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Problem-solving as search

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History Problem-solving as search – early insight of AI. Newell and Simon’s theory of human intelligence and problem-solving. Early examples: 1956: Logic Theorist (Allen Newell & Herbert Simon) 1958: Geometry problem solver (Herbert Gelernter) 1959: General Problem Solver (Herbert Simon & Alan Newell) 1971: STRIPS (Stanford Research Institute Problem Solver, Richard Fikes & Nils Nilsson)

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Real-World Problem-Solving as Search Examples: Route/Path finding: Robots, cars, cell-phone routing, airline routing, characters in video games, … Layout of circuits Job-shop scheduling Game playing (e.g., chess, go) Theorem proving Drug design

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Classic AI Toy Problem: 8-puzzle initial state goal state Notion of “searching a state space” 283 164 75 Pictures from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp 2 8 31 6 4 75

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8-puzzle search tree Pictures from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp 283 164 75 283 164 75 283 1 6 4 75 283 164 75 283 1 64 75 283 1 6 4 75 2 8 3 1 6 4 75 283 1 6 4 75

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What is size of state space?

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What is size of state space for 8-puzzle? Size of state space 9! = 181,440 Size of 15-puzzle state space? 16! = 2 x 10 13 Size of 24-puzzle state space? 25! = 1.5 x 10 25

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What is size of state space for 8-puzzle? Size of state space 9! = 181,440 Size of 15-puzzle state space? 16! = 2 x 10 13 Size of 24-puzzle state space? 25! = 1.5 x 10 25 Can’t do exhaustive search!

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Approximate number of states Tic-Tac-Toe: 3 9 Checkers: 10 40 Rubik’s cube: 10 19 Chess: 10 120

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In general, a search problem is formalized as : state space special start and goal state(s) operators that perform allowable transitions between states cost of transitions All these can be either deterministic or probabilistic.

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State space as a tree/graph Search as tree search Solutions: “winning” state, or path to winning state

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How to solve a problem by searching 1.Define search space Initial, goal, and intermediate states 2.Define operators for expanding a given state into its possible successor states Defines search tree 3.Apply search algorithm (tree search) to find path from initial to goal state, while avoiding (if possible) repeating a state during the search. 4.Solution is path from initial to goal state (e.g., traveling salesman problem) or, simply a goal state, which might not be initially known (e.g., drug design)

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Missionaries and cannibals Three missionaries and three cannibals are on the left bank of a river. There is one canoe which can hold one or two people. Find a way to get everyone to the right bank, without ever leaving a group of missionaries in one place outnumbered by cannibals in that place. From http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

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Missionaries and cannibals Three missionaries and three cannibals are on the left bank of a river. There is one canoe which can hold one or two people. Find a way to get everyone to the right bank, without ever leaving a group of missionaries in one place outnumbered by cannibals in that place. From http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp How to set this up as a search problem?

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Missionaries and cannibals State space: Size? Initial state: Goal state: Operators: Cost of transitions: Search tree:

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Drug design Example: Search for sequence of up to N amino acids that forms protein shape that matches a particular receptor on a pathogen. (Note: There are 20 amino acids to choose from at each locus in the string.)

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Drug design State space: Size? Initial state: Goal state: Operators: Cost of transitions: Search tree:

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Search Strategies A strategy is defined by picking the order of node expansion. Strategies are evaluated along the following dimensions: 1.completeness – does it always find a solution if one exists? 2.optimality – does it always find a optimal (least-cost or highest value) solution? 3.time complexity – number of nodes generated/expanded 4.space complexity – maximum number of nodes in memory Time and space complexity are often measured in terms of: b – maximum branching factor of the search tree d – depth of the least-cost solution m – maximum depth of the state space (may be infinite) Adapted from http://www.cs.uiuc.edu/class/sp06/cs440/Lectures/lec2.pp

