Solving Problems by Searching

1. Solving Problems by Searching CPS 4801-01

2. Outline • Problem-solving agents • Example problems • Basic search algorithms

3. Problem-Solving Agents • Intelligent agents are supposed to maximize their performance measure. • If the agent can adopt a goal and aim at satisfying it.

4. Problem-Solving Agents • “formulate, search , execute” design

5. Problem-Solving Agents • Goal formulation, based on the current situation and the agent’s performance measure, is the first step in problem solving. • Problem formulation is the process of deciding what actions and states to consider, given a goal. • The process of looking for a sequence of actions that reaches the goal is search. • Execution phase: once a solution is found, the actions it recommended can be carried out.

6. Example: Romania • On holiday in Romania; currently in Arad. • Flight leaves tomorrow from Bucharest • Formulate goal: be in Bucharest • Formulate problem: • states: various cities • action: drive between cities • Find solution: • sequence of cities: e.g., Arad, Sibiu, Fagaras, Bucharest

7. Example: Romania

8. Problem Formulation • A problem is defined by five components. • Initial state e.g., “at Arad” • Actions (s)  {a1, a2, a3, … } • e.g., {Go(Sibiu), Go(Timisoara), Go(Zerind)} • Transition model: Result (s,a)  s’ • e.g., Result(In(Arad), Go(Timisoara)) = In(Timisoara) • Goal test (s)  T/F e.g., “at Bucharest” • Path cost (sss)  n (additive) • sum of cost of individual steps, e.g., number of miles traveled, number of minutes to get to destination

9. Example Problems • Toy problems: to illustrate or exercise various problem-solving methods, given a concise, exact description. • Real-world problems: tend not to have a single agreed-upon description.

10. Example: Vacuum World • states? • initial state? • actions? • transition model? • goal test? • path cost?

11. states? location and dirt, 2*22=8, n*2n • initial state? any state • actions? Left, Right, Suck (Up, Down) • transition model? • goal test? no dirt at all locations • path cost? 1 per action (0 for NoOp)

12. Example: The 8-Puzzle • states? • initial state? • actions? • transition model? • goal test? • path cost?

13. states? A state description specifies the location of the eight tiles and the blank one. • initial state? any state • actions? movement of the blank space: Left, Right, Up, Down • transition model? (s,a)s’ • goal test? goal state (given) • path cost? 1 per move

14. The 8-Queens Problem • states? • initial state? • actions? • transition model? • goal test? • path cost?

15. Incremental formulation • states? Any arrangement of 0 to 8 queens • initial state? No queens on the board • actions? Add a queen to any empty square • transition model? Return the board with a queen added • goal test? 8 queens, none attacked • path cost? no interest

16. The 8-Queens Problem • 64*63*…*57 ~= 1.8*1014 • A better formation: • states: All possible arrangement of n queens, one per column in the leftmost n columns, with no queen attacking another. (0<=n<=8) • actions: Add a queen to any square in the leftmost empty column such that it is not attacked by any other queen.

17. Donald Knuth (1964): starting with the number 4, a sequence of factorial, square root, and floor operation will reach any desired positive integer. • states? Positive numbers • initial state? 4 • actions? Apply factorial, square root, or floor operation • transition model? definition of the operations • goal test? desired positive integer • path cost? number of operations

18. Real-World Problems • Route-finding problem • Touring problems • e.g., visit every city in the Romania example at least once, starting and ending in Bucharest. • Traveling salesman problem (TSP) • each city must be visited exactly once • shortest tour

19. Tree Search Algorithms • search tree: initial state as the root, the branches as actions and the nodes as states. • We expand the current state and generate a new set of states. • parent node, child nodes • leaf node: a node with no children (frontier) • search strategy: how to choose which state to expand next • repeated state, redundant paths

20. Tree Search Example • The initial state

21. Tree Search Example • After expanding Arad

22. Tree Search Example • After expanding Sibiu

23. Graph Search Algorithm • explored set: remembers every expanded node • Newly generated nodes that are in the explored set are discarded. • The frontier separates the explored region and the unexplored region.

24. Data Structure • For each node n, • n.State • n.Parent • n.Action: parent generated the node • n.Path-Cost

25. Data Structure • Frontier: queue • Empty? (queue) : T/F • Pop (queue) • Insert (queue) • FIFO, LIFO, priority • Explored set: hash table

26. Search Strategies • A search strategy is defined by picking the order of node expansion • Strategies are evaluated along the following dimensions: • completeness: does it always find a solution if one exists? • time complexity: number of nodes generated • space complexity: maximum number of nodes in memory • optimality: does it always find a least-cost solution? • Time and space complexity are measured in terms of • b:branching factor ormaximum number of successors of any node • d: depth of the least-cost solution • m: maximum length of any path in the state space (may be ∞)

27. Uninformed Search Strategies • Uninformed search strategies use only the information available in the problem definition. • Breadth-first search Uniform-cost search Depth-first search Depth-limited search