# ~/PSoft Recitation 7: Exam Review

## Brandon Rozek PhD Student @ RPI studying Automated Reasoning in AI and Linux Enthusiast.

Updated on

## Question 1: Forward Reasoning

Fill in the blanks using forward reasoning. Don’t forget to:

• Simplify expressions
• State the strongest postcondition
``````{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:

``````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?

``````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:

• Simplify expressions
• State the weakest precondition
``````{                 }
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?