problem solving state space search

State Space Search in Artificial Intelligence

State space search is a problem-solving technique used in Artificial Intelligence (AI) to find the solution path from the initial state to the goal state by exploring the various states. The state space search approach searches through all possible states of a problem to find a solution. It is an essential part of Artificial Intelligence and is used in various applications, from game-playing algorithms to natural language processing.

Introduction

A state space is a way to mathematically represent a problem by defining all the possible states in which the problem can be. This is used in search algorithms to represent the initial state, goal state, and current state of the problem. Each state in the state space is represented using a set of variables.

The efficiency of the search algorithm greatly depends on the size of the state space, and it is important to choose an appropriate representation and search strategy to search the state space efficiently.

One of the most well-known state space search algorithms is the A algorithm. Other commonly used state space search algorithms include breadth-first search (BFS) , depth-first search (DFS) , hill climbing , simulated annealing , and genetic algorithms .

Features of State Space Search

State space search has several features that make it an effective problem-solving technique in Artificial Intelligence. These features include:

Exhaustiveness: State space search explores all possible states of a problem to find a solution.

Completeness: If a solution exists, state space search will find it.

Optimality: Searching through a state space results in an optimal solution.

Uninformed and Informed Search: State space search in artificial intelligence can be classified as uninformed if it provides additional information about the problem.

In contrast, informed search uses additional information, such as heuristics, to guide the search process.

Steps in State Space Search

The steps involved in state space search are as follows:

• To begin the search process, we set the current state to the initial state.
• We then check if the current state is the goal state. If it is, we terminate the algorithm and return the result.
• If the current state is not the goal state, we generate the set of possible successor states that can be reached from the current state.
• For each successor state, we check if it has already been visited. If it has, we skip it, else we add it to the queue of states to be visited.
• Next, we set the next state in the queue as the current state and check if it's the goal state. If it is, we return the result. If not, we repeat the previous step until we find the goal state or explore all the states.
• If all possible states have been explored and the goal state still needs to be found, we return with no solution.

State Space Representation

State space Representation involves defining an INITIAL STATE and a GOAL STATE and then determining a sequence of actions, called states, to follow.

• State: A state can be an Initial State, a Goal State, or any other possible state that can be generated by applying rules between them.
• Space: In an AI problem, space refers to the exhaustive collection of all conceivable states.
• Search: This technique moves from the beginning state to the desired state by applying good rules while traversing the space of all possible states.
• Search Tree: To visualize the search issue, a search tree is used, which is a tree-like structure that represents the problem. The initial state is represented by the root node of the search tree, which is the starting point of the tree.
• Transition Model: This describes what each action does, while Path Cost assigns a cost value to each path, an activity sequence that connects the beginning node to the end node. The optimal option has the lowest cost among all alternatives.

Example of State Space Search

The 8-puzzle problem is a commonly used example of a state space search. It is a sliding puzzle game consisting of 8 numbered tiles arranged in a 3x3 grid and one blank space. The game aims to rearrange the tiles from their initial state to a final goal state by sliding them into the blank space.

To represent the state space in this problem, we use the nine tiles in the puzzle and their respective positions in the grid. Each state in the state space is represented by a 3x3 array with values ranging from 1 to 8 , and the blank space is represented as an empty tile.

The initial state of the puzzle represents the starting configuration of the tiles, while the goal state represents the desired configuration. Search algorithms utilize the state space to find a sequence of moves that will transform the initial state into the goal state.

This algorithm guarantees a solution but can become very slow for larger state spaces. Alternatively, other algorithms, such as A search , use heuristics to guide the search more efficiently.

Our objective is to move from the current state to the target state by sliding the numbered tiles through the blank space. Let's look closer at reaching the target state from the current state.

To summarize, our approach involved exhaustively exploring all reachable states from the current state and checking if any of these states matched the target state.

Applications of State Space Search

• State space search algorithms are used in various fields, such as robotics, game playing, computer networks, operations research, bioinformatics, cryptography, and supply chain management. In artificial intelligence, state space search algorithms can solve problems like pathfinding , planning , and scheduling .
• They are also useful in planning robot motion and finding the best sequence of actions to achieve a goal. In games, state space search algorithms can help determine the best move for a player given a particular game state.
• State space search algorithms can optimize routing and resource allocation in computer networks and operations research.
• In Bioinformatics , state space search algorithms can help find patterns in biological data and predict protein structures.
• In Cryptography , state space search algorithms are used to break codes and find cryptographic keys.
• State Space Search is a problem-solving technique used in AI to find a solution to a problem by exploring all possible states of the problem.
• It is an exhaustive, complete, and optimal search process that can be classified as uninformed or informed.
• State Space Representation is a crucial step in the state space search process as it determines the efficiency of the search algorithm.
• State space search in artificial intelligence has several applications, including game-playing algorithms, natural language processing, robotics, planning, scheduling, and computer vision.

