website/content/ta/spring2022/csci2600/recitation07.md

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---
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title: "PSoft Recitation 7: Exam Review"
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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;
{ }
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if (x > 12) {
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y = -x;
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{ }
} 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
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Fill in the blanks using backward reasoning. Don't forget to:
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- Show your work
- Simplify expressions
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- State the weakest precondition
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```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?