Hello, dear student! In today's lesson, we will explore the concept of Stacks in programming, specifically using PHP. Stacks are key data structures employed in various applications like memory management and algorithm backtracking. Our aim for this session is to understand what Stacks are, learn how to implement and manipulate them in PHP, and explore their complexities. Let's dive in!

First, let's grasp what a Stack is. Picture a stack of boxes that you can only access from the top. That's essentially a Stack: a Last-In, First-Out (LIFO) structure. The principal operations are Push (adding an element to the top of the stack), Pop (removing the topmost element), and Peek (viewing the topmost element without removing it).

In PHP, Stacks can be implemented using arrays due to their dynamic nature. Let’s explore how to create a Stack using an array in PHP:

Here, top represents the position of the current top-most element in the stackArray, initialized to -1 to indicate an empty state.

Let's examine the Push operation, which adds a new element to the top of the Stack.

Before adding an element, the method checks if the stack is full by comparing top with size - 1. If space is available, the element is added; otherwise, a "Stack Overflow" message is displayed. The ++$this->top operation increments the top index before assigning the data.

The Pop operation removes and returns the topmost element from the Stack.

This method checks if the stack is empty by verifying if top is greater than -1. If not empty, it decreases top and returns the element at the top. The top index handles the logical removal by no longer pointing to the removed element.

Lastly, the Peek operation retrieves the topmost element without removing it from the Stack.

The Push, Pop, and Peek operations are performed at one end of the data structure, thus taking constant time, represented as O(1). The space complexity is proportional to the number of elements, represented as O(n).

Congratulations, you've completed the lesson on Stacks! You now understand the basic concept of Stacks, how to implement and use them in PHP, and analyze their complexity. Up next, you're invited to engage in practice exercises to apply these ideas and enhance your problem-solving skills. Get ready to put your knowledge to the test!