State Space Search

State Space Representation in Artificial Intelligence

Problem statement: explain state splace reseach in ai with example.

To find solution to any problem the foremost condition is that it has to be precisely defined or represented. By defining it precisely means to present an abstract problem in real workable states that are really understood. These states are then operated by some set of operators and finally the solution is obtained. The decision of which operator is to be applied is taken by the control strategy used. There are two ways in which the AI problem can be represented.

State Space Representation

• Problem Reduction

State Space Representation consist of defining an INITIAL State (from where to start), the GOAL State (The destination) and then we follow certain set of sequence of steps (called States). Let’s define each of them separately.

State : AI problem can be represented as a well formed set of possible states. State can be Initial State i.e. starting point, Goal State i.e. destination point and various other possible states between them which are formed by applying certain set of rules.

Space : In an AI problem the exhaustive set of all possible states is called space.

Search : In simple words search is a technique which takes the initial state to goal state by applying certain set of valid rules while moving through space of all possible states.

So we can say that to do a search process we need the following.

• Initial State
• Set of valid Rules

So, a set of all possible states for a given problem is known as state space representation of the problem.

For example:    In chess game: The initial position of all the pieces on a chess board defines the initial state. The rules of playing chess defines the set of legal rules and Goal state is defined by any possible board position corresponding to checkmate or a draw state. The point to be noted here is that there can be more than one Goal state possible.

State Space Representation of Tic Tac Toe game is shown is the following figure. Starting from the initial state as we move on applying rules of putting X (cross) or O (Zero) we keep on generating the states, hence the set of states all such generated states is called space, unless we reach one of the goal states that can be a win situation or a draw situation. The new state is generated from the earlier one is by applying the a control strategy.

State Space Representation of Tic Tac Toe Game

State space representation are very advantageous in AI problems as the whole state space is given it becomes easy to find the solution path that leads from initial state to goal state. The basic job is to create such algorithms which can search through the problem space and find out the best solution path. Later chapters will discuss about these search procedures and control strategy to move through the space.

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Search Algorithms in AI

Artificial Intelligence is the study of building agents that act rationally. Most of the time, these agents perform some kind of search algorithm in the background in order to achieve their tasks. 

• A State Space. Set of all possible states where you can be.
• A Start State. The state from where the search begins.
• A Goal State. A function that looks at the current state returns whether or not it is the goal state.
• The Solution to a search problem is a sequence of actions, called the plan that transforms the start state to the goal state.
• This plan is achieved through search algorithms.

Types of search algorithms: 

There are far too many powerful search algorithms out there to fit in a single article. Instead, this article will discuss six of the fundamental search algorithms, divided into two categories, as shown below. 

Note that there is much more to search algorithms than the chart I have provided above. However, this article will mostly stick to the above chart, exploring the algorithms given there. 

Uninformed Search Algorithms: 

The search algorithms in this section have no additional information on the goal node other than the one provided in the problem definition. The plans to reach the goal state from the start state differ only by the order and/or length of actions. Uninformed search is also called Blind search . These algorithms can only generate the successors and differentiate between the goal state and non goal state.  The following uninformed search algorithms are discussed in this section.

• Depth First Search
• Breadth First Search
• Uniform Cost Search

Each of these algorithms will have: 

• A problem graph, containing the start node S and the goal node G.
• A strategy, describing the manner in which the graph will be traversed to get to G.
• A fringe, which is a data structure used to store all the possible states (nodes) that you can go from the current states.
• A tree, that results while traversing to the goal node.
• A solution plan, which the sequence of nodes from S to G.

Depth First Search :

Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. It uses last in- first-out strategy and hence it is implemented using a stack. Example: 

Question. Which solution would DFS find to move from node S to node G if run on the graph below?   

Solution. The equivalent search tree for the above graph is as follows. As DFS traverses the tree “deepest node first”, it would always pick the deeper branch until it reaches the solution (or it runs out of nodes, and goes to the next branch). The traversal is shown in blue arrows. 

Path:   S -> A -> B -> C -> G 

= the depth of the search tree = the number of levels of the search tree.  = number of nodes in level  .  Time complexity: Equivalent to the number of nodes traversed in DFS.  Space complexity: Equivalent to how large can the fringe get.  Completeness: DFS is complete if the search tree is finite, meaning for a given finite search tree, DFS will come up with a solution if it exists.  Optimality: DFS is not optimal, meaning the number of steps in reaching the solution, or the cost spent in reaching it is high. 

Breadth First Search :

Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a ‘search key’), and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level. It is implemented using a queue. Example:  Question. Which solution would BFS find to move from node S to node G if run on the graph below? 

