1. Search Methodologies Fall 2013 Comp3710 Artificial Intelligence Computing Science Thompson Rivers University

2. TRU-COMP3710 Search Methodologies2 Course Outline Part I – Introduction to Artificial Intelligence Part II – Classical Artificial Intelligence Knowledge Representation Searching Search Methodologies Advanced Search Knowledge Represenation and Automated Reasoning Propositinoal and Predicate Logic Inference and Resolution for Problem Solving Rules and Expert Systems Part III – Machine Learning Part IV – Advanced Topics

3. TRU-COMP3710 Search Methodologies3 Chapter Learning Outcomes Decide if DFS (Depth First Search) and/or BFS (Breadth First Search) is complete and/or admissible on a tree and/or a graph. Define what an admissible heuristic is. Write a pseudo A* algorithm. Given two heuristics, decide which one is better. Given expanded states within a running A* algorithm, decide the next state to visit. Design and implement a Java program for the 8-puzzle game, using an A* algorithm. Use of Hill Climbing for the 4-queens game.

4. TRU-COMP3710 Search Methodologies4 Chapter Outline Introduction Problem Solving as Search Data-Driven and Goal-Driven Search Generate and Test Depth-First Search Breadth-First Search Properties of Search Methods Why Humans Use Depth-First Search Implementing Depth-First and Breadth-First Search Example: Web Spidering Depth-First Iterative Deepening Using Heuristics for Search Hill Climing Best-First Search Beam Search Identifying Optimal Paths

5. TRU-COMP3710 Search Methodologies5 Brute Force Search Search methods that examine every node in the search tree – also called exhaustive. It does not say the search order. Generate and Test is the simplest brute force search method: Generate possible solutions to the problem. Test each one in turn to see if it is a valid solution. Stop when a valid solution is found. The method used to generate possible solutions must be carefully chosen. [Q] Can brute force search find any solution? [Q] What if there are so many possible solutions, i.e., states in the search tree? [Q] Any better ideas?

6. TRU-COMP3710 Search Methodologies6 Depth-First Search (DFS) Another exhaustive search method. Follows each path to its deepest node, before backtracking to try the next path. [Q] A search traversal in the next example is ? 1 23 4 5 6 7

7. TRU-COMP3710 Search Methodologies7 Breadth-First Search (BFS) Another exhaustive search method. Follows each path to a given depth before moving on to the next depth. [Q] A search traversal in the next example is ? 1 23 4 5 6 7

8. TRU-COMP3710 Search Methodologies8 It is important to choose the correct search method for a given problem. In some situations, for example, depth first search will never find a solution, even though one exists. [Q] Can you give an example of this case?

9. TRU-COMP3710 Search Methodologies9 Properties of Search Methods 1. Complexity How much time and memory the method uses. 2. Completeness A complete method is one that is guaranteed to find a goal if one exists. 3. Optimality & Admissibility A method is optimal (or admissible) if it is guaranteed to find the best path to the goal. Is an admissible method is complete? Is a complete method is admissible? 4. Irrevocability A method is irrevocable if, like hill climbing, it does not ever back-track. [Q] Assuming a tree with uniform edge values, Is DFS complete? Is DFS admissible? Is BFS complete? Is BFS admissible? [Q] Assuming a tree with non-uniform edge values, [Q] Assuming a graph with uniform edge values,

10. TRU-COMP3710 Search Methodologies10 Implementations Generic search algorithm:(Note that the search space can be a tree or a graph.) Do Select a node;// using an evaluation function Visit the node; Test if the node is a goal, then stop;// using a goal test function Otherwise, expand the child nodes; How to implement depth-first search? Stack How to implement breadth-first search? Queue

11. TRU-COMP3710 Search Methodologies11 Depth-First Iterative Deepening An exhaustive search method based on both depth-first and breadth- first search. Carries out depth-first search to depth of 1, then to depth of 2, 3, and so on until a goal node is found. Efficient in memory use, and can cope with infinitely long branches. Not as inefficient in time as it might appear, particularly for very large trees, in which it only needs to examine the largest row (the last one) once. [Q] The search path in the next example is ? 1 23 4 5 6 7

12. TRU-COMP3710 Search Methodologies12 Best-First Search Picks the most likely path (based on heuristic value) from the partially expanded tree at each stage. Tends to find a shorter path than depth-first or breadth-first search, but does not guarantee to find the best path in a graph. [Q] What is the search path in the next example? Is the result the best path? [Q] What if the costs are all the same? 2 3 1 23 1.5

13. TRU-COMP3710 Search Methodologies13 Beam Search Breadth-first method. Only expands the best few paths at each level. Thus has the memory advantages of depth-first search. Not exhaustive, and so may not find the best solution. May not find a solution at all. All the search methods so far have too large time complexity. [Q] What do we have to do?

