CS 8520: Artificial Intelligence
description
Transcript of CS 8520: Artificial Intelligence
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
CS 8520: Artificial Intelligence
Solving Problems by Searching
Paula Matuszek
Spring, 2013
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are in turn based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
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2CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Problem-Solving Agents• A goal-based agent is essentially solving a
problem– Given some state and some goal,
– Figure out how to get from the current state to the goal
– By taking some sequence of actions.
• The problem is defined as a state space and a goal. Solving the problem consists of searching the state space for the sequence of actions that leads to the goal.
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3CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Example: Romania
• On 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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4CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Example: Romania
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5CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Single-state problem formulation• A problem is defined by five items:
– initial state e.g., "at Arad"
– actions or successor function S(x) = set of action–state pairs • e.g., S(Arad) = {<Arad to Zerind, Zerind>, … }
– transition model: new state resulting from action
– goal test, can be• explicit, e.g., x = "at Bucharest"
• implicit, e.g., Checkmate(x)
– 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 a goal state
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6CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Selecting a state space• Real world is absurdly complex
– therefore state space must be abstracted for problem solving
• (Abstract) state = set of real states
• (Abstract) action = complex combination of real actions– e.g., "Arad to Zerind" represents a complex set of possible
routes, detours, rest stops, etc.
• For guaranteed realizability, any real state "in Arad“ must get to some 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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7CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Vacuum world state space graph
• States? Actions? Transition Models? Goal Test? Path Cost?
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8CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Vacuum world state space graph
• states? integer dirt and robot location • actions? Left, Right, Suck• transition models? In A, in B, clean• goal test? no dirt at all locations• path cost? 1 per action
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9CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Example: The 8-puzzle
• states?
• actions?
• transition model?
• goal test?
• path cost?
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Example: The 8-puzzle
• states? locations of tiles • actions? move blank left, right, up, down• transition model? square with tile moved. • goal test? = goal state (given)• path cost? 1 per move
[Note: optimal solution of n-Puzzle family is NP-hard]
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Search• The basic concept of search views the state space as
a search tree– Initial state is the root node
– Each possible action leads to a new node defined by the transition model
– Some nodes are identified as goals
• Search is the process of expanding some portion of the tree in some order until we get to a goal node
• The strategy we use to choose the order to expand nodes defines the type of search
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Tree search algorithms
• Basic idea:– offline, simulated exploration of state space by
generating successors of already-explored states (a.k.a.~expanding states)
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Tree search example
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Tree search example
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Tree search example
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Implementation: general tree search
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Implementation: states vs. nodes• A state is a (representation of) a physical configuration• A node is a data structure constituting part of a search tree
includes state, parent node, action, path cost g(x), depth
• The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Search strategies• A search strategy is defined by picking the order of node
expansion. (e.g., breadth-first, depth-first)
• Strategies are evaluated along the following dimensions:– completeness: does it always find a solution if one exists?
– time complexity: number of nodes generated
– space complexity: maximum number of nodes in memory
– optimality: does it always find a least-cost solution?
• Time and space complexity are measured in terms of – b: maximum branching factor of the search tree
– d: depth of the least-cost solution
– m: maximum depth of the state space (may be infinite)
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Summary• Problem-solving agents search through a problem or state space for
an acceptable solution.
• A state space can be specified by• An initial state
• Actions
• A transition model or successor function S(x) describing the results of actions
• A goal test or goal state
• A path cost
• A solution is a sequence of actions leading from the initial state to a goal state
• The formalization of a good state space is hard, and critical to success. It must abstract the essence of the problem so that– It is easier than the real-world problem.– A solution can be found.– The solution maps back to the real-world problem and solves it.
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Uninformed search strategies
• Uninformed search strategies use only the information available in the problem definition
• Breadth-first search
• Uniform-cost search
• Depth-first search
• Depth-limited search
• Iterative deepening search
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Implementation: general tree search
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Breadth-first search
• Expand shallowest unexpanded node
• Implementation:– fringe is a FIFO queue, i.e., new successors go
at end
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Breadth-first search• Expand shallowest unexpanded node
• Implementation:– fringe is a FIFO queue, i.e., new successors
go at end
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Breadth-first search• Expand shallowest unexpanded node
• Implementation:– fringe is a FIFO queue, i.e., new successors go
at end
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Breadth-first search• Expand shallowest unexpanded node
• Implementation:– fringe is a FIFO queue, i.e., new successors go
at end
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Properties of breadth-first search• Complete? Yes (if b is finite)
• Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)
• Space? O(bd+1) (keeps every node in memory)
• Optimal? Yes (if cost = 1 per step)
• Space is the bigger problem (more than time)
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Uniform-cost search• Expand least-cost unexpanded node
• Implementation:– fringe = queue ordered by path cost
• Equivalent to breadth-first if step costs all equal
• Complete? Yes, if step cost >= epsilon (otherwise can loop)
• Time and space? O(b(ceiling(C*/ epsilon))) where C* is the cost of the optimal solution and epsilon is the smallest step cost. Can be much worse than breadth-first if many small steps not on optimal path
• Optimal? Yes – nodes expanded in increasing order of g(n)
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Uniform Cost Search
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-first search• Expand deepest unexpanded node
• Implementation:– fringe = LIFO queue, i.e., put successors at front
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Properties of depth-first search
• Complete? No: fails in infinite-depth spaces, spaces with loops– Modify to avoid repeated states along path
• then complete in finite spaces
• Time? O(bm): terrible if m is much larger than d– but if solutions are dense, may be much faster than
breadth-first
• Space? O(bm), i.e., linear space!
• Optimal? No
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Depth-limited search• = depth-first search with depth limit l
• Nodes at depth l have no successors
• Solves problem of infinite depth
• Incomplete
• Recursive implementation:
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Iterative deepening search•Repeated Depth-Limited search, incrementing limit l until a solution is found or failure.
•Repeats earlier steps at each new level, so inefficient -- but never more than doubles cost
•No longer incomplete
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Iterative deepening search l =0
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Iterative deepening search l =1
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Iterative deepening search l =2
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Iterative deepening search l =3
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Properties of iterative deepening
• Complete? Yes
• Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)
• Space? O(bd)
• Optimal? Yes, if step cost = 1
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Summary of Algorithms for Uninformed Search
Criterion Breadth-first
Uniform Cost Depth First
Depth-Limited
Iterative Deepening
Complete? Yes Yes No No Yes
Time? O(bd) O(b(ceilingC*/ε)) O(bm) O(bl) O(bd)
Space? O(bd) O(b(ceilingC*/ε)) O(bm) O(bl) O(bd)
Optimal? Yes Yes No No Yes
Where: b is branching factord is depth of shallowest solution, m is max depth of search treeC* is cost of optimal solution
ε is the smallest step size
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
A Caution: Repeated States
• Failure to detect repeated states can turn a linear problem into an exponential one, or even an infinite one.– For example: 8-puzzle
– Simple repeat -- empty square simply moves back and forth
– More complex repeats also possible.
• Save list of expanded states -- the closed list.
• Add new state to fringe only if it's not in closed list.
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CSC 8520 Spring 2013. Paula Matuszek
Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA.
Summary: Uninformed Search• Problem formulation usually requires abstracting away
real-world details to define a state space that can feasibly be explored
• There exists a variety of uninformed search strategies
• Search strategies can be evaluated by– whether they find a solution if one exists: completeness
– how long they take: time complexity
– how much memory they take: space complexity
– whether they find the best solution: optimality
• Iterative deepening search uses linear space and not much more time than other algorithms: usual choice