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Search Problems Russell and Norvig: Chapter 3, Sections 3.1 – 3.3 CS121 – Winter 2003 Problem-Solving Agent sensors ? environment agent actuators Search Problems 2 Problem-Solving Agent sensors ? environment agent actuators • Actions • Initial state • Goal test Search Problems 3 State Space and Successor Function state space successor function • Actions • Initial state • Goal test Search Problems 4 Initial State state space successor function • Actions • Initial state • Goal test Search Problems 5 Goal Test state space successor function • Actions • Initial state • Goal test Search Problems 6 Example: 8-puzzle 8 2 1 2 3 3 4 7 4 5 6 5 1 6 7 8 Initial state Goal state Search Problems 7 Example: 8-puzzle 8 2 7 3 4 8 2 5 1 6 3 4 7 5 1 6 8 2 8 2 3 4 7 3 4 7 5 1 6 5 1 6 Search Problems 8 Example: 8-puzzle Size of the state space = 9!/2 = 181,440 15-puzzle .65 x 1012 0.18 sec 6 days 24-puzzle .5 x 1025 12 billion years 10 millions states/sec Search Problems 9 Search Problem State space Initial state Successor function Goal test Path cost Search Problems 10 Search Problem State space each state is an abstract representation of the environment the state space is discrete Initial state Successor function Goal test Path cost Search Problems 11 Search Problem State space Initial state: usually the current state sometimes one or several hypothetical states (“what if …”) Successor function Goal test Path cost Search Problems 12 Search Problem State space Initial state Successor function: [state subset of states] an abstract representation of the possible actions Goal test Path cost Search Problems 13 Search Problem State space Initial state Successor function Goal test: usually a condition sometimes the description of a state Path cost Search Problems 14 Search Problem State space Initial state Successor function Goal test Path cost: [path positive number] usually, path cost = sum of step costs e.g., number of moves of the empty tile Search Problems 15 Search of State Space Search Problems 16 Search of State Space Search Problems 17 Search State Space Search Problems 18 Search of State Space Search Problems 19 Search of State Space Search Problems 20 Search of State Space search tree Search Problems 21 Simple Agent Algorithm Problem-Solving-Agent 1. initial-state sense/read state 2. goal select/read goal 3. successor select/read action models 4. problem (initial-state, goal, successor) 5. solution search(problem) 6. perform(solution) Search Problems 22 Example: 8-queens Place 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal. A solution Not a solution Search Problems 23 Example: 8-queens Formulation #1: • States: any arrangement of 0 to 8 queens on the board • Initial state: 0 queens on the board • Successor function: add a queen in any square • Goal test: 8 queens on the board, none attacked 648 states with 8 queens Search Problems 24 Example: 8-queens Formulation #2: • States: any arrangement of k = 0 to 8 queens in the k leftmost columns with none attacked • Initial state: 0 queens on the board • Successor function: add a queen to any square in the leftmost empty column such that it is not attacked by any other queen 2,067 states • Goal test: 8 queens on the board Search Problems 25 실제 n-queen 문제 Neural, Genetic 또는 Heuristic 방법으로 잘 해결 최악의 경우에는 처리 불가능 실제 n이 커지면 답이 매우 많으므로 간단한 Heuristics로도 답을 쉽게 찾음 따라서 n이 커도 답을 잘 찾는다고 해서 인공지능 접근방법이 문제를 해결한다는 증거는 아님 그러나 많은 실제 문제는 알고리즘에서 이야기하는 최악의 경우로는 잘 가지 않음 더구나 대부분 우리가 원하는 답은 최적이 아니라 실제 활용해서 도움이 되는, feasible solution을 원하므로 인공지능 기법이 효과적으로 이용될 수 있음 Search Problems 26 Example: Robot navigation What is the state space? Search Problems 27 Example: Robot navigation Cost of one horizontal/vertical step = 1 Cost of one diagonal step = 2 Search Problems 28 Example: Robot navigation Search Problems 29 Example: Robot navigation Search Problems 30 Example: Robot navigation Cost of one step = ??? Search Problems 31 Example: Robot navigation Search Problems 32 Example: Robot navigation Search Problems 33 Example: Robot navigation Cost of one step: length of segment Search Problems 34 Example: Robot navigation Search Problems 35 Example: Assembly Planning Initial state Complex function: it must find if a collision-free Goal state merging motion exists Successor function: • Merge two subassemblies Search Problems 36 Example: Assembly Planning Search Problems 37 Example: Assembly Planning Search Problems 38 Assumptions in Basic Search The environment is static The environment is discretizable The environment is observable The actions are deterministic open-loop solution Search Problems 39 Search Problem Formulation Real-world environment Abstraction Search Problems 40 Search Problem Formulation Real-world environment Abstraction Validity: Can the solution be executed? Search Problems 41 Search Problem Formulation Real-world environment Abstraction Validity: Can the solution be executed? Does the state space contain the solution? Search Problems 42 Search Problems 43 Search Problems 44 Search Problems 45 Search Problems 46 Search Problem Formulation Real-world environment Abstraction Validity: Can the solution be executed? Does the state space contain the solution? Usefulness Is the abstract problem easier than the real- world problem? Search Problems 47 Search Problem Formulation Real-world environment Abstraction Validity: Can the solution be executed? Does the state space contain the solution? Usefulness Is the abstract problem easier than the real- world problem? Without abstraction an agent would be swamped by the real world Search Problems 48 Search Problem Variants One or several initial states One or several goal states The solution is the path or a goal node In the 8-puzzle problem, it is the path to a goal node In the 8-queen problem, it is a goal node Search Problems 49 Problem Variants One or several initial states One or several goal states The solution is the path or a goal node Any, or the best, or all solutions Search Problems 50 Important Parameters Number of states in state space 8-puzzle 181,440 8-queens 2,057 15-puzzle .65 x 1012 100-queens 1052 24-puzzle .5 x 1025 There exist techniques to solve N-queens problems efficiently! Stating a problem as a search problem is not always a good idea! Search Problems 51 Important Parameters Number of states in state space Size of memory needed to store a state Search Problems 52 Important Parameters Number of states in state space Size of memory needed to store a state Running time of the successor function Search Problems 53 Applications Route finding: airline travel, telephone/computer networks Pipe routing, VLSI routing Pharmaceutical drug design Robot motion planning Video games Search Problems 54 Task Environment Observable Deterministic Episodic Static Discrete Agents Crossword puzzle Fully Deterministic Sequential Static Discrete Single Chess with a clock Fully Strategic Sequential Semi Discrete Multi Poker Partially Strategic Sequential Static Discrete Multi Backgammon Fully Stochastic Sequential Static Discrete Multi Taxi driving Partially Stochastic Sequential Dynamic Continuous Multi Medical diagnosis Partially Stochastic Sequential Dynamic Continuous Single Image-analysis Fully Deterministic Episodic Semi Continuous Single Part-picking robot Partially Stochastic Episodic Dynamic Continuous Single Refinery controller Partially Stochastic Sequential Dynamic Continuous Single Interactive English tutor Partially Stochastic Sequential Dynamic Discrete Multi Figure 2.6 Examples of task environments and their characteristics. Search Problems 55 Summary Problem-solving agent State space, successor function, search Examples: 8-puzzle, 8-queens, route finding, robot navigation, assembly planning Assumptions of basic search Important parameters Search Problems 56 Future Classes Search strategies Blind strategies Heuristic strategies Extensions Uncertainty in state sensing Uncertainty action model On-line problem solving Search Problems 57