Searching

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Overview

• Problem-solving agents

• Example Problems

• Searching

• Measuring performance

• Search strategies

• Partial Information

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Problem Solving Agents• Goal-based agent

Agent

Sensors

Actuators

Environm

ent

Percepts

Actions

What the world is like now

What action I should do now

Goals

What my actions do

How the world evolves

State

What will it be like if I do action A

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Problem Solving

• World is made of up discrete states• A state is the set of characteristics for the world

at a given point in time

• Actions change the world from one state to another. S1

S2 S3 S4

Left Forward

Right

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Problem Solving Steps• Goal formulation

• Determine state or a set of states that will satisfy the agent’s goal

• Problem formulation• Determine actions and states to consider

• Search• Consider possible sequences of actions from current state• Return a solution, a sequence of actions that reaches a goal

state• In general, would like the best sequence of actions w.r.t. the

agent’s performance measure.

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Problem Solving Agent Program

Figure 3.1 page 61

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Example, Route Finding

• Consider a system like MapQuest• Provide instructions for driving from one

location to another

• Agent perspective• Agent simulates driving and identifies best

route given performance measure (time, distance, interstates, etc.)

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Example, Route Finding

Agent Type

Performance Measure

Environment Actuators Sensors

Route Finder

Shortest distance, shortest time, most interstate miles and shortest time, etc.

Map with roads, speed limits, etc.

(Simulated) accelerator, steering, etc.

(Simulated)

Speedometer, odometer, road identifier, etc.

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Example, Route Finding

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Example, Route Finding

• What are the states of the world?• The location of the agent, i.e. the Romanian

cities

• What goal does the agent formulate?• To be in Bucharest

• What is the performance measure?• Shortest distance

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Example, Route Finding

• What is the problem formulation?• States to consider are those between Arad and

Bucharest• Actions, go to a specific city

• Must be adjacent to city agent is located in.

• Search• Try all possible paths between Arad and

Bucharest• Select shortest path

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Formal Definition of Problems• An initial state that the agent starts in.• A description of possible actions for the agent from a

given state.• A successor function, which maps a state to a set of

(action, state) pairs

• A goal test, which determines if a state is a goal state• A path cost, which assigns numeric cost to a given

path.• Optimal solution is one that has the least path cost

amongst all solutions

*:)( statesactionsstatesxS

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Formal Definition for Example

• Initial state• In(Arad)

• Successor function• Maps state to adjacent cities

{<Go(Sibiu), In(Sibiu)>, <Go(Timisoara), In(Timisoara)>, <Go(Zerind), In(Zerind)>}

• Goal test• state is In(Bucharest)

• Path cost• Sum of distances along driving path

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Modeling Problems in Java

public interface Problem { /** The initial problem state */ public State getInitialState();

/** A function that returns a mapping of actions to * a state given a state. */ public Map successor(State state);

/** Return whether or not the state is a goal state */ public boolean goalState(State state);

/** Determine the cost of a path */ public Number pathCost(SearchNode node);}

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When is search a good idea?

• Static• World doesn’t change as agent thinks

• Observable• Assumes initial state is known

• Discrete• Must enumerate courses of action

• Deterministic• Must be able to predict the results of actions

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More Problems

• The 8-puzzle

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Problem Formulation

• State• Location of each tile

• Could be represented with an ordered list

0 1 2

3 4 5

6 7 8

Position numbers

[7,2,4,5,0,6,8,3,1]

Keep track of blank location

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Problem Formulation

• Initial state• Any state can be the initial state

• Any permutation of 0 through 8

• Goal test• State = [1,2,3,4,5,6,7,8,0]

• Path cost• Each step costs 1, so total number of steps

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Problem Formulation

• Successor function• View the blank as movable, so any state

reachable by moving blank left, right, up, or down

[7,2,4,5,0,6,8,3,1]

[7,2,4,0,5,6,8,3,1] [7,2,4,5,6,0,8,3,1] [7,0,4,5,2,6,8,3,1] [7,2,4,5,3,6,8,0,1]

L R U D

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Successor Functionb = position of blank;successors = {}; // empty setif (b-1 % 3 != 2) { create new state by swapping state[b] and state[b-1]; add new state to successors;}if (b+1 % 3 != 0) { create new state by swapping state[b] and state[b+1]; add new state to successors;}if (b-3 >= 0) { create new state by swapping state[b] and state[b-3]; add new state to successors;}if (b+3 < 9) { create new state by swapping state[b] and state[b+3]; add new state to successors;}return successors;

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More Problems

• The phasor measurement unit (PMU) problem

• Consider power system graphs (PSGs)• Busses (nodes)• Lines (edges)

• PMUs observe characteristics of a PSG for monitoring

• Place the fewest number of PMUs on a PSG

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How A PMU Observes a PSG

• Any bus with a PMU is observed

• Any line incident with a bus containing a PMU is observed

• Any bus incident with an observed line is observed

• Any line between two observed busses is observed

• If all the lines incident with an observed bus are observed, save one, then all of the lines incident to that bus are observed.