Solution. The equivalent search tree for the above graph is as follows. As BFS traverses the tree “shallowest node first”, it would always pick the shallower branch until it reaches the solution (or it runs out of nodes, and goes to the next branch). The traversal is shown in blue arrows. 

Path: S -> D -> G 

  = the depth of the shallowest solution.  = number of nodes in level  .  Time complexity: Equivalent to the number of nodes traversed in BFS until the shallowest solution.  Space complexity: Equivalent to how large can the fringe get.  Completeness: BFS is complete, meaning for a given search tree, BFS will come up with a solution if it exists.  Optimality: BFS is optimal as long as the costs of all edges are equal. 

Uniform Cost Search: 

UCS is different from BFS and DFS because here the costs come into play. In other words, traversing via different edges might not have the same cost. The goal is to find a path where the cumulative sum of costs is the least.  Cost of a node is defined as: 

Example:  Question. Which solution would UCS find to move from node S to node G if run on the graph below? 

Solution. The equivalent search tree for the above graph is as follows. The cost of each node is the cumulative cost of reaching that node from the root. Based on the UCS strategy, the path with the least cumulative cost is chosen. Note that due to the many options in the fringe, the algorithm explores most of them so long as their cost is low, and discards them when a lower-cost path is found; these discarded traversals are not shown below. The actual traversal is shown in blue. 

Path: S -> A -> B -> G  Cost: 5  Let  = cost of solution.  = arcs cost.  Then  effective depth  Time complexity:    , Space complexity:   

Advantages: 

• UCS is complete only if states are finite and there should be no loop with zero weight.
• UCS is optimal only if there is no negative cost.

Disadvantages: 

• Explores options in every “direction”.
• No information on goal location.

Informed Search Algorithms: 

Here, the algorithms have information on the goal state, which helps in more efficient searching. This information is obtained by something called a heuristic.   In this section, we will discuss the following search algorithms. 

• Greedy Search
• A* Tree Search
• A* Graph Search

Search Heuristics: In an informed search, a heuristic is a function that estimates how close a state is to the goal state. For example – Manhattan distance, Euclidean distance, etc. (Lesser the distance, closer the goal.) Different heuristics are used in different informed algorithms discussed below. 

Greedy Search:

In greedy search, we expand the node closest to the goal node. The “closeness” is estimated by a heuristic h(x).  Heuristic: A heuristic h is defined as-  h(x) = Estimate of distance of node x from the goal node.  Lower the value of h(x), closer is the node from the goal.  Strategy: Expand the node closest to the goal state, i.e. expand the node with a lower h value.  Example: 

Question. Find the path from S to G using greedy search. The heuristic values h of each node below the name of the node.   

Solution. Starting from S, we can traverse to A(h=9) or D(h=5). We choose D, as it has the lower heuristic cost. Now from D, we can move to B(h=4) or E(h=3). We choose E with a lower heuristic cost. Finally, from E, we go to G(h=0). This entire traversal is shown in the search tree below, in blue. 

Path:   S -> D -> E -> G 

Advantage: Works well with informed search problems, with fewer steps to reach a goal.  Disadvantage: Can turn into unguided DFS in the worst case.   

A* Tree Search:

• Here, h(x) is called the forward cost and is an estimate of the distance of the current node from the goal node.
• And, g(x) is called the backward cost and is the cumulative cost of a node from the root node.
• A* search is optimal only when for all nodes, the forward cost for a node h(x) underestimates the actual cost h*(x) to reach the goal. This property of A* heuristic is called admissibility . 

Strategy: Choose the node with the lowest f(x) value.  Example:  

Question. Find the path to reach from S to G using A* search. 

Solution. Starting from S, the algorithm computes g(x) + h(x) for all nodes in the fringe at each step, choosing the node with the lowest sum. The entire work is shown in the table below.  Note that in the fourth set of iterations, we get two paths with equal summed cost f(x), so we expand them both in the next set. The path with a lower cost on further expansion is the chosen path.   

A* Graph Search :

• A* tree search works well, except that it takes time re-exploring the branches it has already explored. In other words, if the same node has expanded twice in different branches of the search tree, A* search might explore both of those branches, thus wasting time
• A* Graph Search, or simply Graph Search, removes this limitation by adding this rule: do not expand the same node more than once. 
• Heuristic. Graph search is optimal only when the forward cost between two successive nodes A and B, given by h(A) – h (B), is less than or equal to the backward cost between those two nodes g(A -> B). This property of the graph search heuristic is called consistency . 

Question. Use graph searches to find paths from S to G in the following graph. 

the Solution. We solve this question pretty much the same way we solved last question, but in this case, we keep a track of nodes explored so that we don’t re-explore them. 

Path:   S -> D -> B -> E -> G  Cost:   7 

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