Introduction to Artificial Intelligence Class 1 Planning & Search

1. Introduction to Artificial Intelligence Class 1 Planning & Search Henry Kautz Winter 2007

2. Outline of Course • Heuristic search • Constraint satisfaction • Automated planning • Propositional logic • First-order logic and logic programming • Knowledge engineering • Probabilistic reasoning: directed and undirected graphical models • Learning graphical models • Decision trees and ensemble methods • Neural networks

3. Planning • Input • Description of initial state of world • Description of goal state(s) • Description of available actions • Optional: Cost for each action • Output • Sequence of actions that converts the initial state into a goal state • May wish to minimizelength or cost of plan

4. Classical Planning • Atomic time • Deterministic actions • Complete knowledge • No numeric reward (just goals) • Only planner changes world

5. Route Planning • State = intersection • Operators = block between intersections • Operator cost = length of block

6. Blocks World • Control a robot arm that can pick up and stack blocks. • Arm can hold exactly one block • Blocks can either be on the table, or on top of exactly one other block • State = configuration of blocks • { (on-table G), (on B G), (clear B), (holding R) } • Operator = pick up or put down a block • (put-down R) put on table • (stack R B) put on another block

7. State Space • Planning = Finding (shortest) paths in state graph put-down(R) stack(R,B) pick-up(R) pick-up(G) stack(G,R)

8. STRIPS Representation • (define (domain prodigy-bw) • (:requirements :strips) • (:predicates • (on ?x ?y) • (on-table ?x) • (clear ?x) • (arm-empty) • (holding ?x))

9. Problem Instance • (define (problem bw-sussman) • (:domain prodigy-bw) • (:objects A B C) • (:init • (on-table a) (on-table b) (on c a) • (clear b) (clear c) (arm-empty)) • (:goal • (and (on a b) (on b c)))) goal may be a partial description

10. Operator Schemas • (:action stack • :parameters (?obj ?under_obj) • :precondition • (and (holding ?obj) (clear ?under_obj)) • :effect • (and (not (holding ?obj)) • (not (clear ?under_obj)) • (clear ?obj) • (arm-empty) • (on ?obj ?under_obj))) add effects – make true delete effects – make false

11. Blocks World Blackbox Planner Demo

12. Search Algorithms • Today: Space-State Search • Depth-First • Breadth-First • Best-First • A* • Next Class: • Local Search • Constraint Satisfaction

13. A General Search Algorithm • Search( Start, Goal_test, Criteria ) • Open = { Start }; Closed = { }; • repeat • if (empty(Open)) return fail; • select Node from Open using Criteria; • if (Goal_test(Node)) return Node; • for each Child of node do • if (Child not in Closed) • Open = Open U { Child }; • Closed = Closed U { Node }; Closed list optional

14. Breadth-First Search • Search( Start, Goal_test, Criteria ) • Open: fifo_queue; • Closed: hash_table; • enqueue(Start, Open); • repeat • if (empty(Open)) return fail; • Node = dequeue(Open); • if (Goal_test(Node)) return Node; • for each Child of node do • if (not find(Child, Closed)) • enqueue(Child, Open) • insert(Child, Closed) Criteria = shortest distance from Start

15. Depth-First Search • Search( Start, Goal_test, Criteria ) • Open: stack; • Closed: hash_table; • push(Start, Open); • repeat • if (empty(Open)) return fail; • Node = pop(Open); • if (Goal_test(Node)) return Node; • for each Child of node do • if (not find(Child, Closed)) • push(Child, Open) • insert(Child, Closed) Criteria = longest distance from Start

16. Best-First Search • Search( Start, Goal_test, Criteria ) • Open: priority_queue; • Closed: hash_table; • enqueue(Start, Open, heuristic(Start)); • repeat • if (empty(Open)) return fail; • Node = dequeue(Open); • if (Goal_test(Node)) return Node; • for each Child of node do • if (not find(Child, Closed)) • enqueue(Child, Open, heuristic(Child)) • insert(Child, Closed) Criteria = shortest heuristic estimate of distance to goal

17. Properties • Depth First • Simple implementation (stack) • Might not terminate • Might find non-optimal solution • Breadth First • Always terminates if solution exists • Finds optimal solutions • Visits many nodes • Best First • Always terminates if heuristic is “reasonable” • Visits many fewer nodes • May find non-optimal solution

18. Best-First with Manhattan Distance ( x+  y) Heuristic 53nd St 52nd St G 51st St S 50th St 10th Ave 9th Ave 8th Ave 3rd Ave 2nd Ave 7th Ave 6th Ave 5th Ave 4th Ave

19. Non-Optimality of Best-First 53nd St 52nd St S G 51st St 50th St 10th Ave 9th Ave 8th Ave 3rd Ave 2nd Ave 7th Ave 6th Ave 5th Ave 4th Ave

20. A* • Criteria: minimize (distance from start) + (estimated distance to goal) • Implementation: priority queue • f(n) = g(n) + h(n) • f(n) = priority of a node • g(n) = true distance from start • h(n) = heuristic distance to goal

21. Optimality of A* • Suppose the estimated distance is alwaysless than or equal to the true distance to the goal • heuristic is a lower bound • heuristic is admissible • Then: when the goal is removed from the priority queue, we are guaranteed to have found a shortest path!

22. Maze Runner Demo

23. Observations on A* • Perfect heuristic: If h(n) = h*(n) (true distance) for all n, then only the nodes on the optimal solution path will be expanded. • Null heuristic: If h(n) = 0 for all n, then this is an admissible heuristic and A* acts like breath-first search. • Comparing heuristics: If h1(n)  h2(n)  h*(n) for all non-goal nodes, then h2 is as least as good a heuristic as h1 • Every node expanded by A* using h2 is also expanded by A* using h1 • if h1(n)<h1(n) for some n, then h2 is stronger than h1 • Combining heuristics: if h1(n) and h2(n) are admissible, then h3(n) = MAX(h1(n),h2(n)) is admissible • Why?

24. Search Heuristics • “Optimistic guess” at distance to a solution • Some heuristics are domain specific • Manhattan distance for grid-like graphs • Euclidean distance for general road maps • Rubik’s Cube • Admissible, but weak: # cubits out of place / 8 • Better: MAX( Sum( Manhattan distance edge cubits )/4, Sum( Manhattan distance corner cubits )/4 )

25. Planning Heuristics • A useful non-admissible heuristic for planning is the number of goals that need to be achieved • Why not admissible? • Good admissible heuristics for planning can be created by relaxing the operators, e.g.: • Eliminate preconditions, or • Eliminate negative preconditions & effects • Use the length of the solution to the relaxed problem as a heuristic for the length of the solution to the original problem

26. Homework Shakey the robot has to bring coffee to Prof. Kautz. In order to make the coffee, Shakey will need to gather coffee filters, coffee, and Prof. Kautz's mug and bring them to the coffee maker in the kitchen. The coffee and filters are in the supply room, but it is locked. To unlock the supply room, Shakey will need to get the key from Prof. Kautz's office. Represent this problem in STRIPS notation What is the true value of the start state? What is the heuristic value of the start start, based on rhe relaxed problem with no preconditions on actions?