14. TRU-COMP3710 Search Methodologies14 Heuristics Heuristic: a rule or other piece of information that is used to make methods such as search more efficient or effective. In search, often use a heuristic evaluation function, f(n): f(n) tells you the approximate distance (or cost) from a node, n, to a goal node. f(n) may not be 100% accurate, but it should give better results than pure guesswork.

15. TRU-COMP3710 Search Methodologies15 Heuristics – How Informed? In general, the more informed a heuristic is, the better it will perform. Heuristic h is called more informed than j, if: h(n)  j(n) for all nodes n. In general, a search method using h will search more efficiently than one using j. A heuristic should reduce the number of nodes that need to be examined. [Q] Can you solve the 8-puzzle game?8-puzzle game [Q] Can you find any good heuristic for the 8-puzzle game? [Q] Can you decide which heuristic is better (or admissible)? [Q] Which one do you want to move first? h 1 : the sum of Manhatan distances h 1 (left) = 5 h 2 : the number of miss tiles h 2 (left) = 4 Which one is more informed?

16. TRU-COMP3710 Search Methodologies16 Optimal Paths An optimal path through a tree is one that is the shortest possible path from root to goal. In other words, an optimal path has the lowest cost (not necessarily at the shallowest depth, if edges have costs associated with them). The British Museum algorithm is a general approach to find an optimal path by examining every possible path, and selecting the one with the least cost. There are more efficient ways. We need to use some good heuristics. [Q] Which ones then?

17. TRU-COMP3710 Search Methodologies17 Heuristics – admissible Definition: A heuristic h(n) is admissible if, for every node n, h(n) ≤ h * (n), where h * (n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic. Example: h SLD (n) (never overestimates the actual road distance) on a map h(n) = 0 for all n [Q] Can you give admissible heuristics for the 8-puzzle game? h 1 : the sum of Manhatan distances h 1 (left) = 5 h 2 : the number of miss tiles h 2 (left) = 4

18. TRU-COMP3710 Search Methodologies18 Optimal Path – A* Algorithms Uses the cost function: f(node) = g(node) + h(node). g(node) is the cost of the path so far leading to the node. h(node) is an underestimate of how far node is from a goal state. (This is the definition of an admissible heuristic.) f is a path-based evaluation function. An A* algorithm expands paths from nodes that have the lowest f value until the algorithm visits the goal node. Theorem: If h(n) is admissible, A * using is optimal. [Q] Which admissible heuristic is better? [Q] When two admissible heuristics satisfy h 1 (n)  h 2 (n) for all nodes n, which one is better?

19. TRU-COMP3710 Search Methodologies19 Optimal Path – A* Algorithms A* algorithms are optimal, i.e., they are guaranteed to find the shortest path to a goal node, provided h(node) is always an underestimate. (This is the definition of an admissible heuristic.) A* methods are also optimally efficient – they expand the fewest possible paths to find the right one. If h is not admissible, the method is called A, rather than A*. Algorithm: next node <- initial node; while (next node != goal) expand nodes; select next node to visit;// a non-visited node // having the smallest f-value

20. TRU-COMP3710 Search Methodologies20 Optimal Path – Uniform Cost Search [Q] zero is an admissible heuristic? Also known as branch and bound. Like A*, but uses: f(node) = g(node). g(node) is the cost of the path leading up to node. I.e., h(node) = 0. Even when a goal node is found (at the child node expansion; still not visited yet), the method needs to continue to run in case a preferable solution is found, till a goal node is visited.

