ProblemSolving(L-2).pdf

ProblemSolving(L-2).pdf

• 1.
  Problem Solving, Search andControl Strategies
• 2.
  Problem Solving, Searchand Control Strategies 1. General Problem Solving • Problem solving definitions: • problem space, • problem solving, • state space, • state change, • structure of state space, • problem solution, • problem description; • Examples of problem definition.
• 3.
  Problem Solving, Searchand Control Strategies • Problem definitions: A problem is defined by its elements and their relations. To provide a formal description of a problem, we need to do following: a. Define a state space that contains all the possible configurations of the relevant objects, including some impossibleones. b. Specify one or more states, that describe possible situations, from which the problem-solving process may start. These states are called initial states. c. Specify one or more states that would be acceptable solution to the problem. These states are called goal states. d. Specify a set of rules that describe the actions (operators) available. The problem can then be solved by using the rules, in combination with an appropriate control strategy, to move through the problem space until a path from an initial state to a goal state is found.
• 4.
  Problem Solving, Searchand Control Strategies • Problem definitions: This process is known as search.  Search is fundamental to the problem-solving process.  Search is a general mechanism that can be used when more direct method is not known.  Search provides the framework into which more direct methods for solving subparts of a problem can be embedded. A very large number of AI problems are formulated as search problems.
• 5.
  Problem Solving, Searchand Control Strategies Problem Space • A problem space is represented by directed graph, where nodes represent search state and paths represent the operators applied to change the state. • To simplify a search algorithms, it is often convenient to logically and programmatically represent a problem space as a tree. • A tree usually decreases the complexity of a search at a cost. Here, the cost is due to duplicating some nodes on the tree that were linked numerous times in the graph; e.g., node B and node D shown in example below.
• 6.
  Problem Solving, Searchand Control Strategies Problem Space A tree is a graph in which any two vertices are connected by exactly one path. Alternatively, any connected graph with no cycles is a tree.
• 7.
  Problem Solving, Searchand Control Strategies • States A state is a representation of elements at a given moment. A problem is defined by its elements and their relations. At each instant of a problem, the elements have specific descriptors and relations; the descriptors tell - how to select elements ? Among all possible states, there are two special states called :  Initial state is the start point  Final state is the goal state
• 8.
  Problem Solving, Searchand Control Strategies • State Change: Successor Function A Successor Function is needed for state change. The successor function moves one state to another state. Successor Function : ◊ Is a description of possible actions; a set of operators. ◊ Is a transformation function on a state representation, which converts that state into another state. ◊ Defines a relation of accessibility among states. ◊ Represents the conditions of applicability of a state and corresponding transformation function
• 9.
  Problem Solving, Searchand Control Strategies • State Space A State space is the set of all states reachable from the initial state. Definitions of terms : ◊ A state space forms a graph (or map) in which the nodes are states and the arcs between nodes are actions. ◊ In state space, a path is a sequence of states connected by a sequence of actions. ◊ The solution of a problem is part of the map formed by the state space.
• 10.
  Problem Solving, Searchand Control Strategies • Structure of a State Space The Structures of state space are trees and graphs.  Tree is a hierarchical structure in a graphical form; and  Graph is a non-hierarchical structure. ◊ Tree has only one path to a given node; i.e., a tree has one and only one path from any point to any other point. ◊ Graph consists of a set of nodes (vertices) and a set of edges (arcs). Arcs establish relationships (connections) between the nodes; i.e., a graph has several paths to a given node. ◊ operators are directed arcs between nodes. Search process explores the state space. In the worst case, the search explores all possible paths between the initial state and the goal state.
• 11.
  Problem Solving, Searchand Control Strategies • Problem Solution In the state space, a solution is a path from the initial state to a goal state or sometime just a goal state. ◊ A Solution cost function assigns a numeric cost to each path; It also gives the cost of applying the operators to the states. ◊ A Solution quality is measured by the path cost function; and An optimal solution has the lowest path cost among all solutions. ◊ The solution may be any or optimal or all. ◊ The importance of cost depends on the problem and the type of solution asked.
• 12.
  Problem Solving, Searchand Control Strategies 1. Examples of Problem Definitions
• 13.
  Problem Solving, Searchand Control Strategies • Example 2 :