Last time: ProblemSolving

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Last time Problem-Solving

• Problem solving
• Goal formulation
• Problem formulation (states, operators)
• Search for solution
• Problem formulation
• Initial state
• ?
• ?
• ?
• Problem types
• single state accessible and deterministic
  environment
• multiple state ?
• contingency ?
• exploration ?

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Last time Problem-Solving

• Problem solving
• Goal formulation
• Problem formulation (states, operators)
• Search for solution
• Problem formulation
• Initial state
• Operators
• Goal test
• Path cost
• Problem types
• single state accessible and deterministic
  environment
• multiple state ?
• contingency ?
• exploration ?

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Last time Problem-Solving

• Problem solving
• Goal formulation
• Problem formulation (states, operators)
• Search for solution
• Problem formulation
• Initial state
• Operators
• Goal test
• Path cost
• Problem types
• single state accessible and deterministic
  environment
• multiple state inaccessible and deterministic
  environment
• contingency inaccessible and nondeterministic
  environment
• exploration unknown state-space

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Last time Finding a solution
Solution is ??? Basic idea offline, systematic
exploration of simulated state-space by
generating successors of explored states
(expanding)

• Function General-Search(problem, strategy)
  returns a solution, or failure
• initialize the search tree using the initial
  state problem
• loop do
• if there are no candidates for expansion then
  return failure
• choose a leaf node for expansion according to
  strategy
• if the node contains a goal state then return
  the corresponding solution
• else expand the node and add resulting nodes to
  the search tree
• end

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Last time Finding a solution
Solution is a sequence of operators that bring
you from current state to the goal state. Basic
idea offline, systematic exploration of
simulated state-space by generating successors of
explored states (expanding).

• Function General-Search(problem, strategy)
  returns a solution, or failure
• initialize the search tree using the initial
  state problem
• loop do
• if there are no candidates for expansion then
  return failure
• choose a leaf node for expansion according to
  strategy
• if the node contains a goal state then return
  the corresponding solution
• else expand the node and add resulting nodes to
  the search tree
• end

Strategy The search strategy is determined by ???
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Last time Finding a solution
Solution is a sequence of operators that bring
you from current state to the goal state Basic
idea offline, systematic exploration of
simulated state-space by generating successors of
explored states (expanding)

• Function General-Search(problem, strategy)
  returns a solution, or failure
• initialize the search tree using the initial
  state problem
• loop do
• if there are no candidates for expansion then
  return failure
• choose a leaf node for expansion according to
  strategy
• if the node contains a goal state then return
  the corresponding solution
• else expand the node and add resulting nodes to
  the search tree
• end

Strategy The search strategy is determined by
the order in which the nodes are expanded.
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Last time search strategies

• Uninformed Use only information available in the
  problem formulation
• Breadth-first
• Uniform-cost
• Depth-first
• Depth-limited
• Iterative deepening
• Informed Use heuristics to guide the search
• Best first
• A

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Evaluation of search strategies

• Search algorithms are commonly evaluated
  according to the following four criteria
• Completeness does it always find a solution if
  one exists?
• Time complexity how long does it take as a
  function of number of nodes?
• Space complexity how much memory does it
  require?
• Optimality does it guarantee the least-cost
  solution?
• Time and space complexity are measured in terms
  of
• b max branching factor of the search tree
• d depth of the least-cost solution
• m max depth of the search tree (may be
  infinity)

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Last time uninformed search strategies

• Uninformed search
• Use only information available in the problem
  formulation
• Breadth-first
• Uniform-cost
• Depth-first
• Depth-limited
• Iterative deepening

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Um algoritmo robusto e limpo

