Informed search algorithms. Section 3.5 Russell & Norvig

Transcription

1 Informed search algorithms Section 3.5 Russell & Norvig

2 Outline Review Informed search Greedy search 2nd Term 2011 Informed Search - Lecture 1 2

3 Goals of this part of the Course Primary: Show how search permeates much of AI problem-solving puzzles & games scheduling problems planning Formulate search problems Create heuristics using abstraction Reduce search needed to solve problems 2nd Term 2011 Informed Search - Lecture 1 3

4 The Importance of Research in Search Search: core AI problem-solving approach. However, search is intractable. Improvements to search algorithms affect many other areas of AI. 2nd Term 2011 Informed Search - Lecture 1 4

5 Ubiquity of Search Solving puzzles Planning Playing games Machine Learning Pattern matching Satisfying constraints Vision processing Almost all problem-solving 2nd Term 2011 Informed Search - Lecture 1 5

6 State Space Search Search is done through space of states. Edges connect states and have associated costs and direction. Problem is specified by an initial state and a goal test. Solution: path from initial state to a state that satisfies the goal test. Solution cost = sum of edge costs along solution path. 2nd Term 2011 Informed Search - Lecture 1 6

7 Formal Definition of solution to problem /* solution(+problem,?solution) A seq of states is a solution to a problem iff The first state in the seq is the init state of the problem, The last state in the seq satisfies the goal test, & each state in the seq is a neighbor of its preceding state */ solution(problem(state, Goal), [State]) :- call(goal, State). %% executes Goal(State) solution(problem(state, Goal), [State RestOfSolution]) :- neighbor(state, Neighbor), solution(problem(neighbor, Goal), RestOfSolution). The above definition is a valid prolog program which can both check whether a given list of states is a solution to a given problem or if just given the problem will generate a solution to it. Need to define domain (neighbor/1) and your goal predicate Goal/1. 2nd Term 2011 Informed Search - Lecture 1 7

8 Example Domain Definition: neighbor(losangeles, sanfrancisco). neighbor(losangeles, sandiego). neighbor(sanfrancisco, portland). neighbor(sanfrancisco, lasvegas). neighbor(portland, seattle). Goal Definition: reachedhome(seattle).?- solution(problem(losangeles, reachedhome), Solution). Solution = [losangeles,sanfrancisco,portland,seattle]? 2nd Term 2011 Informed Search - Lecture 1 8

9 Run through example do example in emacs under SICStus 2nd Term 2011 Informed Search - Lecture 1 9

10 Search Space 2nd Term 2011 Informed Search - Lecture 1 10

11 What happens if clauses in different order 2nd Term 2011 Informed Search - Lecture 1 11

12 Tree Search Keeps record of current path and choice points along path (to visit if current path abandoned). [Can check for duplicate states along current path, avoid loops.] No global duplicate state checking. When goal state is found, solution is simply current path. 2nd Term 2011 Informed Search - Lecture 1 12

13 Naive solution implementation Prolog has its own search procedure for executing a program: depth-first search. Our naive solution s search strategy is Prolog s and has all the advantages & disadvantages of depth-first search. 2nd Term 2011 Informed Search - Lecture 1 13

14 Status of Tree Search Advantages: Only needs to store current path Linear memory costs Can use simpler logic (lower costs per node) Disadvantages Non-optimal solution Repeats search for duplicate states Incomplete (for infinite graphs) 2nd Term 2011 Informed Search - Lecture 1 14

15 Graph Search Primarily, does a type of breadth-first search. Does global check for duplicate states. Keeps whole search graph in memory. When goal state is found, solution needs to be extracted from search graph. 2nd Term 2011 Informed Search - Lecture 1 15

16 Graph search Notes: 1. Fringe is the set of leaf nodes 2. Remove-Front is the search strategy 3.Avoid redundant searches for duplicate states 2nd Term 2011 Informed Search - Lecture 1 16

17 Graph version of solution /* solution(+problem, -Solution) */ solution(problem(initialstate, Goal), Solution) :- solution(goal, [node(initialstate, nil)], [ ], Solution). /* solution(+goal, +Fringe, +Closed, -Solution) */ solution(goal, [node(state, ParentState) _], Closed, Solution) :- call(goal, State), extractsolution(parentstate, Closed, [State], Solution). solution(goal, [node(state, Parent) RestNodes], Closed, Solution) :- findall(neighbornode, newneighbornode(state, Closed, NeighborNode), NeighboringNodes), updateclosed(state, Closed, NewClosed), orderfringe(restnodes, NeighboringNodes, NewFringe), solution(goal, NewFringe, [node(state, Parent) NewClosed], Solution).d 2nd Term 2011 Informed Search - Lecture 1 17

