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### Problems, Problem spaces and Search

• 1. ArtificiAl intelligence 15cS562 Ravi Kumar B N Assistant Professor Dept. of CSE BMSIT&M
• 2. PROBLEMS, PROBLEM SPACES AND SEARCH Ravi Kumar B N, Asst.Prof,CSE,BMSIT 2
• 3. Contents Ravi Kumar B N, Asst.Prof,CSE,BMSIT 3 • Defining the problem as a State Space Search • Production Systems • Control Strategies • Breadth First Search • Depth First Search • Heuristic Search • Problem Characteristics • Is the Problem Decomposable? • Can Solution Steps be ignored or undone? • Production system characteristics • Issues in the design of search programs
• 4. To Build a System to Solve a Particular Problem, The Following Four Things are Needed 1. Define the problem precisely- specify both initial and final situations(state) 2. Analyze the problem 3. Isolate and represent the task knowledge that is necessary to solve the problem 4. Choose the best problem solving technique and apply it Ravi Kumar B N, Asst.Prof,CSE,BMSIT 4
• 5. State space search Problem = Searching for a goal state It is a process in which successive configurations or states of an instance are considered , with the goal of finding a goal state with a desired property . State space- a set of states that a problem can be in. - The group consisting of all the attainable states of a problem ex: Customers in a line would have state space {0,1,2….} Ravi Kumar B N, Asst.Prof,CSE,BMSIT 5
• 6. Search Problem S: the full set of states S0 :the initial state A:SS set of operators G : the set of final states. G is subset of S Search problem: Find a sequence of actions which transforms the agent from the initial state to goal state. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 6
• 7. Representing search problems Using directed graph - The states are represented as nodes - The allowed actions are represented as arcs. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 7
• 8. Problem Formulation •A single state problem formulation is defined by four items Initial state, successor function, goal test and path cost •Problem formulation means choosing a relevant set of states to consider, and a feasible set of operators for moving from one state to another •Search is the process of imagining sequences of operators applied to the initial state and checking which sequence reaches a goal state. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 8
• 9. Examples. Problem: On holiday in Singapur; currently in Mysur. Flight leaves tomorrow from Bangalore. Find a short route to drive to Bangalore. Formulate problem: states: various cities actions: drive between cities solution: sequence of cities Path Cost: distance travelled Ravi Kumar B N, Asst.Prof,CSE,BMSIT 9
• 10. Vacuum world state space Ravi Kumar B N, Asst.Prof,CSE,BMSIT 10 States: Dirt and Robot Location Actions: Left, right, clean Goal test: No dirt at all locations Path cost: 1 per action
• 11. The 8 - Puzzle Ravi Kumar B N, Asst.Prof,CSE,BMSIT 11
• 12. The 8 - Puzzle Ravi Kumar B N, Asst.Prof,CSE,BMSIT States: Locations of tiles Actions: Move blank left, right, up, down Goal test: Given Path cost: 1 per move 12
• 13. State space search: Playing Chess • Each position can be described by an 8 by 8 array. • Initial position is the game opening position. • Goal position is any position in which the opponent does not have a legal move and his or her king is under attack. • Legal moves can be described by a set of rules: -Left sides can be described by a set of rules -Right sides describe the new resulting state Ravi Kumar B N, Asst.Prof,CSE,BMSIT 13
• 14. Playing chess contd… • State space is a set of legal positions. • Starting at the initial state. • Using the set of rules to move from one state to another. • Attempting to end up in a goal state. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 14
• 15. Playing Chess Contd.. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 15 One Legal Chess move
• 16. Playing Chess Contd.. • Writing the rules like above leads to very large number • These rule poses serious practical difficulties - No person could ever supply a complete set of rules. It would take too long and could certainly not be done without mistakes - No program could easily handle all those rules. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 16
• 17. Playing Chess Contd.. Another way to describe the chess moves White pawn at Square(file e, rank 2) Move a pawn from AND Square(file e, rank 2) Square(file e, rank 3) is empty  to AND Square(file e, rank 4) Square(file e, rank 4) is empty Ravi Kumar B N, Asst.Prof,CSE,BMSIT 17
• 18. The Water Jug Problem Ravi Kumar B N, Asst.Prof,CSE,BMSIT 18 Given two jugs, a 4-gallon and 3-gallon, neither has any measuring maskers on it. There is a pump that can be used to fill the jugs with water. How can you get exactly 2 gallons of water into the 4-gallon jug? 1 Gallon = 3.785 Liter