Function UniformCost-Search(problem, Queuing-Fn)
returns a solution, or failure open ?
make-queue(make-node(initial-stateproblem)) clo
sed ? empty loop do if open is empty then
return failure currnode ? Remove-Front(open) i
f Goal-Testproblem applied to State(currnode)
then return currnode children ?
Expand(currnode, Operatorsproblem) while
children not empty see next slide
end closed ? Insert(closed,
currnode) open ? Sort-By-PathCost(open) end
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Um algoritmo robusto e limpo

see previous slide children ?
Expand(currnode, Operatorsproblem) while
children not empty child ? Remove-Front(childre
n) if no node in open or closed has childs
state open ? Queuing-Fn(open, child) else
if there exists node in open that has childs
state if PathCost(child) lt PathCost(node)
open ? Delete-Node(open, node) open ?
Queuing-Fn(open, child) else if there exists
node in closed that has childs state if
PathCost(child) lt PathCost(node) closed ?
Delete-Node(closed, node) open ?
Queuing-Fn(open, child) end see previous
slide
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Informed search (busca com informação)

• Informed search
• Uso de heurísticas para guiar a busca
• Best first
• A
• Heuristica
• Hill-climbing
• Simulated annealing

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Best-first search

• Idéia
• usar uma função de avaliação para cada nó
  estimação de desejabilidade
• Expandir nó não expandido mais desejável.
• Implementação
• QueueingFn insere successores em ordem
  decrescente de desejabilidade
• Casos especiais
• greedy search
• A search

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Romania com custo de cada passo em km
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Greedy search (gula é um pecado capital?)

• Função de estimação
• h(n) estimação do custo de nó ao objetivo
  (heuristica)
• Por exemplo
• hSLD(n) distância em linha reta do nó a
  Bucharest
• Greedy search expande primeiro o nó que
  aparentemente é o mais próximo do objetivo, de
  acordo com h(n).

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Propriedades do Greedy Search

• Completo?
• Tempo?
• Memória?
• Ótimo?

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Properties of Greedy Search

• Completo? Não pode ficar parado em loops
• e.g., Iasi gt Neamt gt Iasi gt Neamt gt
• Completo espaço finito com teste de estado
  repetido.
• Tempo? O(bm) mas uma boa heurística pode dar
• uma melhora dramática
• Memória? O(bm) mantém todos os nós em memória
• Ótimo? Não.

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A search

• Idéia evitar expandir caminhos que já são caros
• função de avaliação f(n) g(n) h(n) com
• g(n) custo do caminho para atingir o nó
• h(n) custo estimado ao objetivo, do nó
• f(n) custo estimado total do caminho pelo nó
  ao objetivo
• A search usa uma heurística admissível, isto é,
• h(n) ? h(n) onde h(n) é o custo verdadeiro a
  partir de n.
• Por exemplo hSLD(n) nunca sobre-estima
  distância real da estrada.
• Teorema A search é ótimo

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Properties of A

• Complete?
• Time?
• Space?
• Optimal?

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Properties of A

• Complete? Yes, unless infinitely many nodes with
  f ? f(G)
• Time? Exponential in (relative error in h) x
  (length of solution)
• Space? Keeps all nodes in memory
• Optimal? Yes cannot expand fi1 until fi is
  finished

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Prova do lema caminho máximo
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Otimalidade de A (prova mais usual)
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Otimalidade de A (prova standard)

• Suponha que um objetivo G sub-ótimo2 foi gerado e
  está na fila. Seja n um nó não expandido num
  caminho mais curto para um objetivo G ótimo.

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Heurísticas admissíveis
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Heurísticas admissíveis
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Problema relaxado

• Heurísticas admissíveis podem ser derivadas do
  custo exato de uma solução para uma versão
  relaxada do problema.
• Se as regras do 8-puzzle forem relaxadas de
  maneira que uma casa possa mover a qualquer
  lugar, então h1(n) produz a solução mais curta.
• Se as regras são relaxadas de modo que uma casa
  possa se mover a qualquer posição adjacente,
  então h2(n) produz a solução mais curta.

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Next time

• Iterative improvement
• Hill climbing
• Simulated annealing