21. 21 Let’s use the heuristic of Manhatan distances. \ Let’s expand the current node, then 283, 104, 765g = 1; h = 4; f = 5 283, 164, 075g = 1; h = 6; f = 7 283, 164, 750g = 1; h = 6; f = 7 283, 104, 765 (g=1) is selected and visited. Let’s expand this new node. 283, 164, 075g = 1; h = 6; f = 7 previous ones in the queue 283, 164, 750g = 1; h = 6; f = 7 203, 184, 765g = 1 + 1; h = 3; f = 5 new ones 283, 014, 765g = 1 + 1; h = 5; f = 7 283, 140, 765g = 1 + 1; h = 5; f = 7 283, 164, 705 visited already

22. 22 283, 104, 765 (g=1) is selected and visited. Let’s expand this new node. 283, 164, 075g = 1; h = 6; f = 7 previous ones in the queue 283, 164, 750g = 1; h = 6; f = 7 203, 184, 765g = 2; h = 3; f = 5 new ones 283, 014, 765g = 2; h = 5; f = 7 283, 140, 765g = 2; h = 5; f = 7 283, 104, 765 (g=2) is selected and visited. Let’s expand this new node. 283, 164, 075g = 1; h = 6; f = 7 283, 164, 750g = 1; h = 6; f = 7 283, 014, 765g = 2; h = 5; f = 7 283, 140, 765g = 2; h = 5; f = 7 023, 184, 765g = 2+1; h = 2; f = 5 230, 184, 765g = 2+1; h = 4; f = 7 283, 104, 765 visited already 023, 184, 765 (g=3) is selected and visited. Let’s expand this new node.

23. 23 023, 184, 765 (g=3) is selected and visited. Let’s expand this new node. 283, 164, 075g = 1; h = 6; f = 7 283, 164, 750g = 1; h = 6; f = 7 283, 014, 765g = 2; h = 5; f = 7 283, 140, 765g = 2; h = 5; f = 7 230, 184, 765g = 3; h = 4; f = 7 123, 084, 765g = 3+1; h = 1; f = 5 283, 104, 765 visited already 123, 084, 765 (g=4) is selected and visited. Let’s expand this new node. 283, 164, 075g = 1; h = 6; f = 7 283, 164, 750g = 1; h = 6; f = 7 283, 014, 765g = 2; h = 5; f = 7 283, 140, 765g = 2; h = 5; f = 7 230, 184, 765g = 3; h = 4; f = 7 023, 184, 765 visited already 123, 804, 765g = 4+1; h = 0; f = 5 123, 784, 065g = 4+1; h = 2; f = 7 123, 804, 765 (g=5) is selected and visited. This is the goal node.

24. TRU-COMP3710 Search Methodologies24 Optimal Path – Uniform Cost Search From Arad to Bucharest Using the heuristic h ZERO

25. TRU-COMP3710 Search Methodologies25 Arad(0): Zerind(75); Sibiu (140); Timisoara(118) [Arad <-] Zerind(75): Oradea(75+71); Sibiu(140); Timisoara(118) [Arad <-] Timisoara(118): Lugoj(118+111); Oradea(146); Sibiu(140) [Arad <-] Sibiu(140): Oradea(140+151); Rimnicu(140+80); Fagaras(140+99); Lugoj(229); Oradea(146) [Zerind <-] Oradea(146): Rimnicu(220); Fagaras(239); Lugoj(118+111) [Sibiu <-] Rimnicu(220): Craiova(220+146); Pitesti(220+97); Fagaras(239); Lugoj(229) [Timisoara <-] Lugoj(229): Mehadia(229+70); Craiova(366); Pitesti(317); Fagaras(239) [Sibiu <-] Fagaras(239): Bucharest(239+211); Mehadia(299); Craiova(366); Pitesti(317) [Lugoj <-] Mehadia(299): Dobreta(299+75); Bucharest(450); Craiova(366); Pitesti(317) [Rimnicu <-] Pitesti(317): Bucharest(317+101); Dobreta(374); Bucharest(450); Craiova(366) [Rimnicu <-] Craiova(366): Dobreta(366+120); Bucharest(418); Dobreta(374) [Mehadia <-] Dobreta(374): Bucharest(418) [Pitesti <-] Bucharest(418) How many nodes are visited?

