What is this lecture about? Solving Problems by Search. Example: Romania. Example: Romania (continued)

Transcription

1 What is this lecture about? Artificial Intelligence lp 4 VT2010 Solving problems by first translating the real world into abstract states. Then systematically explore the state-space by defining a search tree. There exist both general and specific search methods for exploring the entire or parts of the search tree. Solving Problems by Search March 16, 2010 Birgit Grohe Uninformed Search Informed Search 2 Example: Romania Example: Romania (continued) Russel Norvig, Figure 3.2, page 63 Suppose you are on holidays in Romania, currently in Arad. Your flight leaves tomorrow in Bucharest. You have no geographical knowledge about Romania. Goal formulation: Be in Bucharest Problem formulation: states (cities), operators (drive between cities) Solution: Can be represented as a sequence of states (cities) Example: (Arad, Sibiu, Fagaras, Bucharest) 4 3

2 Problem formulation A problem is defined by four components: Example: 8-Puzzle Russel Norvig, Figure 3.4, page 65 Initial state: e.g. atarat Operators: e.g. Arat Sibiu, Sibiu Arat Goal test: e.g. x = atbucharest Path cost: e.g. sum of distances or number of operators executed A solution is a sequence of operators leading from the initial state to the goal state. 6 5 Example: 8-Puzzle (continued) Problem formulation for the 8-puzzle: Example: 8-Queens Problem Given 8 queens, place them on a chess board such that no queen attacks any other. Russel Norvig, Figure 3.5, page 66 States: Integer locations of the tiles Operators: Move blank left, right, up or down Goal test: = goal state (explicitly given in this case) Path cost: 1 per move A solution is a sequence of moves leading from an initial state to the goal state. Note: Finding the optimal (shortest sequence) of the n Puzzle is NP-complete. 8 7

3 Further Examples Travelling Salesperson probem (TSP) VLSI design Robot navigation Various internet search applications Bioinformatics (e.g. protein design)... Example: 8-Queens Problem (continued) Problem formulation for the 8-queens problem: States: Any arrangement of 0-8 queens on the board; initial state is the empty board. Operators: Place a queen on the board Goal test: 8 queens on the board, none attacked (partly explicit, partly implicit goal formulation!) Path cost: not interesting A solution is a sequence of queen placements starting from the emptly board and leading to a goal state Searching for Solutions - Search Trees Example: A Search Tree for Romania Russel Norvig, Figure 3.6, page 70 To be able to easily describe and analyze the solution process from the inital state to the goal state using the operators, we use a search tree. Russel Norvig, Figure 3.8, page

4 Properties of BFS: Breadth-first Search (BFS) Completeness: Does it find the optimal solution if one exists? Yes. Optimality: Does it find the least cost solution? Only in special cases (cost per step = 1). Time Complexity: Nr of nodes generated O(b d+1 ) Space Complexity: Max nr of nodes in memory O(b d+1 ) BAD! d depth of a solution closest to the root m maximum depth of the search tree (m may be ) b branching factor, assume b is finite Uninformed Search Stategies Assume that we do not have any knowledge about the structure of the state space, e.g. about the geography of Romania. Strategies for exploring the corresponding search tree systematically are defined by the order in which the nodes in the tree are explored. Breadth-first search (BFS) Depth-first seach (DFS) Variants of BFS and DFS: Uniform-cost search, Depth limited search, iterative deepening etc Depth-first Search (DFS) Properties of BFS: Comparing Uninformed Search Strategies Russel Norvig, Figure 3.17, page 81 Completeness: If m is finite, yes. Otherwise no. Optimality: No. Time Complexity: O(b m ), BAD if m >> d Space Complexity: O(bm) NICE! d depth of a solution closest to the root m maximum depth of the search tree (m may be ) b branching factor, assume b is finite 16 15

5 Informed Search Make use of knowledge about the structure of the state space / search tree. Select next node in the search tree based on an evaluation function f(n). f(n) = g(n) + h(n) where g(n) is the path cost from start node to current node n and h(n) estimated cost of the ceapest path from n to the goal. Informed Search Greedy best-first search A search Local search methods Hill-climbing, Simulated annealing, Genetic algorithms A Search Greedy Best-first Search Optimal and complete, if h(n) chosen carefully, i.e. if h(n) is an admissible heuristic (never overestimates the real cost). Example Romania: h(n) = h SLD (n) is an admissible heuristic. Russel Norvig, Figure 4.3, page 98 A very simple and optimistic heuristic with f(n) = h(n). In general neither optimal nor complete. Example Romania: h SLD (n) = straight line distance from node n to the goal. Russel Norvig, Figure 4.1 and Figure 4.2, pages

6 Local Search for Optimization Problems Local Search Methods Main problem: Can get stuck in local optimum. Russel Norvig, Figure 4.10, page 111 Instead of systematically exploring the entire search tree, move around in a state space landscape. A local search method starts in a state and moves to a neighboring state according to some strategy: Hill-climbing: Moves only to neighboring states with better cost value. Simulated annealing: Moves to neighboring states with worse cost with some probability according to a cooling scheme (inspired from cooling down metals in physics). Genetic algorithms: Combines several states to new states, also using mutation (inspired from reproduction) References S. Russel, P. Norvig: Artificial Intelligence Cormen, Leiserson, Rivest, Stein: Introduction to Algorithms Brassard, Bratley: Fundamentals of Algorithmics 23