18 Status of Graph Search Possible Advantages: Complete Optimal Only searches subspaces once These advantages depend upon strategy Disadvantages: Exponential memory costs More complex logic 2nd Term 2011 Informed Search - Lecture 1 18

19 Outline Review Best-first search Greedy search 2nd Term 2011 Informed Search - Lecture 1 19

20 Search strategies A search strategy is defined by picking the order of node expansion Let g(n) be the distance n s state is from the initial state. Depth-first search strategy is pick node with highest g-value. Breadth-first search strategy is pick node with lowest g-value. 2nd Term 2011 Informed Search - Lecture 1 20

21 Best-first search strategy Given a set of nodes on the fringe of a search, which one is best to expand next? Based on what criteria? Criteria: expand best nodes first, i.e., those along an optimal solution path How do we do that? Use additional information to suggest such nodes. 2nd Term 2011 Informed Search - Lecture 1 21

22 Informed Search Strategies Informed Search Strategies use information beyond the problem desc. We will only look at functions that guess distance from a state to nearest goal state. h(n) is the function that guesses how far n s state is from its nearest goal state. 2nd Term 2011 Informed Search - Lecture 1 22

23 Romania with step costs in km 2nd Term 2011 Informed Search - Lecture 1 23

24 Best-first search Idea: use a function f(n) for each node f(n) is an estimate of "desirability of a node Expand most desirable unexpanded node Implementation: Order the nodes in fringe in decreasing order of desirability (normally, higher f is then less desirable) Uninformed Search: Depth-first: f(n) = -g(n) Breadth-first: f(n) = g(n) 2nd Term 2011 Informed Search - Lecture 1 24

25 Best-first informed search strategies Greedy Search A* Search Iterative Deepening A* (IDA*) Weighted A* Search 2nd Term 2011 Informed Search - Lecture 1 25

26 Outline Review Best-first search Greedy search 2nd Term 2011 Informed Search - Lecture 1 26

27 Greedy search Evaluation function: f(n) = h(n) h(n) = estimate of cost from n to 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 2nd Term 2011 Informed Search - Lecture 1 27

28 Greedy best-first search example 2nd Term 2011 Informed Search - Lecture 1 28

29 Greedy best-first search example 2nd Term 2011 Informed Search - Lecture 1 29

30 Greedy best-first search example 2nd Term 2011 Informed Search - Lecture 1 30

31 Greedy best-first search example 2nd Term 2011 Informed Search - Lecture 1 31

32 Why greedy search is attractive With a decent enough heuristic, goes almost directly to goal. Best case: time and space are linear So, why not always do greedy search? 2nd Term 2011 Informed Search - Lecture 1 32

33 Properties of greedy best-first search Complete? No, has same problem with infinite graphs as depth-first search Time? O(b m ), but a good heuristic can give dramatic improvement Space? O(b m ) -- keeps all nodes in memory Optimal? No 2nd Term 2011 Informed Search - Lecture 1 33

34 Greedy Search in Prolog /* solution(+heuristic, +Goal, +Fringe, +Closed, -Solution) */ solution(_heuristic, Goal, [Node _], Closed, Solution) :- node(node, State, ParentState, _FValue), test(goal, State), extractsolution(parentstate, Closed, [State], Solution). solution(heuristic, Goal, [Node RestNodes], Closed, Solution) :- nodestate(node, State), findall(neighbornode, newneighbornode(state, Heuristic, [Node Closed], NeighborNode), NeighboringNodes), orderfringe(restnodes, NeighboringNodes, NewFringe), solution(heuristic, Goal, NewFringe, [Node Closed], Solution). 2nd Term 2011 Informed Search - Lecture 1 34

35 Summary Search strategy defines a traversal of the search space, e.g., pick lowest f(n). Informed search strategies use information outside of problem description. One such type of information is estimated distance to nearest goal: h(n). Greedy search: f(n) = h(n). 2nd Term 2011 Informed Search - Lecture 1 35

36 Challenge Can you create state space representation for following domains: scheduling taxi service in Auckland playing chess getting a degree at UofA enjoying your life You need to represent states of the world, actions that change states, problems, and solutions. 2nd Term 2011 Informed Search - Lecture 1 36

37 Next Time Look at: A* search IDA* Heuristics 2nd Term 2011 Informed Search - Lecture 1 37