• 19. The Water Jug Problem The state space for this problem can be described as the set of ordered pairs of integers (x,y) such that x = 0, 1,2, 3 or 4 and y = 0,1,2 or 3; x represents the number of gallons of water in the 4-gallon jug and y represents the quantity of water in 3- gallon jug The start state is (0,0) The goal state is (2,n) Ravi Kumar B N, Asst.Prof,CSE,BMSIT 19
• 20. Production rules for Water Jug Problem The operators to be used to solve the problem can be described as follows: Ravi Kumar B N, Asst.Prof,CSE,BMSIT 20
• 21. Production rules Ravi Kumar B N, Asst.Prof,CSE,BMSIT 21
• 22. To solve the water jug problem •Required a control structure that loops through a simple cycle in which some rule whose left side matches the current state is chosen • the appropriate change to the state is made as described in the corresponding right side •the resulting state is checked to see if it corresponds to goal state. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 22
• 23. One solution to the water jug problem Shortest such sequence will have a impact on the choice of appropriate mechanism to guide the search for solution. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 23
• 24. Production Systems To Structure AI Programs A production system consists of: •A set of rules, each consisting of a left side that determines the applicability of the rule and a right side that describes the operation to be performed if that rule is applied. •One or more knowledge/databases that contain whatever information is appropriate for the particular task. Some parts of the database may be permanent, while other parts of it may pertain only to the solution of the current problem. •A control strategy that specifies the order in which the rules will be compared to the database and a way of resolving the conflicts that arise when several rules match at once. •A rule applier Ravi Kumar B N, Asst.Prof,CSE,BMSIT 24
• 25. Control Strategies  How to decide which rule to apply next during the process of searching for a solution to a problem?  The two requirements of good control strategy are that • It should cause motion. • It should be systematic Ravi Kumar B N, Asst.Prof,CSE,BMSIT 25
• 26. BFS Tree for Water Jug problem Ravi Kumar B N, Asst.Prof,CSE,BMSIT 26 (0,0) (4,0) (0,3) (4,3) (0,0) (1,3) (4,3) (0,0) (3,0)
• 27. Breadth First Search Ravi Kumar B N, Asst.Prof,CSE,BMSIT 27
• 28. Breadth First Search Algorithm: 1. Create a variable called NODE-LIST and set it to initial state 2. Until a goal state is found or NODE-LIST is empty do a. Remove the first element from NODE-LIST and call it E. If NODE- LIST was empty, quit b. For each way that each rule can match the state described in E do: i. Apply the rule to generate a new state ii. If the new state is a goal state, quit and return this state iii. Otherwise, add the new state to the end of NODE-LIST Ravi Kumar B N, Asst.Prof,CSE,BMSIT 28
• 29. Depth First Search Ravi Kumar B N, Asst.Prof,CSE,BMSIT 29
• 30. Depth First Search Algorithm: 1.If the initial state is a goal state, quit and return success 2.Otherwise, do the following until success or failure is signaled: a. Generate a successor, E, of initial state. If there are no more successors, signal failure. b. Call Depth-First Search, with E as the initial state c. If success is returned, signal success. Otherwise continue in this loop. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 30
• 31. Backtracking • In this search, we pursue a single branch of the tree until it yields a solution or until a decision to terminate the path is made. • It makes sense to terminate a path if it reaches dead-end, produces a previous state. In such a state backtracking occurs • Chronological Backtracking: Order in which steps are undone depends only on the temporal sequence in which steps were initially made. • Specifically most recent step is always the first to be undone. • This is also simple backtracking. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 31
• 32. Advantages of Depth-First Search • DFS requires less memory since only the nodes on the current path are stored. • By chance, DFS may find a solution without examining much of the search space at all Ravi Kumar B N, Asst.Prof,CSE,BMSIT 32
• 33. Advantages of BFS • BFS will not get trapped exploring a blind alley. • If there are multiple solutions, then a minimal solution will be found. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 33
