1/12: Problem Solving (Search)

1. 1/12: Problem Solving (Search) • Administrative • Assignment 1 due Friday • Current reading: • Chapters 1-3, Rich & Knight • Chapters 1-3, Touretzky • Chapter 12, Rich & Knight • Today • Search from Chapter #2

2. Problem-Solving in AI • Stages of problem solving • formulate the problem (subjective) • goal formulation • problem formulation • solve the formulated problem (objective)

3. The Classical AI Problem • The agent • no observational power • The environment • no exogenous events or other agents • actions are deterministic transitions • start state is known with certainty • The reward • reach a goal state (sometimes at minimum cost) • for a single trial

4. To describe a problem • Problem definition • Define the state space • Define the actions and their transition functions • define action/path costs • Define the initial state and goal region • Some examples • Eight puzzle • Missionaries and cannibals • Tower of Hanoi (see with emacs “Esc-x hanoi”)

5. M M M M M M ? C C C C C C Two example problems 1 2 7 1 2 3 4 3 6 4 5 6 5 8 7 8

6. Problem solving as graph search • “Every problem is a graph search problem, as long as you can figure out the right graph to search” • Graph vertices = states • Graph arcs = action transitions • Initial vertex and goal vertices identified • Objective: find any/cheapest path from the initial state to any goal state • Note: this is taking us far afield from reactive/observable problems!

7. Graph search in AI • The size of the state space is such that the graph is infinite or at least too big to hold in memory • incremental construction of the graph is the name of the game

8. Graph Search in Theory and Practice

9. Incremental Graph Search: Nondeterministic Version • This is an abstract way to look at the search problem. It highlights where the crucial choice is, and thus where we want to apply “intelligence” • Set N  initial state • Loop • if N is a goal node, terminate with success • choose an action A to apply in N • if there are no such actions, fail • set N  A(N)

10. Nondeterministic Search • All the smarts is in choose • A variety of ways to implement the “algorithm” • most common is to maintain a list of the “frontier” nodes • a heuristic function ranks them according to how “promising” each looks • promising means something like “how long is the shortest path from here to a goal” • at each iteration the highest ranked node is removed from the frontier, checked for goal-hood, its successors generated

11. Search: Lisp Implementation (defun basic-search (initial-state final-state-checker state-generator state-comparator) (really-search (list initial-state) final-state-checker state-generator state-comparator)) Function Inputs Output initial-state state final-state-checkerstateBoolean state-generatorstate (list-of state) state-comparatorstatestateBoolean

12. Lisp Implementation (cont.) (defun really-search (frontier final-state-checker state-generator state-comparator) (cond ((null frontier) NIL) (T (let ((next-state (car frontier))) (cond ((funcall final-state-checker next-state) next-state) (T (let ((new-states (funcall state-generator next-state))) (really-search (sort (append new-states (cdr frontier)) state-comparator) final-state-checker state-generator state-comparator))))))))

13. Search Strategies • Uninformed • breadth first • depth first • iterative deepening • bi-directional • Informed • greedy • A* • Memory bounded • IDA* • SMA*

14. Breadth-First Search • Strategy: always prefer the shortest path • consider all solutions of length k before considering any solution of length k+1 • Advantages • complete: will always find a solution if there is one • Disadvantages • memory intensive: frontier grows exponentially with b and d • always takes exponential time to find long solutions

15. Depth-First Search • Strategy: always prefer the longest path • exhaust a single node on the frontier before considering any of its siblings • Advantages • space efficient: frontier grows only with d and not with b • can find a long solution very quickly (if one chooses well) • Disadvantages • can work on an arbitrary bad path for arbitrarily long • prone to looping (exploring cycles in the graph) • as a practical matter, incomplete • Possible solutions: • loop detection • depth limit

16. Backward and Bi-Directional Search • Search so far has started at initial and generated successors. • Suppose instead we stared at goal and generated predecessors. • or suppose we started both at initial and at goal, and went in both directions • The argument: fan-out versus fan-in • Difficulties • computing predecessors • bookkeeping to determine when a bi-directional search succeeds

17. Iterative-Deepening Search • Strategy • use a bounded depth-first approach • but incrementally increase the depth bound if no solution is found depth frontier size 610348 expansions wasted 1 6 2 31 3 156 4 781 5 3906 6 19531 7 97656 8 488281 9 2441406 20% wasted effort (assuming proper depth were known) 3051754 total expansions

18. Informed Search Methods • “Informed” is exactly the ranking function in the search code • except comparison versus ranking • Usual interpretation: • h’(n)estimated cost of cheapest path from n to any goal • Need to balance the cost so far with expected cost to goal: • g(n) actual minimum cost of getting to n • h’(n)estimated minimum cost getting from n to goal

19. A* Search • An informed method for finding the minimum-cost path from initial to a goal • The ranking function is simply • f’(n) = g(n) + h’(n)estimated minimum cost to a goal • how does this limit the agent’s reward structure?? • What are the implications of getting h’ wrong? • if h’(n)=h(n) for all n • if h’(n)  h(n) for all n but strictly less than for some n • if h’(n) > h(n) for some n

20. A*: Properties of the Cost Estimate • If h’(n) is exactlyh(n) for all n, then the search immediately converges to the optimal solution. • If h'(n)=0 for all n, the search is breadth-first. • If h' never overestimates h, and goal states are “correctly identified” then the first goal node found will be optimal. • (All provided that goal states are correctly identified.) • If h' sometimes overestimates h, then the goal node found may be sub-optimal.

21. A* Search: Concluded • An h’ that never overestimates h is called an admissible search heuristic. • A* search defined to be best-first with an admissible h • Graceful degradation: suppose h’ is not admissible, but doesn’t miss by much? • If the probability that h'(s)>h(s) is less than , then the probability that A* will return an answer that is sub-optimal by more than a factor of  is small.

22. IDA* • IDA* combines Iterative Deepening and A* • recall ID did depth-first for increasing depth bounds d = 0, 1, ... until a solution was found • IDA does depth-first search using f-bounds instead of depth bounds • depth-first(fb) considers all nodes n such that f(n)<= fb • when a node’s f value exceeds fb it is pruned, but • the f-bound for the next iteration is the minimum over the f values for all the pruned states • Problems when f values are closely spaced • alternative is to increment depth bound by some fixed or adaptive 