These all follow from physical laws.

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Problem Formulation

• State• Locations of PMUs and which parts of PSG are observed

• Initial state• No PMUs and nothing observed

• Successor function• States obtained by placing a PMU on a node without a PMU

• Goal test• The entire PSG is observed

• Path cost• The number of PMUs placed

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Searching

• Problem solving involves searching through the state space

• The state space is organized with a state tree• Start with initial state, then apply successor

function repeatedly• More generally, we have a search graph, as states

can be reached by different paths– 8-puzzle: Consider moving blank left and then right

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Search Nodes

• Nodes are a data structure with 5 components

Action: rightDepth: 6Path-Cost: 6

Parent-Node

State

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Search Nodes

public class SearchNode { private SearchNode parent; private State state; private Action action; private int depth; private Number cost;

// Include any necessary constructors, access, setter // and other methods needed by the node class.}

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Tree Search

Expand

Goal test?

Fringe - set of states to be expanded

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Tree Search

• A search strategy determines the order in which nodes are expanded

Figure 3.8 page 71

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Organizing the Fringe

• The fringe is implemented using a queue• Actually a priority queue

• Operations

MakeQueue(element, …); // create queue containing elementsEmpty?(queue); // return true if queue is emptyFirst(queue); // return first element, but leave queuedRemoveFirst(queue); // return first element, dequeueInsert(element,queue); // insert in ordered location into queueInsertAll(elements, queue); // insert multiple elements.

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Tree Search

Figure 3.9 page 72

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Tree Search in Javapublic class TreeSearch { private PriorityQueue fringe; private Problem problem;

/** * Construct a tree search object for the specified problem. */ public TreeSearch(Problem problem, PriorityQueue fringe) { this.problem = problem; this.fringe = fringe; /* Create the root of the search tree */ SearchNode root = new SearchNode(null, // parent problem.getInitialState(), // state null, // action 0, // depth null); // cost fringe.add(root); }

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Tree Search in Java /** * Carry out the search, returning a collection of * actions to perform. */ public Collection execute() { while (!fringe.isEmpty()) { SearchNode node = fringe.removeFirst(); if (problem.goalState(node)) { return solution(node); } expand(node); } return null; // throw exception instead? }

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Tree Search in Java /** * Expand the current search node. */ public void expand(SearchNode node) { Map successors; successors = problem.successor(node.getState()); for (Iterator iter = successors.keySet().iterator(); iter.hasNext(); ) { Action act = (Action) iter.next(); State state = (State) successors.get(act); SearchNode newNode = new Node(node, state, act, node.getDepth() + 1); problem.pathCost(newNode); fringe.add(newNode); } }

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Tree Search in Java /** * Return a sequence of actions for the agent to take. */ public Collection solution(SearchNode node) { LinkedList actions = new LinkedList();

while (node != null) { actions.addFirst(node.getAction()); node = node.getParent(); } return actions; }}

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Measuring Problem Solving Performance

• Completeness• Is the algorithm guaranteed to find a solution?

• Optimality• Does the strategy find an optimal solution?

• Time complexity• How long does it take to find a solution?

• Space complexity• How much memory is needed to perform the search?

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Complexity and Big-Oh

• Read Appendix A.• T(n) is the complexity of an algorithm.• O(f(n)) is defined by

T(n) is O(f(n)) if T(n) <= kf(n) for some k, for all n > n0

• What this means is that the growth in time (or space) complexity is bounded above by some constant times f(n).

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Complexity and Big-Oh

• Example• If T(n) = 2n + 2, then T(n) is O(n) using k = 3

and n0 = 2.

• Use O() to say something about long term behavior of algorithms• O(n) is always better than O(n2) as n

approaches infinity.

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Complexity and Search

• What is n for search strategies?• Usually a measure of the search tree/graph size

• Search tree size is measured by• b, the branching factor, the maximum number

of successors of any node

• d, the depth of the shallowest goal node

• m, the maximum length of any path

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Complexity

• Time complexity is in terms of the number of nodes expanded

• Space complexity is in terms of the maximum number of nodes stored in memory

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Cost

• Search cost, which is generally time complexity but may also include space complexity

• Total cost, which is the search cost plus the path cost of the solution

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Search Strategy

• An uninformed strategy is given no information other than the problem definition.• The algorithms are given no insight into the structure of

the problem

• All non-goal states are equal

• Informed strategies take advantage of additional a priori knowledge of the problem• Identify more promising non-goal states.