• 34. Problem Characteristics Inorder to choose the most appropriate method for a particular problem, it is necessary to analyze the problem along several key dimensions: • Is the problem decomposable into a set of independent smaller or easier subproblems? • Can solution steps be ignored or at least undone if they prove unwise? • Is the problem’s universe predictable? • Is a good solution to the problem obvious without comparison to all other possible solutions? • Is the desired solution a state of the world or a path to a state? • Is a large amount of knowledge absolutely required to solve the problem or is knowledge important only to constrain the search? • Can a computer that is simply given the problem return the solution or will the solution of the problem require interaction between the computer and a person? Ravi Kumar B N, Asst.Prof,CSE,BMSIT 34
• 35. Is the problem Decomposable? • Whether the problem can be decomposed into smaller problems? • Using the technique of problem decomposition, we can often solve very large problems easily. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 35
• 37. Blocks World Problem Non Decomposable Ravi Kumar B N, Asst.Prof,CSE,BMSIT 37 Following operators are available: CLEAR(x) [ block x has nothing on it]- > ON(x, Table) CLEAR(x) and CLEAR(y) -> ON(x,y) [ put x on y] C A B A B C Start: ON(C,A) Goal: ON(B,C) and ON(A,B) ON(B,C) ON(B,C) and ON(A,B) ON(B,C) ON(A,B) CLEAR(A) ON(A,B) CLEAR(A) ON(A,B)
• 38. Can Solution Steps be ignored or undone? Suppose we are trying to prove a math theorem. We can prove a lemma. If we find the lemma is not of any help, we can still continue. 8-puzzle problem Chess: A move cannot be taken back. Important classes of problems: • Ignorable ( theorem proving) • Recoverable ( 8-puzzle) • Irrecoverable ( Chess) The recoverability of a problem plays an important role in determining the complexity of the control structure necessary for the problem’s solution. • Ignorable problems can be solved using a simple control structure that never backtracks • Recoverable problems can be solved by a slightly more complicated control strategy that does sometimes make mistakes • Irrecoverable problems will need to be solved by systems that expends a great deal of effort making each decision since decision must be final. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 38
• 39. Is the universe Predictable? • Certain Outcome ( ex: 8-puzzle) • Uncertain Outcome ( ex: Bridge) • For solving certain outcome problems, open loop approach ( without feedback) will work fine. • For uncertain-outcome problems, planning can at best generate a sequence of operators that has a good probability of leading to a solution. We need to allow for a process of plan revision to take place. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 39
• 40. Is a good solution absolute or relative? • Any path problem • Best path problem • Any path problems can often be solved in a reasonable amount of time by using heuristics that suggest good paths to explore. • Best path problems are computationally harder. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 40
• 41. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 41
• 42. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 42
• 43. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 43
• 44. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 44
• 45. Is the solution a state or a path? Finding a consistent interpretation For the sentence “The bank president ate a dish of pasta salad with the fork”. We need to find the interpretation but not the record of the processing. Water jug : Here it is not sufficient to report that we have solved , but the path that we found to the state (2,0). Thus a statement of a solution to this problem must be a sequence of operations ( Plan) that produces the final state. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 45
• 46. Is the solution a state or a path? A path solution problem can be reformulated as a state – solution problem by describing a state as a partial path to a solution. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 46
• 47. What is the role of knowledge? Two examples: • Playing Chess: Knowledge is required to constrain the search for a solution • Newspaper story understanding: Lot of knowledge is required even to be able to recognize a solution. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 47
• 48. What is the role of knowledge? Consider a problem of scanning daily newspapers “to decide which are supporting the democrats and which are supporting the republicans in some upcoming election”. We need lots of knowledge to answer such questions as: • The names of the candidates in each party • The facts that if the major thing you want to see done is have taxes lowered, you are probably supporting the republicans • The fact that if the major thing you want to see done is improved education for minority students, you are probably supporting the democrats. • etc Ravi Kumar B N, Asst.Prof,CSE,BMSIT 48
• 49. Does the task require Interaction with a person? The programs require intermediate interaction with people for additional inputs and to provided reassurance to the user. There are two types of problems: • Solitary • Conversational Decision on using one of these approaches will be important in the choice of problem solving method. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 49
