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+# AI - lesson 06
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+#### Francesco Arrigoni
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+###### 28 October 2015
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+## Informed search strategies
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+
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+### Evaluation function $f(n)$ types
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+
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+### Greedy best first
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+$$f(n)=h(n)$$
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+$h(n)$ is called *heuristic function* and is an estimate of __how far__ is a node __from the goal__
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+
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+#### example
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+- The nodes of the graph are __cities__ and the heuristic function can be the cartesian distance of the cities
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+
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+- In the game of 15 an heuristic can be an estimate of the number of moves needed to complete the game
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+
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+By definition the heuristic function is defined over __nodes__, but commonly are defined over a __state__, in fact
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+the heuristic function of two nodes referring to the same node is the same.
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+
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+According to how heuristic functions are defines, there can be loops in __greedy best first__
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+
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+#### Optimality
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+
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+This strategy is __not optimal__ in general
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+
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+#### Complexity
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+
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+Setting a not clever heuristic function, this strategy is equivalent to the depht first
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+But with a good heuristic function, we can exploit this to achieve better results.
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+__time__:$O(b^m)$
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+__space__:$O(b^n)$
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+
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+### $A^*$
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+
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+In this case the __evaluation function__ is defined as $f(n)=g(n)+h(n)$
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+$g(n)$ is the cost for going from the root to node n
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+$h(n)$ is an estimation of the costs for going from node n to the goal g
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+
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+Doing so $f(n)$ is the estimated cost of the solution passing through n
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+
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+#### Optimality
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+
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+$A^*$ is optimal when using __tree search__, that is when the heuristic function is *admissible*
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+$h^*()$ is admissible when $\forall n\;h(n)\le h^*(n)$ The heuristic function never overestimates the cost of reaching a node.
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+
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+Returning to the road example:
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+The straight distance (line of sight) between two cities is a valid heuristic by definition, in fact the real distance
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+can not be smaller than this.
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+
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+##### example
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+
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+$$f(g_2)=g(g_2)+h(g_2)>c^*$$
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+$$f(n)=g(n)+h(n)\le c^*$$
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+$$f(n)\le c^*<f(g_2)$$
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+
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+An Heuristic function is __consistent__ if for every node that is a successor of n
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+$$\foralln,n' h(n)\le c(n,n')+h(n')$$
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+
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+__Consistency__ of heuristic function implies __admissibilitu__
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+
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+- $f(n) is not decreasing along every path
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+
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+$A^*$ will choose nodes from the frontier with value that in a non decreasing order.
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+
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+#### Complexity
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+- Time complexity $O(\square^{|h-h^*|})
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+
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+If the heuristic function is always zero, $A^*$ degerates in uniform cost.
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+
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+If we have a perfect heuristic function, the time complexity is constant
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+$A^*$ is called __optimally efficient__, this means that given a fixed heuristic function. $A^*$ is guaranteed to expand the
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+minimum number of nodes
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+
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+
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