26. TRU-COMP3710 Search Methodologies26 Optimal Path – more informed heuristic From Arad to Bucharest Using the heuristic h SLD Additional information

27. TRU-COMP3710 Search Methodologies27 Arad(0): Zerind(75+374); Sibiu (140+253); Timisoara(118+329) [Arad <-] Sibiu(140): Oradea(140+151+380); Fagaras(140+99+176); Rimnicu(140+80+193); Zerind(75+374); Timisoara(118+329) [Sibiu <-] Rimnicu(220): Craiova(220+146+160); Pitesti(220+97+100); Oradea(291+380); Fagaras(239+176); Zerind(75+374); Timisoara(118+329) [Sibiu <-] Fagaras(239): Bucharest(239+211+0); Craiova(366+160); Pitesti(317+100); Oradea(291+380); Zerind(75+374); Timisoara(118+329) [Rimnicu <-] Pitesti(317): Craiova(317+138+160); Bucharest(317+101+0); Bucharest(450+0); Craiova(366+160); Oradea(291+380); Zerind(75+374); Timisoara(118+329) [Pitesti <-] Bucharest(418) How many nodes are visited?

28. TRU-COMP3710 Search Methodologies28 283164705(0+5): 283104765(1+4); 283164075(1+6); 283164750(1+6) 283104765(1+4): 203184765(2+3); 283014765(2+5); 283140765(2+5); 283164075(1+6); 283164750(1+6) 203184765(2+3): 023184765(3+2); 230184765(3+4); 283014765(2+5); 283140765(2+5); 283164075(1+6); 283164750(1+6) 023184765(3+2): 123084765(4+1); 230184765(3+4); 283014765(2+5); 283140765(2+5); 283164075(1+6); 283164750(1+6) 123084765(4+1); 123784065(5+2); 123804765(5+0); 230184765(3+4); 283014765(2+5); 283140765(2+5); 283164075(1+6); 283164750(1+6) 123804765(5+0) h 1 : the sum of Manhatan distances h 1 (left) = 5 [Q] How many times do you need to move?

29. TRU-COMP3710 Search Methodologies29 [Q] How to implement the 8-puzzle game? A bit detail algorithm:

30. TRU-COMP3710 Search Methodologies30 Optimal Path – Greedy Search Like A*, but uses: f(node) = h(node). Hence, always expands the node that appears to be closest to a goal, regardless of what has gone before. We humans usually use this approach. Not optimal, and not guaranteed to find a solution. Can easily be fooled into taking poor paths. [Q] Can you give an example?

31. TRU-COMP3710 Heuristic Search31 Local Search Algorithms In the previous problems, we know what a goal state is. However, in many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution. [Q] Can we make any good heuristic? [Q] Can we use A* algorithms? Why? State space = set of "complete" configurations Find configuration satisfying constraints, e.g., n-queens. (We do not have to consider of the order of move.) In such cases, we can use local search algorithms. Keep a single "current" state, and try to improve it. [Q] Isn’t it the greedy approach?

32. TRU-COMP3710 Heuristic Search32 Example: n-queens Put n queens on an n × n board with no two queens on the same row, column, or diagonal [Q] What is a goal state? How do we solve? [Q] Can we use a local search algorithm?

33. TRU-COMP3710 Search Methodologies33 Hill Climbing An informed, irrevocable local search method. Easiest to understand when considered as a method for finding the highest point in a three dimensional search space: Check the height one foot away from your current location in each direction; North, South, East and West. As soon as you find a position whose height is higher than your current position, move to that location, and restart the algorithm.

34. TRU-COMP3710 Search Methodologies34 Hill Climbing – Foothills Difficulties for hill-climbing methods. A foothill is a local maximum.

35. TRU-COMP3710 Search Methodologies35 Hill Climbing – Plateaus Difficulties for hill-climbing methods. Flat areas that make it hard to find where to go next.

36. TRU-COMP3710 Search Methodologies36 Hill Climbing – Ridges Difficulties for hill-climbing methods B is higher than A. At C, the hill-climber can’t find a higher point North, South, East or West, so it stops.

37. TRU-COMP3710 Search Methodologies37 Summary A* algorithms with heuristics Greedy search [Q] When a goal is known? Local search [Q] When a goal is know? [Q] How to improve?