• Heuristics, pruning, …

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Uninformed Search Strategies

• Breadth-first search• Uniform-cost search

• Depth-first search

• Depth-limited search

• Iterative deepening search

• Bidirectional search

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Breadth First Search

• Fringe is organized in FIFO queue

1

2 3 4

5 6

1

2

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Breadth First Search

• Complete?• Yes

• Optimal?• Finds shallowest goal node. Only if step costs are all

the same

• Time complexity?• O(bd+1)

• Space complexity?• O(bd+1)

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Uniform Cost Search

• Order fringe by increasing path cost

A

S 140 T 118 Z 75

O 141 A 150

A

Z 75

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Uniform Cost Search

• Complete?• Yes, if each step costs is at least e > 0

• Optimal?• Yes, as above

• Time complexity?• O(bc), where C is cost of optimal solution c = ceil(C/e)• c could be larger than d!

• Space complexity?• O(bc)

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Depth First Search

• Fringe organized in descending order of depth (i.e. deepest node first)

A

S 140 T 118 Z 75

R 220 F 239

A

S140

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Depth First Search• Complete?

• No.

• Optimal?• No.

• Time complexity?• O(bm)

• Space complexity?• O(bm)

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Depth First Search

• Can reduce memory using backtracking• Store only one successor at a time

• Go back if path fails and generate next successor

• Feasible if actions can be easily undone

• Good for problems with large state descriptions.

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Depth-limited search• Organize as depth first, but limit the depth in the tree

searched to level l• Depth-first is when l is infinity

• Complete?• Only if l >= d

• Optimal• Only if l = d

• Time• O(bl)

• Space• O(bl)

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Iterative Deepening Search

• Execute depth limited search for l = 1, 2, 3, …

• Complete?• Yes

• Optimal• Yes, if step costs the same

• Time Complexity?

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Time Complexity of IDS

• Derived on the board.• N(IDS) = (d)b + (d-1)b2 + …+(1)bd

• N(BFS) = b + b2 + … + bd + (bd+1-b)

• Time complexity• O(bd)

• Space complexity• O(bd)

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Bidirectional Search

• Search both forward and backward• Need to have a predecessor function

• Complete• Yes, if BFS used

• Optimal• Yes, with identical step costs and BFS used

• Time• O(bd/2)

• Space• O(bd/2)

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Repeated States

• Repeated states can turn a linear problem into an exponential one!

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Graph Search

• States are open (not visited) or closed (visited)

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Graph Search

• Uses more memory• Proportional to size of state space

• DFS and IDS no longer linear space

• Implement closed list with hash table

• Could lose optimality• Could arrive at a state by a more expensive path

• Particularly a problem with IDS

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Partial Information

• Sensorless problems• Agent doesn’t know initial state

• Each action could lead to several possible successor states

• Contingency problems• Partially observable environments

• Nondeterministic actions

• Exploration• States and actions are unknown

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Sensorless Problems

• The initial state of the agent is completely unknown• One of several possible states

• Agent knows available actions, but can’t see results.

• Can we still design an agent that performs rationally?

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View sets of statesas a single worldstate.

Senseless Monkey

Down

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Exercise (in class)

• Draw the complete state diagram for the senseless monkey. Ignore sleep action.

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Answer

D U

E

U

E

EE

E

U

U

U

U

D

D

D

D

D

E

U,ED,ED

D

D,EU,E

U

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Sensorless Problems

• A set of states in the sensorless problem is a goal state if every state in the set is a goal state

• In general, if the original state space has S states, the sensorless state space has 2S possible states (power set). Not all may be reachable though.

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Contingency Problems

• Information is gained from sensors after acting

• Example: Lawn environment. Mower doesn’t know locations of all obstacles.

• Formulate action sequences like• [Counter, if (Obstacle) then Counter else

Forward, …]• Handled with planning (Chapter 12).

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Summary

• Represent problems in search spaces• Nodes represent physical states• Edges represent actions taken

• Problems have 4 components• Initial state• Description of actions (successor function)• Goal Test• Path cost (e.g., via step cost per action)

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Summary

• Search strategies defined by the order fringe nodes are expanded• Priority queue

• In general, iterative deepening is best approach• However, should study characteristics of each problem

to select best strategy

• Sensorless problems can be solved by mapping to sets of physical states