As Chapter 4 pointed out, you can define ADT operations in a mathematically formal way by...

As Chapter 4 pointed out, you can define ADT operations in a mathematically formal way by using axioms. For example, the following axioms formally define the ADT stack, wherestack is an arbitrary stack anditem is an arbitrary stack item.

(aStack.createStack()).isEmpty()= true

(aStack.push(item)).isEmpty() =false

(aStack.createStack()) .pop() = error

(aStack.push(item)).pop() = aStack

(aStack.createStackf)) .peek() = error

(aStack.push(item)).peek() =item

You can use these axioms, for example, to prove that the stack defined by the sequence of operations

Create an empty stack

Push a 5

Push a 7

Push a 3

Pop (the 3)

Push a 9

Push a 4

Pop (the 4)

which you can write as

(((((((aStack.createStackf)) .push(5)).push(7)).push(3)). pop()).push(9)).push(4)).pop()

is exaedy the same as the stack defined by the sequence

Create an empty stack

Push a 5

Push a7

Push a 9

which you can write as

(((aStack.createStack()).push(5)).push(7)).push(9) .

Similarly, you can use the axioms to show drat

(((((((aStack.createStack()).push(l)).push(2)).pop()).push(3)).pop())«pop())•isEmpty()

is true.

a. The following representation of a stack as a sequence ofpush operations without anypop operations is called a canonical form:

(...(aStack.createStack()).push()).push()...).push()

Prove that any stack is equal to a stack that is in canonical form.

b. Prove that the canonical form is unique. That is, a stack is equal to exactly one stack that is in canonical form.

c. Use the axioms to show formally that

((((((((((aStack.createStack()).push(6)).push(9)). pop()).pop()).push(2)).pop()).push(3)).push(l)). pop()).peek()

equals 3.