• 50. Does the task require Interaction with a person? Solitary Problem: in which there is no intermediate communication and no demand for an explanation of the reasoning process. Conversational Problem: In which intermediate communication is to provide either additional assistance to the computer or additional information to the user. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 50
• 51. Problem Classification • There are several broad classes into which the problems fall. • These classes can each be associated with generic control strategy that is appropriate for solving the problems: • Most diagnostic task : ex: medical diagnostics, diagnosis of faults in mechanical devices • Propose and Refine: ex: design and planning Ravi Kumar B N, Asst.Prof,CSE,BMSIT 51
• 52. Production System Characteristics Production Systems are good way to describe the operations that can be performed in a search for a solution to a problem. 1.Can production systems, like problems, be described by a set of characteristics that shed some light on how they can easily be implemented? 2.If so, what relationships are there between problem types and the types of production systems best suited to solving the problems? Ravi Kumar B N, Asst.Prof,CSE,BMSIT 52
• 53. Production System Characteristics To answer to the first question is yes. Consider the following definitions of Classes of Production systems: • Monotonic Production System • Non-Monotonic Production system • Partially commutative Production system: • Commutative Production system- both monotonic and partially commutative. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 53
• 54. Monotonic Production Systems Production system in which the application of a rule never prevents the later application of another rule that could also have been applied at the time the first rule was applied. i.e., rules are independent. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 54
• 55. Commutative Production system A partially Commutative production system has a property that if the application of a particular sequence of rules transform state x into state y, then any permutation of those rules that is allowable, also transforms state x into state y. A Commutative production system is a production system that is both monotonic and partially commutative. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 55
• 56. Four Categories of Production System Monotonic NonMonotonic Partially Commutative Theorem proving Robot Navigation Not Partially Commutative Chemical Synthesis Bridge Ravi Kumar B N, Asst.Prof,CSE,BMSIT 56
• 57. Partially Commutative, Monotonic These production systems are useful for solving ignorable problems. Example: Theorem Proving They can be implemented without the ability to backtrack to previous states when it is discovered that an incorrect path has been followed. This often results in a considerable increase in efficiency, particularly because since the database will never have to be restored, It is not necessary to keep track of where in the search process every change was made. They are good for problems where things do not change; new things get created. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 57
• 58. Non Monotonic, Partially Commutative • Useful for problems in which changes occur but can be reversed and in which order of operations is not critical. • Example: Robot Navigation, 8-puzzle, blocks world • Suppose the robot has the following ops: go North (N), go East (E), go South (S), go West (W). To reach its goal, it does not matter whether the robot executes the N-N-E or N-E-N. Ravi Kumar B N, Asst.Prof,CSE,BMSIT 58
• 59. Not partially Commutative Problems in which irreversible change occurs Example: chemical synthesis The ops can be :Add chemical x to the pot, Change the temperature to t degrees. These ops may cause irreversible changes to the potion being brewed. The order in which they are performed can be very important in determining the final output. (X+y) +z is not the same as (z+y) +x Non partially commutative production systems are less likely to produce the same node many times in search process. When dealing with ones that describe irreversible processes, it is partially important to make correct decisions the first time, although if the universe is predictable, planning can be used to make that less important.Ravi Kumar B N, Asst.Prof,CSE,BMSIT 59
• 60. Issues in the design of search programs • The direction in which to conduct the search ( forward versus backward reasoning). • How to select applicable rules ( Matching) • How to represent each node of the search process ( knowledge representation problem) Ravi Kumar B N, Asst.Prof,CSE,BMSIT 60
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