ISC 4322/6300 – GAM 4322 Artificial Intelligence

1. ISC 4322/6300 – GAM 4322Artificial Intelligence Lecture 3 Informed Search and Exploration Instructor: Alireza Tavakkoli September 10, 2009 University Of Houston- Victoria Computer Science Department

2. Uninformed Search • Problem Solving Agents • A sequence of actions • Desirable state • Definition • Problem • Solution • Approach • Uninformed search mechanisms • Analysis of algorithms concepts • Asymptotic complexity • NP-completeness

3. Problem Solving • What is a goal? • A set of world states • What actions to take? • Level of detail • Problem formulation • In our example • Driving to a particular town • States are being in a particular town • Goal • Driving to Bucharest

4. Searching for Solutions • Search tree • Search nodes • Root • Initial state search node • Expansion of states • Applying the successor function Search Strategy

5. Comparing Search Strategies

6. Summary of Algorithms

7. Material • Chapter 4 Section 1 – 3 • Exclude memory-bounded heuristic search • Outline • Informed search strategies • Greedy best-first search • A* search • Heuristics • Local search algorithms • Hill-climbing search • Simulated annealing search • Local beam search • Genetic algorithms

8. Informed Search 282 538 • So far the techniques made use only of information that could be stored on a graph, i.e., the successor function • Often, however, we have available problem-specific knowledge that can help. • When additional information is used... we talk about “Informed Search”. • How can we take advantage of such additional information?

9. Informed Search Strategies • Informed search • Has more knowledge beyond the problem definition • Best-First Search • Expand the seemingly best node • Cost to the goal is minimum • Evaluation function f(n) • Data structure • Priority queue  a fringe in ascending order of f-values • Heuristic function h(n) • If n is the goal then h(n)= 0 • Variants • Greedy Best-First Search • A* Algorithm

10. Romania w/ SLD in Km

11. Greedy Best-First Search • Greedy best-first search expands the node that appears to be closest to goal • Evaluation function f(n) = h(n) (heuristic) • = estimate of cost from n to goal • e.g., hSLD(n) = straight-line distance from n to Bucharest • The heuristic function can not be computed from the problem definition • The heuristic function is relevant to the problem goal • Straight line distance is likely related to the road distance

12. Greedy Best-First Search Example

13. Greedy Best-First Search Example

14. Greedy Best-First Search Example

15. Greedy Best-First Search Example

16. Properties of GBFS • Complete? • No – can get stuck in loops • e.g., Iasi  Neamt  Iasi  Neamt  • Time? • O(bm) • but a good heuristic can give dramatic improvement • Space? • O(bm) • keeps all nodes in memory • Optimal? • No

17. A* Search • Idea: avoid expanding paths that are already expensive • Evaluation function f(n) = g(n) + h(n) • g(n) = cost so far to reach n • h(n) = estimated cost from n to goal • f(n) = estimated total cost of path through n to goal

18. A* search example

19. A* search example

20. A* search example

21. A* search example

22. A* search example

23. A* search example

24. Admissible Heuristics • 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: hSLD(n) (never overestimates the actual road distance) • Optimality • Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal • Tree search allows for repeated states

25. Optimality of A* (proof) • Suppose some suboptimal goal G2has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. • f(G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f(G) = g(G) since h(G) = 0 • f(G2) > f(G) from above

26. Optimality of A* (proof) • Suppose some suboptimal goal G2has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. • f(G2) > f(G) from above • h(n) ≤ h*(n) since h is admissible • g(n) + h(n) ≤ g(n) + h*(n) • f(n) ≤ f(G) < f(G2) Hence f(G2) > f(n), and A* will never select G2 for expansion

27. A* Using Graph Search • Case 2: GRAPH-SEARCH • (avoid repeated states) • Admissibility is not sufficient. • 2 solutions to retain optimality: • once you found a repeated state, discard the more expensive path (extra book-keeping, expensive) • make sure the first path followed to a repeated state is the optimal path • Consistency • If h(n) is consistent • then A* is optimal • Admissible = Monotonic • Satisfies the triangular inequality