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Search methods Uninformed search: 1.Breadth-first 2.Depth-first 3.Depth-limited 4.Iterative deepening depth-first 5.Bidirectional Informed (or heuristic) search (deterministic or stochastic): 1.Greedy best-first 2.A* (and many variations) 3.Hill climbing 4. Simulated annealing 5. Genetic algorithm 6. Tabu search 7. Ant colony optimization Adversarial search: 1. Minimax with alpha-beta pruning

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Uninformed strategies Breadth-first: Expand all nodes at depth d before proceeding to depth d+1 Depth-first: Expand deepest unexpanded node Depth-limited: Depth-first search with a cutoff at a specified depth limit Iterative deepening: Repeated depth-limited searches, starting with a limit of zero and incrementing once each time http://www.cse.unl.edu/~choueiry/S03-476-876/searchapplet/index.html

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Breadth-first: Complete? Optimal? Time? Space? Depth-first: Complete? Optimal? Time? Space? Depth-limited: Complete? Optimal? Time? Space? Iterative deepening: Complete? Optimal? Time? Space? Uninformed Search Properties

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Informed (heuristic) Search What is a “heuristic”? Examples: 8 puzzle Missionaries and Cannibals Tic Tac Toe Traveling Salesman Problem Drug design

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Best-first greedy search 1.current state = initial state 2. Expand current state 3. Evaluate offspring states s with heuristic h(s), which estimates cost of path from s to goal state 4.current state = argmin s h(s) for s offspring(current state) 5.If current state ≠ goal state, go to step 2. http://alumni.cs.ucr.edu/~tmatinde/projects/cs455/TSP/heuristic/ Travellinganimation.htm

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Search Terminology Completeness solution will be found, if it exists Optimality least cost solution will be found Admissable heuristic h s, h never overestimates true cost from state s to goal state Best first greedy search: Complete? Optimal? 8-puzzle heuristics: Hamming distance, Manhattan distance: Admissible? Example of non-admissable heuristic for 8-puzzle?

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A* Search Uses evaluation function f (n)= g(n) + h(n) where n is a node. 1.g is a cost function Total cost incurred so far from initial state at node n 2.h is an heuristic Best first search is A* with g = 0.

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h 1 (start state) = h 2 (start state) =

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A* Pseudocode give code and show example on 8-puzzle

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A* Pseudocode create the open list of nodes, initially containing only our starting node create the closed list of nodes, initially empty while (we have not reached our goal) { consider the best node in the open list (the node with the lowest f value) if (this node is the goal) { then we're done } else { move the current node to the closed list and consider all of its successors for (each successor) { if (this suceessor is in the closed list and our current g value is lower) { update the successor with the new, lower, g value change the sucessor’s parent to our current node } else if (this successor is in the open list and our current g value is lower) { update the suceessor with the new, lower, g value change the sucessor’s parent to our current node } else this sucessor is not in either the open or closed list { add the successor to the open list and set its g value } } } } Adapted from: http://en.wikibooks.org/wiki/Artificial_Intelligence/Search/Heuristic_search/Astar_Search#Pseudo-code_A.2A

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A* search is complete, and is optimal if h is admissible

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Proof of Optimality of A* Suppose a suboptimal goal G 2 has been generated and is in the OPEN list. Let n be an unexpanded node on a shortest path to an optimal goal G 1. G1G1 start G2G2 n f(G 2 ) = g(G2) since h(G 2 ) = 0 g(G 1 ) since G 2 is suboptimal f(G 2 ) f(n) since h is admissible Since f(G 2 ) f(n), A* will never select G 2 for expansion

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Variations of A* IDA* (iterative deepening A*) ARA* (anytime repairing A*) D* (dynamic A*)

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(From http://aima.eecs.berkeley.edu/slides-pdf/)

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Example of Simulated Annealing Netlogo simulation

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Simulated Annealing is complete (if you run it for a long enough time!)

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Genetic Algorithms Similar to hill-climbing, but with a population of “initial states”, and stochastic mutation and crossover operations for search.

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