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Problems, Problem Spaces, and Search • Define the Problem precisely • Analyse the problem • Isolate and represent the task knowledge • Choose the best problemsolving technique Chapter 2 1 Defining the Problem as a State Space Search Example - Water Jug Problem You are given two jugs, a 4-gallon one and a 3-gallon one. Neither has any measring markers 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? initial state (0,0) goal state (2,n) Chapter 2 2 Operators 1 if x<4 -> (4,y) 2 if y<3 -> (x,3) 3 if x>0 -> (x-d,y) 4 if y>0 -> (x,y-d) 5 if x>0 -> (0,y) 6 if y>0 -> (x,0) 7 if x+y>=4 and y>0 -> (4,y-(4-x)) 8 if x+y>=3 and x>0 -> (x-(3-y),3) 9 if x+y<=4 and y>0 -> ((x+y),0) 10 if x+y<=3 and x>0 -> (0,x+y) 11 (0,2) -> 12 (2,y) -> Chapter 2 (2,0) (0,y) 3 Formal Description of a Problem 1- Define a state space 2- Specify initial states 3- Specify goal states 4- Specify set of rules that describe the actions available Chapter 2 4 Production System • A set of Rules • One or more databases • A control stsretegy • A Rule applier Chapter 2 5 Control Strategies The requirements of a good control strategy are : • it causes motion • it is systematic Chapter 2 6 Algorithm : Breadth-First Search (0,0) ---------------------| | (4,0) (0,3) ------------------------------| | | | | | (4,3) (0,0) (1,3) (4,3) (0,0) (3,0) Algorithm : Depth-First Search (0,0) ------------------| (4,0) ---------| (4,3) Chapter 2 7 Advantage of Depth-First Search • • Less Memory By chance, it may find a solution without examining much of the search space at all Advantage of Breadth-First Search It will not get trapped exploring a blind alley If there is a solution then BFS is guaranteed to find it If there are multiple solution, a minimal solution will be found Chapter 2 8 Heurestic Search Example The traveling Salesman Problem A Simple motion-causing and systemic control structure can solve the problem in time propootional to: (N-1) ! Applying nearest neighbor heurestic N2 A heurestic function is a function that maps from problem state descriptions to measures of desirability , usually represented as numbers Chapter 2 9 Problem Characteristics Is the Problem Decomposable? Integration, Blocks World Can Solution Steps Be Ignored or Undone ? Theorem Proving, 8-Puzzle, Chess Is the Universe Predictable? 8-Puzzle, Bridge Is a Good Solution Absolute or Relative ? Answering question, Salesman Problem Is the Solution a State or a Path? NLUnderstanding, Water Jug What is the Role of Knowledge? Chess, newspaper story understanding Does the Task Require Interaction witha Person? Theorem proofing, Medical Diagnosis Chapter 2 10 Production System Characteristics • A Monotonic Production System is a 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 selected • A Nonmonotonic Production System is one in which the above condition is not true • A Partially commutative Production System is a production system with the property that if the application of a particular sequence of rules transforms 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. Chapter 2 11 Issues in the Design of Search Programs • Forward versus backward reasoning • Matching rules against state • Representing each node of the search process Additional Problems The Missionaries and Cannibals Problem The Tower of Hanoi The Monkey and Banana Problem Cryptarithmetic Chapter 2 12