SP10 cs188 lecture 4 -- CSPs (2PP)

SP10 cs188 lecture 4 -- CSPs (2PP) - CS 188: Artificial...

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1 CS 188: Artificial Intelligence Spring 2010 Lecture 4: A* wrap-up + Constraint Satisfaction 1/28/2010 Pieter Abbeel – UC Berkeley Many slides from Dan Klein Announcements s Project 0 (Python tutorial) is due today s If you don’t have a class account yet, pick one up after lecture s Written 1 (Search) is due today s Project 1 (Search) is out and due next week Thursday s Section/Lecture
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2 Recap: Search s Search problem: s States (configurations of the world) s Successor function: a function from states to lists of (state, action, cost) triples; drawn as a graph s Start state and goal test General Tree Search s Important ideas: s Fringe s Expansion s Exploration strategy s Main question: which fringe nodes to explore? Detailed pseudocode is in the book!
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3 A* Review s A* uses both backward costs g and forward estimate h: f(n) = g(n) + h(n) s A* tree search is optimal with admissible heuristics (optimistic future cost estimates) s Heuristic design is key: relaxed problems can help Admissible Heuristics s A heuristic h is admissible (optimistic) if: where is the true cost to a nearest goal s Example: s Coming up with admissible heuristics is most of what’s involved in using A* in practice. 15
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4 Optimality of A*: Blocking Notation: s g(n) = cost to node n s h(n) = estimated cost from n to the nearest goal (heuristic) s f(n) = g(n) + h(n) = estimated total cost via n s G*: a lowest cost goal node s G: another goal node Optimality of A*: Blocking Proof: s What could go wrong? s We’d have to have to pop a suboptimal goal G off the fringe before G* s This can’t happen: s Imagine a suboptimal goal G is on the queue s Some node n which is a subpath of G* must also be on the fringe (why?) s n will be popped before G
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5 Tree Search: Extra Work! s Failure to detect repeated states can cause exponentially more work. Why? Graph Search s Very simple fix: never expand a state twice s Can this wreck completeness? Optimality?
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6 Optimality of A* Graph Search Proof: s New possible problem: nodes on path to G* that would have been in queue aren’t, because some worse n’ for the same state as some n was dequeued and expanded first (disaster!) s Take the highest such n in tree s Let p be the ancestor which was on the queue when n ’ was expanded s Assume f(p) < f(n) s f(n) < f(n’) because n’ is suboptimal s p would have been expanded before n s Contradiction! Consistency s Wait, how do we know parents have better f-values than their successors? s Couldn’t we pop some node n , and find its child n’ to have lower f value? s YES: s What can we require to prevent these inversions? s Consistency: s Real cost must always exceed reduction in heuristic A B G 3 h = 0 h = 10 g = 10
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7 A* Graph Search Gone Wrong S A B C G 1 1 1 2 3 h=2 h=1 h=4 h=1 h=0 S (0+ 2 ) A (1+ 4 ) B (1+ 1 ) C (2+ 1 ) C (3+ 1 ) G (6+ 0 ) State space graph Search tree C is already in the closed-list, hence not placed in the priority queue Consistency 3 A C G h=4 h=1 1 The story on Consistency:
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SP10 cs188 lecture 4 -- CSPs (2PP) - CS 188: Artificial...

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