# AI: A Modern Approach, Chpts. 3-4 Russell and Norvig

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1 AI: A Modern Approach, Chpts. 3-4 Russell and Norvig Sequential Decision Making in Robotics CS 599 Geoffrey Hollinger and Gaurav Sukhatme (Some slide content from Stuart Russell and HweeTou Ng) Spring,

2 2 Decision Making Problem

3 Problem types Deterministic, fully observable Agent knows exactly which state it will be in solution is a sequence Non-observable Agent may have no idea where it is solution (if any) is a sequence Partially observable observations provide new information about current state solution is a contingent plan or a policy often interleave search, execution Unknown state space exploration problem 3

4 Vacuum world Observable, start in #5. [Right; Suck] Non-observable, start in: {1;2;3; 4;5;6;7;8} e.g., Right goes to {2;4;6;8} [Right; Suck; Left; Suck] Partially observable, start in #5, suck can dirty a clean carpet, local sensing only [Right; if dirt then Suck] Possible actions: left, right, suck 4

5 Vacuum world state space 5 States: Successor function: Actions: Goal test: Path cost:

6 Vacuum world state space 6 States: cross product of dirtiness and robot locations Successor function: Left/Right changes location, suck changes dirtiness Actions: Left, Right, Suck, NoOp Goal test: no dirt Path cost: 1 per action (0 for NoOp)

7 7 Non-observable vacuum world

8 8-puzzle state space 8 States: State transition Actions: Goal test: Path cost:

9 [Note: optimal solution of n-puzzle family is NP-hard] 8-puzzle state space 9 States: integer locations of tiles (ignore intermediate positions) State transitions: tile locations change Actions: move tile left, right, up, down Goal test: goal state (given) Path cost: 1 per move

10 Basic tree search algorithm Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) 10

11 Search strategies A strategy is defined by picking the order of node expansion 11 Strategies are evaluated along the following dimensions: Completeness: does it always find a solution if one exists? Time complexity: number of nodes generated/expanded 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: maximum branching factor of the search tree d: depth of the least-cost solution m:maximum depth of the state space (may be 1) For instance, a strategy might have time or space complexity O(bd) or O(b d )

12 Uninformed search strategies Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search b = branching factor d = solution depth l = depth limit m = max tree depth C* = cost of optimal ε = min action cost 12

13 Avoiding repeated states Key idea: Maintaining a closed list can be exponentially more efficient 13

14 Best-first search Idea: use an evaluation function for each node estimate of desirability Expand most desirable unexpanded node Implementation: fringe is a queue sorted in decreasing order of desirability Special cases: Greedy search A* search 14

15 Map of Romania Straight line distances provide estimate of cost to goal 15

16 Greedy search Evaluation function h(n) (heuristic) estimate of cost from n to the closest goal E.g., h SLD (n) = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal

17 Greedy search Complete: No Can get stuck in loops Complete in finite space with repeated-state checking Time: O(b m ) but better with a good heuristic Space: O(b m ) keeps all nodes in memory Optimal: No 17

18 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 to goal from n f(n) = estimated total cost of path through n to goal A* search uses an admissible heuristic i.e., h(n) h*(n) where h*(n) is the true cost from n. (Also require h(n) 0, so h(g) = 0 for any goal G.) e.g., h SLD (n) never overestimates the actual road distance One dirty trick Inflating the heuristic 18

19 A* example 393= = =

20 A* Optimality proof Suppose some suboptimal goal G 2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G. f(g 2 ) = g(g 2 ) since h(g 2 ) = 0 g(g 2 ) > g(g) since G 2 is suboptimal f(g) = g(g) since h(g) = 0 f(g 2 ) > f(g) from above h(n) h*(n) since h is admissible g(n) + h(n) g(n) + h * (n) f(n) f(g) 20 Hence f(g 2 ) > f(n), and A * will not have selected G 2 for expansion

21 A* Optimality proof Complete: Yes unless there are infinitely many nodes with f f(g) Time: Exponential in [relative error in h * length of soln.] Space: Keeps all nodes in memory Optimal: Yes A* expands all nodes with f(n) < C A* expands some nodes with f(n) = C A* expands no nodes with f(n) > C 21

22 Admissible heuristics E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) 22 If h 2 (n) h 1 (n) for all n (both admissible) then h 2 dominates h 1 and is better for search Given any admissible heuristics h a, h b, h(n) = max(h a (n); h b (n)), is also admissible and dominates h a, h b

23 Which heuristics are admissible? Consider an 8-connected grid Euclidean distance to goal Manhattan distance to goal Euclidean distance divided by 2 Euclidean distance times 2 Distance to goal along optimal path Which heuristics dominate? 23

24 Relaxed problems Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere then h 1 (n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square then h 2 (n) gives the shortest solution Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem 24

25 Relaxed version of TSP Well-known example: travelling salesperson problem (TSP) Find the shortest tour visiting all cities exactly once Minimum spanning tree can be computed in O(n 2 ) and is a lower bound on the shortest (open) tour This also leads to a factor 2 approximation algorithm (much more on this later) 25

26 Fun questions Iterative deepening A* with a zero heuristic reduces to? A* with a zero heuristic and uniform costs reduces to? 26

27 Important things to understand Completeness Optimality Time/space complexity State and action spaces State transitions Full, partial, and non-observability Informed/admissible heuristics Relaxed problems To look up if you don t understand: Complexity classes (NP-hard, etc.) 27

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