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INGENIERA ELECTRNICA YTELECOMUNICACIONES

REDES NEURONALES

UNIVERSIDAD DE CUENCAECUADOR

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Ing. Kenneth S. Palacio Baus, [email protected]

2014

Captulo 3:

Solving problems by searching


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OutlineOutlineProblem-solving agentsProblem typesProblem formulation


Basic search algorithms

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2014 [email protected]


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ProblemProblem--solving agentssolving agentsSolving problems by searching

Remember:

Intelligent agents are supposed to maximize their performancemeasure.

This cha ter describes one kind of oal-based a ent.


We limit ourselves to the simplest kind of task environment, forwhich the solution to a problem is always:

A fixed sequence of actions .

We will use the concepts of asymptoticcomplexity --O( ) notation-- andNP-completeness.

Investigar

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Goal formulation , based on the current situation and theagents performance measure, is the first step in problem solving.


Problem formulation is the process of deciding what actionsand states to consider, given a goal.

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2014 [email protected] 5


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Example: RomaniaExample: RomaniaOn holiday in Romania; currently in Arad.Flight leaves tomorrow from Bucharest

Formulate goal :

be in Bucharest


Formulate problem : states : various cities actions : drive between cities

Find solution : sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

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Example: RomaniaExample: RomaniaSolving problems by searching

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Suppose the agent has a map of Romania. The point of a map is toprovide the agent with information about the states it might get itself intoand the actions it can take.


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Example: RomaniaExample: RomaniaSolving problems by searching

An agent with several immediate optionsof unknown value can decide what to doby first examining future actions that


eventually lead to states of known value.

The environment is known, so the agent

knows which states are reached by eachaction.


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Example:Example: RomaniaRomaniaWellWell--defined problems anddefined problems andsolutionssolutions


A problem can be defined formally by fivecomponents:


1. The initial state that the agent starts in.For example, the initial state for our agent

in Romania might be described asIn(Arad).


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2. A description of the possible actions available to the agent.Given a particular state s, ACTIONS(s) returns the set of actionsthat can be executed in s. We say that each of these actions is


.

For example, from the state In(Arad), the applicable actions are{Go(Sibiu),Go(Timisoara),Go(Zerind)}.

3. A description of what each action does; the formal name for thisis the transition model , specified by a function RESULT(s,a) thatreturns the state that results from TRANSITION MODEL doingaction a in state s.


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We also use the term successor to refer to any statereachable SUCCESSOR from a given state by a singleaction.


For example, we haveRESULT( In(Arad), Go(Zerind)) = In(Zerind).

Together, the initial state, actions, and transition model implicitly

define the state space of the problem the set of all statesreachable from the initial state by any sequence of actions.The state space forms a directed network or graph in which thenodes are states and the links between nodes are actions.


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4. A path in the state space is a sequence of states connected by asequence of actions.


The goal test , which determines whether a given state is a goalstate.

Sometimes the GOAL TEST is an explicit set of possible goalstates, and the test simply checks whether the given state is oneof them.

The agents goal in Romania is the singleton set{In(Bucharest)}..


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A path cost function that assigns a numeric cost to each path.The problem-solving agent chooses a cost function that reflects itsown performance measure.


or t e agent try ng to get to uc arest, t me s o t e essence, so t e cost oa path might be its length in kilometers.

The step cost of taking action a in state s to reach state s isdenoted by c(s, a, s ).


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5. Asolution to a problem is an action sequence that leads from


.

Solution quality is measured by the path cost function, and anoptimal solution has the lowest path cost among all solutions.


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ProblemProblem types:types:Properties of the environmentProperties of the environment

Deterministic, fully observable single-state problem Agent knows exactly which state it will be in; solution is a sequence So, each action has exactly one outcome.

Non-observable sensorless problem (also called

conformant roblem


Agent may have no idea where it is; solution is a sequence

Non-deterministic and/or partially observablecontingency problem

percepts provide new information about current state solution is a contingent plan or a policy often interleave search, execution

Unknown state space exploration problem



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Example: vacuum worldExample: vacuum world



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Single-state, start in #5.Solution?





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Single-state, start in #5.Solution? [Right, Suck]

Sensorless, start in{1,2,3,4,5,6,7,8} e.g.,



g t goes to , , , Solution?



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Example: vacuum worldExample: vacuum worldSensorless, start in{1,2,3,4,5,6,7,8} e.g.,Rightgoes to { 2,4,6,8}Solution?[Right,Suck,Left,Suck]


Contingency Nondeterministic: Suckmay

dirty a clean carpet Partially observable: location, dirt at current location. Percept: [L, Clean],i.e., start in #5 or #7

Solution?



