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And-Or Graphs

Explore the concept of And-Or Graphs and their use in solving complex problems that can be decomposed into sub-problems. Learn about the links between nodes that indicate relations between problems, and how to specify start and goal nodes. Discover different search mechanisms and representations in And-Or Graphs.

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And-Or Graphs

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  1. And-Or Graphs CSCE 580 Spr03 Instructor: Marco Valtorta Hrishikesh J. Goradia Seang-Chan Ryu

  2. And-Or Graphs • Technique for solving problems that can be decomposed into sub problems • Links between nodes indicates relations between problems • Or node: one of its successor node has to be solved • And node: all of its successor node has to be solved • Problem can be specified by two thinks: start node, goal nodes. • Goal nodes: trivial (or primitive) problems • Cost can be attached to arcs or nodes

  3. And-Or Graphs • Difference between state-state representation and and or representation. solution: path vs. tree

  4. Search in and-or graphs a b c d e f g h i

  5. Search in and-or graphs • Use Prolog’s own search mechanism • Only get answer yes or no not solution graph. • Hard to extend to use cost as well • Infinite loop if there is a cycle a :- b. a :- c. b :- d, e. e :- h. c :- f, g. f :- h, i. d. g. h.

  6. Search in and-or graphs • Other representation :- op (600, xfx, --->). :- op (500, xfx, :). a ---> or : [b,c]. b ---> and : [d,e]. c ---> and : [f,g]. e ---> or : [h]. f ---> or : [h,i]. goal( d). goal( g). goal( h).

  7. Search in and-or graphs • Depth first search. • Modification of depth first search (restricting with maximum depth) – andor.pl • Iterative deepening. (increase the maximum depth as the search goes along).

  8. Applying and-or graphs • Adversarial situations, such as game playing, can often be represented as and-or trees • Nodes represent the situation • Arcs represent possible actions for each player • Each path represents the sequence of choices made alternately by the two opponents • In the case of game trees, one is interested in a winning strategy.

  9. Tic-tac-toe game • Max size of the state space for a complete solution = 9!

  10. Tic-tac-toe game • For pragmatic reasons, we will consider a subset of the tic-tac-toe problem. • The adjacent figure shows the initial state of our tic-tac-toe program. • Max. state space for our new problem = 4!

  11. Tic-tac-toe game • The winning positions (shown in red) are the goal states. • tictactoe.pl shows the complete Prolog code for our problem.

  12. Final comments… • For use in real-world applications, the AND-OR graphs technique computes the best state instead of a complete path to the terminal state. These graphs use minmax search and are called game trees. • The AND-OR technique for searching can be easily extended to accommodate cost in the graph problems

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