28. Consistent heuristics • A heuristic is consistent if for every node n, every successor n' of n generated by any action a, • h(n) ≤ c(n,a,n') + h(n') • If h is consistent, we have 1- f(n') = g(n') + h(n')  2- f(n') = g(n) + c(n,a,n') + h(n'), 3- h(n) ≤ c(n,a,n') + h(n')  4- g(n)+ c(n,a,n') + h(n') ≥ g(n) + h(n)  From 2 & 4- f(n') ≥ f(n) • i.e., f(n) is non-decreasing along any path. • Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal • Graph search does not allow for repeated states

29. Optimality of A* • A* expands nodes in order of increasing f value • Gradually adds "f-contours" of nodes • Contour i has all nodes with f=fi, where fi < fi+1

30. Properties of A* • Complete? • Yes (unless there are infinitely many nodes with f ≤ f(G) ) • Time? • Exponential • Space? • Keeps all nodes in memory • Optimal? • Yes

31. Admissible Heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? • h2(S) = ?

32. Admissible Heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? 8 • h2(S) = ? 3+1+2+2+2+3+3+2 = 18 • Both heuristics are admissible.

33. Dominance • If h2(n) ≥ h1(n) for all n (both admissible) • then h2dominatesh1 • h2is better for search • Effective branching factor b* • Good heuristics lead to b*1 • Typical search costs (average number of nodes expanded): • Check table 4.8 • d=12 IDS = 3,644,035 nodes b*= 2.78 A*(h1) = 227 nodes b*= 1.42 A*(h2) = 73 nodes b*= 1.24 • d=24 IDS = too many nodes b*= NA A*(h1) = 39,135 nodes b*= 1.48 A*(h2) = 1,641 nodes b*= 1.26

34. Relaxed Problems • A problem with fewer restrictions on the actions is called a relaxed problem • The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem • “The optimal solution to the original problem is, by definition, also a solution to the relaxed problem and therefore, must be at least as expensive as the optimal solution to the relaxed problem. • 8-Puzzle relaxed problems • If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution • If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution

35. Local Search Algorithms • In many optimization problems, the path to the goal is irrelevant • the goal state itself is the solution • State space = set of "complete" configurations • Find configuration satisfying constraints • e.g., n-queens • In such cases, we can use local search algorithms • keep a single "current" state, try to improve it • Advantages: • Very little memory requirements • Often find reasonable solutions in large spaces • Suitable for optimization problems

36. Example: n-queens • Put n queens on an n × n board with no two queens on the same row, column, or diagonal

37. Hill-climbing search • State space landscape

38. Hill-climbing search • "Like climbing Everest in thick fog with amnesia"

39. Hill-climbing search • Problem: depending on initial state, can get stuck in local maxima

40. Hill-climbing search: 8-queens problem • How many successor each state has? • 56 • h = number of pairs of queens that are attacking each other, either directly or indirectly • h = 17 for the above state

41. Hill-climbing search: 8-queens problem • Greed algorithms work well • They make quick progress • However • They get stuck in local maxima, minima • They don’t look ahead! Almost a perfect solution with h=1 Took only 5 steps from the previous Slide!

42. When does hill-climbing fail? • Local maxima/minima • Not found a global maxima, but the neighboring stares are even worse! • Ridges • All maxima are the same and everything else is worse! • Plateaux • All neighboring states have the same evaluation value!

43. Simulated Annealing Search • Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency

44. Properties of Simulated Annealing • One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1 • Widely used in VLSI layout, airline scheduling, etc

45. Local Beam Search • Keep track of k states rather than just one • Start with k randomly generated states • At each iteration, all the successors of all k states are generated • If any one is a goal state, stop; else select the k best successors from the complete list and repeat.

46. Genetic Algorithms • A successor state is generated by combining two parent states • Start with k randomly generated states (population) • A state is represented as a string over a finite alphabet (often a string of 0s and 1s) • Evaluation function (fitness function). Higher values for better states. • Produce the next generation of states by selection, crossover, and mutation

47. Genetic algorithms • Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) • 24/(24+23+20+11) = 31% • 23/(24+23+20+11) = 29% etc

48. Genetic algorithms

49. Questions? • AI: A Modern Approach : • Chapter 4