (k > m) {k = k-m;} else {m = m-k;}

}{k=GCD(K,M)}We discharge it using the loop invariant: GCD(K, M) = gcd(k,m)

Week 9: Exercise 4

First row from left to right: yes, yes, noSecond row from left to right: yes, yes, yes

Week 9: Exercise 5

For the inner loop, the ranking function is x-y and termination can be proved by establishing the following Hoare triple.

{x-y=V&&y<x&&y>0} y = 2*y {x-y < V} For the outer loop, the ranking function is x and termination can be proved by establishing the following Hoare triple

{x>=0&&x=V} y := 1;while (y < x) { y = 2*y; }x = x – 1;{x < V}

which can be proved using a loop invariant (i.e., x is unchanged) for the inner loop.

Week 10: Exercise 1

(Monday or Wednesday or Thursday)and(!Wednesday)and(!Friday)and(!Tuesday and !Thursday)

SAT solving would lead to a solution: Monday = true; the rest = false;

Week 10: Exercise 2

The assertion can be demonstrated possible by symbolically executing the following path. !a && b && !a && c && y = 1 && z = 2 && !(x+y+z!=3)The solution is: a = false; b = true; c = true

Week 10: Exercise 3

Example: x+y<=4

Week 10: Exercise 4

Symbolic execution the left most path, apply interpolantion and get y>=1 at node 3. Skip the last alternative; backtrack …

2

3 3

4 4 4

2

3 3

4 4 4

1x = x+1

[y>=1]x = x+2

[y<1]x = x+4

*

y<1

y>=1

4 4

[y>=1]x = x+2 y<1

[y<1]x = x+4 [y<1]x = x+4 [y<1]x = x+4y>=1 y>=1 y>=1

Week 10: Exercise 5

The only exception that could rise at array[x]=2 when x < 0. The following proves that if that statement is never reached, x >= 0. Base case: assume input is false, the following is true: {x>=0}rec(){x>=0} which is basically {x>=0}x=x+1{x>=0} (*).Induction step: assume that {x>=0}rec(){x>=0}.

{x>=0}rec(){x>=0} implies {x>=0}rec(); rec();{x>=0} (**).{x>=0} method-body {x>=0} since (*) and (**){x>=0} rec() {x>=0}

Week 11: Exercise 1

We can prove that the error is not occurring using a Hoare triple as below: {x=0.1&&y=0}loop-body{y>=0}, which can be proved using the following loop invariant: y>=0.

Week 11: Exercise 2

Sample: y=1, orig(x) > 0, x > y, return = *x, return = orig(*x) + y

Week 11: Exercise 3

The same tree (the proof) will be built.

L0

L1

L2

L3

new!=old

L4L5

new=old+1

L1

L6 L2

L5

L1subsumed by new!=old

Week 12: Exercise 1

Week 12: Exercise 2

1: if (*) { 3: got_lock = 0;4: if (*) {5: lock();6: got_lock ++;7: }8: if (got_lock) {9: unlock();10: }11: if (*) {goto 3;}12: }14: lock();15: old = new;16: if (*) {17: unlock();18: new ++;19: }20: if (new != old) {goto 14;}21: unlock ();

1: if (*) { 3: skip;4: if (*) {5: locked=true6: skip;7: }8: if (*) {9: locked=false;10: }11: if (*) {goto 3;}12: }14: locked=true;15: skip;16: if (*) {17: locked=false;18: skip;19: }20: if (*) {goto 14;}21: locked=true;

Week 12: Exercise 3

r

gy

Week 12: Exercise 4

2

3

4

5

6

7

8

{!locked, ne}

{locked, ne}

{locked, !ne}

{!locked, !ne}

{!locked, ne}

{!locked, ne} 8{locked, !ne}

{locked, !ne}