State Space Representation in Artificial Intelligence
- 1. State Space Representation of Problems -Dr. Mehak Saini
- 9. Figure: A tiny search problem. The figure on the left shows the MoveGen function in the format Node → (List of neighbours). The figure on the right shows the corresponding state space which is implicit and not given upfront. The algorithm itself works with the MoveGen function and also the GoalTest function. Observe that all moves are bidirectional. The tiny search problem has seven nodes, including the start node S, the goal node G, and five other nodes named A, B, C, D, and E. Without any loss of generality, let us assume that the nodes are states in a state space. The algorithms apply to the solution space as well.
- 10. The search space that an algorithm explores is implicit. It is generated on the fly by the MoveGen function, as described in Algorithm The candidates generated are added to what is traditionally called OPEN, from where they are picked one by one for inspection. The candidate that the search algorithm picks from OPEN determines its behaviour Algorithm SimpleSearch picks node N from the head of OPEN. If N is the goal, it terminates with N, else it calls MoveGen and adds the neighbours of N to OPEN in some manner. The way this is done determines the behaviour of the search algorithm.
- 11. The search tree depicts the space as viewed by a given search algorithm. Each variation of the algorithm generates its own search tree. Given that we have decided to extract the next candidate from the head of the OPEN list, we need to decide how to add the new nodes to OPEN.
- 12. State space Search space (or search tree) - State space refers to the set of all possible configurations or states that a problem can take. It encompasses all the possible scenarios that can be reached from the initial state to the goal state. - Search space or search tree is a representation of the paths taken during the search process. It outlines the various states explored, as well as the decisions made during the search. It can be considered a subset of the state space, as not all states are explored in a search process.
- 14. DFS v/s BFS
- 15. Parameter BFS (Breadth First Search) DFS (Depth First Search) Data structure used Queue (FIFO) Stack (LIFO) Compeleteness Complete (if the branching factor is finite) Incomplete (may get stuck in infinite paths) Quality of solution Guaranteed to find the optimal solution (shortest path in unweighted graphs) May not find the optimal solution (depends on path exploration) Space Complexity O(b^d) (where b is the branching factor and d is the depth of the shallowest solution) O(bm) (where b is the branching factor and m is the maximum depth of the search space) Time Complexity O(b^d) O(b^m)
- 16. Which works best, where? DFS is more efficient when the solution is far from the root, and BFS is better when finding the shortest path matters.
- 17. (i) Node (List of Neighbors) (ii) State space diagram
- 18. DFS Algorithm DFS picks node N from the head of OPEN. If N is the goal it terminates with N, else it calls MoveGen and concatenates the list of neighbours of N with OPEN.
- 19. (i) Search Tree of DFS (ii)OPEN -> Stack Algorithm DFS dives headlong into the first branch it sees. The fact that it does so on the leftmost branch is only because the first node in the list returned by MoveGen always ends up in at the head of OPEN. OPEN behaves like a stack with the LIFO property. The behaviour of depth first search can thus be seen to embody the newest nodes first strategy. In the search tree that translates to deepest nodes first, giving the algorithm its name. As can be seen in the Figure, the search oscillates between nodes S and A. This keeps on happenning until the machine crashes. For the 8-puzzle this would mean moving the same tile back and forth. This happens because even though the state space is finite, the search tree is infinite.
- 20. How to stop going around in circles? The problem faced by a search algorithm is similar to a person trying to solve an actual maze (or Labyrinth, which is a confusing structure in which one finds it difficult to reach the end as one keeps on going in circles. Here, the person doesn’t have a map or top-view of the area. There is lack of a global perspective of the kind humans have when, for example, poring over a city map. However, when on the ground at a crossroads, one often does not have a sense of direction looking only at the immediate neighbourhood). This happens because the algorithm has access only to the immediate neighbours, and needs to somehow find a path to the goal. There is lack of a global perspective here. Solution: Maintain another list apart from OPEN, that stores the nodes already inspected. This list is traditionally called CLOSED.
- 21. When nodes already on CLOSED are not added to OPEN, the search space accessible to DFS shrinks dramatically to the subtree with solid edges. It finds the path to the goal as shown. Note that the search tree depends on the left to right order of depth first search, which is why it does not go beyond the nodes D, B, and C on the right. All their neighbours would already be on CLOSED.
- 22. Search tree for the given state space diagram.for DFS
- 23. BFS
- 24. Search tree for the given state space diagram for BFS
- 34. Advantages of state space representation It allows for a formal definition of a problem as the need to convert some given situation into some desired situation using a set of permissible operations. It permits us to define the process of solving a particular problem as a combination of known techniques and search. Search is the general technique of exploring the state space to try to find some path from the current state to a goal state. Search is a very important process in the solution of hard problems for which no direct techniques are available.
- 35. Limitations of state space representation It is not possible to visualize all the states for a given problem. The resources of the computer system are limited to handle the huge state space representations. It doesn’t provide the order of the search that can give a solution to the problem.
- 36. Production systems and control strategies Production system is a model of computation used for automated problem solving and rule-based systems in AI. It consists of: 1. Set of production rules: These are condition-action pairs of the form "If condition, then action." Each rule represents a way to transform one state into another. 2. Working memory: This contains the current state of the problem, represented by facts or symbols. 3. Control strategy: The mechanism for selecting which rule to apply at any given time. Control Strategies: These determine how to choose and apply production rules. Common strategies include: - Forward chaining: Start with the initial state and apply rules to reach the goal state. This is data-driven. - Backward chaining: Start with the goal state and work backward by applying rules to reach the initial state. This is goal-driven. - Conflict resolution: If multiple rules are applicable, a strategy (e.g., priority, specificity) is used to decide which rule to apply next. The combination of production rules, working memory, and control strategies defines how a system moves through its state space to find a solution to a given problem.
- 37. Production systems Production systems provide structures that facilitate describing and performing the search process. A production system consists of :- A set of rules One or more Knowledge/databases A control strategy A rule applier
- 38. Production system rules Forward rules: The left side determine the applicability of the rule and the right side describes the operation to be performed, if the rule is applied. Backward rules: The right side determine the applicability of the rule and the left side describes the operation to be performed, if the rule is applied.
- 39. Tasks of a Control Strategy It specifies the order in which the rules will be compared to the DB. It decides which rule to apply next during the process of searching for a solution to a problem. It provides a way of resolving the conflicts that arise when several rules match at once.
- 40. Requirements of a good control strategy • It causes motion : In water Jug problem, if we apply a simple control strategy of starting each time at the top of the list of rules and choosing the first applicable and then we will never move towards solution. • It is systematic: If we choose another control strategy saying that choose at random from among the applicable rules then definitely it cause motion and eventually will lead to a solution. But this control strategy is not systematic. We may arrive at the same state several times & thus waste time.
- 41. Forward vs Backward reasoning • Forward Reasoning : Control strategy starts with known fact and works towards a conclusion. • Backward Reasoning: Goal directed control strategy. Begin with final conclusion, draw sub goals, generate further sub goals until all sub goals are satisfied.
- 42. A Systematic control strategy Construct a tree with the initial state as its root. Generate all the offspring of the root by applying each of the applicable rules to the initial state. Now for each leaf node, generate all its successors by applying all the rules that are applicable: Continue this process until some rules produces a goal state. This process is called Breadth First Search