mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-12-28 12:53:59 +00:00
133 lines
No EOL
2.5 KiB
Markdown
133 lines
No EOL
2.5 KiB
Markdown
---
|
|
title: "PSoft Recitation 7: Exam Review"
|
|
date: 2022-02-22T21:28:34-04:00
|
|
draft: false
|
|
---
|
|
|
|
## Question 1: Forward Reasoning
|
|
|
|
Fill in the blanks using forward reasoning. Don't forget to:
|
|
|
|
- Carry your variables forward
|
|
- Show your work
|
|
- Simplify expressions
|
|
- State the strongest postcondition
|
|
|
|
```java
|
|
{x > 1}
|
|
y = x;
|
|
{ }
|
|
x = x + 5;
|
|
{ }
|
|
y = 2 * y;
|
|
{ }
|
|
if (x > 12) {
|
|
y = -x;
|
|
{ }
|
|
} else {
|
|
y = -6 * y;
|
|
{ }
|
|
}
|
|
{ }
|
|
```
|
|
|
|
## Question 2: Reasoning about loops
|
|
|
|
Consider the following Dafny code:
|
|
|
|
```csharp
|
|
method until_parity(y: int) returns (index: int)
|
|
requires y < 0
|
|
ensures index == (1 - y) / 2 || index == (-y / 2)
|
|
{
|
|
var p := y;
|
|
index := 0;
|
|
while (p != 0 && p != 1)
|
|
decreases -p
|
|
invariant y <= p <= 1
|
|
invariant index == (p - y) / 2
|
|
{
|
|
p := p + 2;
|
|
index := index + 1;
|
|
}
|
|
}
|
|
```
|
|
|
|
### Q2.1: Loop Invariants
|
|
|
|
Prove that `index == (p - y) / 2` using induction
|
|
|
|
### Q2.2 Postcondition Verification
|
|
|
|
Show that the postcondition is provable from the loop invariant and loop condition.
|
|
|
|
### Q2.2 (Bonus) Decrementing Function
|
|
|
|
Prove that `-p` is the decrementing function.
|
|
|
|
## Question 3: Dafny Invariants
|
|
|
|
What is the missing invariant to make this code verify in Dafny?
|
|
|
|
```csharp
|
|
method copy(in_arr: array<int>) returns (out_arr: array<int>)
|
|
ensures in_arr.Length == out_arr.Length
|
|
ensures forall j :: 0 <= j < in_arr.Length ==> in_arr[j] == out_arr[j]
|
|
{
|
|
out_arr := new int[in_arr.Length];
|
|
var i := 0;
|
|
while i < in_arr.Length
|
|
invariant 0 <= i <= in_arr.Length
|
|
// INVARIANT MISSING HERE
|
|
{
|
|
out_arr[i] := in_arr[i];
|
|
i := i + 1;
|
|
}
|
|
}
|
|
```
|
|
|
|
## Question 4: Backwards Reasoning
|
|
|
|
Fill in the blanks using backward reasoning. Don't forget to:
|
|
|
|
- Show your work
|
|
- Simplify expressions
|
|
- State the weakest precondition
|
|
|
|
```java
|
|
{ }
|
|
w = 2 * w;
|
|
{ }
|
|
z = -w;
|
|
{ }
|
|
y = v + 1;
|
|
{ }
|
|
x = min(y, z);
|
|
{ x < 0 }
|
|
```
|
|
|
|
|
|
|
|
## Question 5: Hoare Triple Validity
|
|
|
|
Assume the following are true:
|
|
|
|
```
|
|
{b} code {y}
|
|
a -> b
|
|
b -> c
|
|
x -> y
|
|
y -> z
|
|
```
|
|
|
|
For the following Hoare triples state whether or not they are valid.
|
|
|
|
If valid, why? If not valid, provide counterexample.
|
|
|
|
*Hint: Recall Liskov Principle of Substitutability*
|
|
|
|
**Q5.1:** Is `{a} code {y}` valid?
|
|
|
|
**Q5.2:** Is `{b} code {x}` valid?
|
|
|
|
**Q5.3:** Is `{b} code {z}` valid? |