B-search-problems

B-search-problems - (Where reasoning consists of exploring...

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1 1 Search Problems (Where reasoning consists of exploring alternatives) R&N: Chap. 3, Sect. 3.1–2 + 3.6 2 Declarative knowledge creates alternatives: Which pieces of knowledge to use? How to use them? Search is a about exploring alternatives . It is a major approach to exploit knowledge Search Knowledge rep. Planning Reasoning Learning Agent Robotics Perception Natural language ... Expert Systems Constraint satisfaction 3 Example: 8-Puzzle 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Initial state Goal state State : Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board 4 8-Puzzle: Successor Function 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Search is about the exploration of alternatives SUCC(state) Æ subset of states The successor function is knowledge about the 8-puzzle game, but it does not tell us which outcome to use, nor to which state of the board to apply it. 5 Across history, puzzles and games requiring the exploration of alternatives have been considered a challenge for human intelligence: Chess originated in Persia and India about 4000 years ago Checkers appear in 3600-year-old Egyptian paintings Go originated in China over 3000 years ago So, it’s not surprising that AI uses games to design and test algorithms 6
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2 7 (n 2 -1)-puzzle 1 2 3 4 5 6 7 8 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 .... 8 15-Puzzle Introduced (?) in 1878 by Sam Loyd, who dubbed himself “America’s greatest puzzle-expert” 9 15-Puzzle Sam Loyd offered $1,000 of his own money to the first person who would solve the following problem: 12 14 11 15 10 13 9 5 6 7 8 4 3 2 1 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 ? 10 But no one ever won the prize !! 11 Stating a Problem as a Search Problem State space S Successor function: x S SUCCESSORS (x) 2 S Initial state s 0 Goal test: x S GOAL? (x) =T or F Arc cost S 1 3 2 12 State Graph Each state is represented by a distinct node An arc (or edge) connects a node s to a node s’ if s’ SUCCESSORS (s) The state graph may contain more than one connected component
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3 13 Solution to the Search Problem A solution is a path connecting the initial node to a goal node (any one) I G 14 Solution to the Search Problem A solution is a path connecting the initial node to a goal node (any one) The cost of a path is the sum of the arc costs along this path An optimal solution is a solution path of minimum cost There might be no solution ! I G 15 How big is the state space of the (n 2 -1)-puzzle? 8-puzzle Æ ?? states 16 How big is the state space of the (n 2 -1)-puzzle? 8-puzzle Æ 9! = 362,880 states 15-puzzle Æ 16! ~ 2.09 x 10 13 states 24-puzzle Æ 25! ~ 10 25 states But only half of these states are reachable from any given state (but you may not know that in advance) 17 Wlg, let the goal be: A tile j appears after a tile i if either j appears on the same row as i to the right of i, or on another row below the row of i.
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