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SingleSingle--state problem formulationstate problem formulationA problem is defined by four items:

1. initial state e.g., "at Arad

2. actions or successor function S(x)= set of actionstate pairs e.g., S(Arad) ={, }


3. goal test , can be explicit, e.g., x = "at Bucharest" implicit, e.g., Checkmate(x)

4. path cost (additive) e.g., sum of distances, number of actions executed, etc. c(x,a,y)is the step cost , assumed to be 0

A solution is a sequence of actions leading from the initial state to agoal state



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Selecting a state spaceSelecting a state spaceReal world is absurdly complex

state space must be abstracted for problem solving

(Abstract) state = set of real states

(Abstract) action = complex combination of real actions

" "


. ., , ,rest stops, etc.

For guaranteed realizability, any real state "in Arad must get tosome real state "in Zerind

(Abstract) solution = set of real paths that are solutions in the real world

Each abstract action should be "easier" than the original problem



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Vacuum world state space graphVacuum world state space graph


states?actions?goal test?path cost?



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Vacuum world state space graphVacuum world state space graph


states? integer dirt and robot location

actions? Left, Right, Suckgoal test? no dirt at all locationspath cost? 1 per action



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Example: The 8Example: The 8--puzzlepuzzleSolving problems by searching

states?actions?goal test?path cost?



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Example: The 8Example: The 8--puzzlepuzzleSolving problems by searching

states? locations of tilesactions? move blank left, right, up, downgoal test? = goal state (given)

path cost? 1 per move

[Note: optimal solution of n-Puzzle family is NP-hard]



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Example:Example: The 8The 8--puzzlepuzzleSolving problems by searching



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Example: robotic assemblyExample: robotic assemblySolving problems by searching

states? : real-valued coordinates of robot jointangles, parts of the object to be assembled

actions?: continuous motions of robot jointsgoal test? : complete assemblypath cost? : time to execute



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Some RealSome Real--world problemsworld problemsSolving problems by searching

Touring problems

Traveling sales person problem (TSP)


Robot navigation

Automatic assembly sequencing


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SearchSearch algorithmsalgorithmsThe process of looking for a sequence ofactions that reaches the goal is calledsearch.


A search algorithm takes a problem asinput and returns a solution in the formof an action sequence.

Once a solution is found, the actions it recommends canbe carried out. This is called the execution phase.



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Tree search algorithmsTree search algorithmsA solution is an action sequence, sosearch algorithms work by consideringvarious possible action sequences.


The possible action sequences starting at the initialstate form a search tree with the initial state at theroot;

The branches are actions and the nodes correspond tostates in the state space of the problem



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Tree search exampleTree search exampleSolving problems by searching



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Implementation: general tree searchImplementation: general tree search




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Search-Tree includes the path from Arad to Sibiu andback to Arad again!

We say that In(Arad) is a repeated state in the searchtree, generated in this case by a loopy path.


Loops can cause certain algorithms to fail, makingotherwise solvable problems unsolvable.

Fortunately, there is no need to consider loopy paths.

Loopy paths are a special case of the more generalconcept of redundant paths, which exist whenever there ismore than one way to get from one state to another.


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Theres never any reason to keep more than onepath to any given state, because any goal statethat is reachable by extending one path is alsoreachable by extending the other.


Algorithms that forget their history are doomedto repeat it.

The way to avoid exploring redundant paths is to

remember where one has been.


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We augment the TREE-SEARCH algorithm with a datastructure called the explored set (also known as theclosed list ), which remembers every expanded node.

Newly generated nodes that match previously generatednodesones in the explored set or the frontiercan be


scar e ns ea o e ng a e o e ron er.


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Graph searchGraph searchSolving problems by searching



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Implementation:Implementation: statesstates vs. nodesvs. nodesSolving problems by searching



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Implementation:Implementation: statesstates vs. nodesvs. nodes

A state is a (representation of) a physical configurationA node is a data structure constituting part of a searchtree includes state , parent node , action , path cost g(x),depth


The Expand function creates new nodes, filling in thevarious fields and using the SuccessorFn of theproblem to create the corresponding states.



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Implementation:Implementation: statesstates vs. nodesvs. nodesSolving problems by searching

Given the components for a parent node, it is easy to see how tocompute the necessary components for a child node.

The function CHILD-NODE takes a parent node and an actionand returns the resulting child node:

502014 kenneth palacio@ucuenca edu ec

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