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In artificial intelligence and computer science, a Search Problem  refers to a type of problem-solving task that involves finding a sequence of actions or steps to reach a goal state from an initial state. Search problems are commonly encountered in various domains, and they are often addressed using search algorithms. Search problems are fundamental in the study of artificial intelligence and are applicable in various domains, including robotics, planning, game playing, and route planning, among others.

The key components of a search problem include: 

     i) State Space: 

The set of all possible states that the system can be in. Each state represents a configuration or situation of the system. 

     ii) Initial State: 

The starting point of the problem, indicating where the search process begins. 

     iii) Actions: 

The set of possible actions or operations that can be performed to transition from one state to another. 

     iv) Transition Model: 

Describes the result of applying a specific action in a particular state, indicating how the system transitions from one state to another. 

     v) Goal Test: 

A criterion or condition to determine whether a given state is a goal state. The goal state represents the desired outcome or solution to the problem. 

     vi) Path Cost: 

The numerical cost associated with a sequence of actions or path from the initial state to a goal state. 

Search algorithms, such as depth-first search, breadth-first search, A* search, and others, are designed to explore the state space systematically or intelligently to find a solution to the search problem. The efficiency and optimality of these algorithms can vary based on factors like the structure of the state space, the complexity of the actions, and the goal test criteria.

The approach of search algorithm commonly used in artificial intelligence, especially in problem-solving,  pathfinding, puzzle-solving, and navigating state spaces to find optimal solutions is often associated with algorithms like breadth-first search or depth-first search. The basic steps of this approach are: 

     i) Start with a frontier that contains an initial state. 

This is the starting point of the search. The frontier typically represents the set of states or nodes that the algorithm has yet to explore. 

     ii) Start with an empty explored set. 

An explored set is a common step to search algorithms to avoid unnecessary re-exploration of nodes and prevent infinite loops in certain cases. The explored set keeps track of the nodes that have already been visited. 

     iii) Repeat: 

        iii.a) If the frontier is empty, then no solution. 

This check ensures that the algorithm terminates if it exhaustively explores all possible paths without finding a solution. If the frontier is empty and no solution has been found, the algorithm concludes that there is no solution. 

        iii.b) Remove a node from the frontier. 

This is part of the exploration process. The algorithm selects a node from the frontier for further examination.  A stack, which is a last-in, first-out (LIFO) data structure, can be used in this step. That is, the last element added to the stack is the first one to be removed. In the search algorithms, using a stack for the frontier typically provides a depth-first search (DFS) strategy. In DFS, the algorithm explores as far as possible along each branch before backtracking. The stack naturally supports this behavior because the last node added to the frontier is the one that will be explored next. In this case, the frontier is treated as a stack.

        iii.c) If the node contains the goal state, return the solution. 

This step checks whether the current node represents a solution to the problem. If it does, the algorithm returns the solution and terminates. 

        iii.d) Expand the node, add resulting nodes to the frontier if they are not in the explored set.

This is the process of generating new nodes by applying possible actions or transitions from the current node. The new nodes are added to the frontier for future exploration. 

Figure 3649a shows a pathfinding from point 1 to point 5. In the script, the initial frontier is represented by a list where each element is a path. The frontier list is initialized with a single path containing only the start node, namely "frontier = [[start]]" in the script, which initializes the frontier list with a single path [start], where start is the starting node. The find_path function then uses this initial frontier to explore and expand paths until it finds a path from the start node to the goal node or exhausts all possibilities. 

The lines responsible for repeating the process in the find_path function are within the while loop. The loop continues until the frontier is empty or a path from the start node to the goal node is found:

        while frontier: This line initiates a loop that continues as long as the frontier list is not empty. 

        path = frontier.pop(0): This line removes and retrieves the first path from the frontier list. This path represents the current exploration path. 

        current_node = path[-1]: This line gets the last node in the current path, representing the current state of the search. 

       if current_node == goal:: This condition checks if the current node is the goal node. If true, it means a path from the start node to the goal node has been found, and the function returns the path. 

      The subsequent lines handle the expansion of the current path. It iterates over the neighbors of the current node, creates new paths by appending each neighbor to the current path, and adds these new paths to the frontier for further exploration. 

During the search process, when the algorithm has expanded the initial frontier and generated paths that include both nodes "3"and "4", those paths will be present in the frontier (the frontier is a list where each element is a path). The point at which both "3" and "4" are in the frontier depends on the order in which the paths are explored. When the algorithm explores the paths that include nodes "3" and "4", it will create new paths by appending "3" and "4" to the existing paths in the frontier. The order in which these paths are added to the frontier determines when both "3" and "4" will be present. For example, if the algorithm explores the path containing node "3" before the path containing node "4", then both "3" and "4" will be in the frontier simultaneously at some point during the exploration.

In the script, the explored set is used to keep track of the nodes that have been visited. The if neighbor not in explored condition ensures that a node is only added to the frontier if it hasn't been explored before. 

Figure 3649a. Pathfinding from point 1 to point 5 (code). 

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