Solving problems by searching - Emory University...Example: Romania On holiday in Romania; currently...
Transcript of Solving problems by searching - Emory University...Example: Romania On holiday in Romania; currently...
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Solving problems by searching
Chapter 3
Some slide credits to Hwee Tou Ng (Singapore)
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Outline
Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms Heuristics
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Intelligent agent solves problems by?
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Problem-solving agents
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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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Example: Romania
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Romania: problem type?
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Romania: Problem type
Deterministic, fully observable single-state problem
Agent knows exactly which state it will be in; solution is a sequence
Non-observable sensorless problem (conformant problem)
Agent may have no idea where it is; solution is a sequence Nondeterministic and/or partially observable
contingency problem percepts provide new information about current state often interleave} search, execution
Unknown state space exploration problem
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Single-state problem formulation
A problem is defined by four items:
1. initial state e.g., "at Arad"2. actions or successor function S(x) = set of action–state pairs
e.g., S(Arad) = {<Arad Zerind, Zerind>, … }1. goal test, can be
explicit, e.g., x = "at Bucharest" implicit, e.g., Checkmate(x)
1. 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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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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Tree search example
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Tree search example
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Tree search example
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Implementation: general tree search
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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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Breadth-first search
Expand shallowest unexpanded node Implementation:
fringe is a FIFO queue, i.e., new successors go at end
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Breadth-first search
Expand shallowest unexpanded node Implementation:
fringe is a FIFO queue, i.e., new successors go at end
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Breadth-first search
Expand shallowest unexpanded node Implementation:
fringe is a FIFO queue, i.e., new successors go at end
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Breadth-first search
Expand shallowest unexpanded node Implementation:
fringe is a FIFO queue, i.e., new successors go at end
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Example: Romania (Q)
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Search strategies
A search strategy is defined by picking the order of node expansion
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 ∞)
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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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Uniform-cost search
Video
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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 ≥ ε Time? # of nodes with g ≤ cost of optimal solution,
O(bceiling(C*/ ε)) where C* is the cost of the optimal solution
Space? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε))
Optimal? Yes – nodes expanded in increasing order of g(n)
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Comparison of Searches
So far: Breadth-first search Uniform-cost (cheapest) search New: Depth-first searchOptimal?
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Depth-first search
Expand deepest unexpanded node Implementation:
fringe = LIFO queue, i.e., put successors at front
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Why depth-first?
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Properties of depth-first search
Complete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path
complete in finite spaces
Time? O(bm): terrible if m >> 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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Summary of algorithms
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Limitations
Video
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Best-first search
Idea: use an evaluation function f(n) for each nodeestimate of "desirability"
Expand most desirable unexpanded node
Implementation:Order the nodes in fringe in decreasing order of desirability
Special cases:greedy best-first searchA* search
–
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Romania with step costs in km
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Greedy best-first search
Evaluation function f(n) = h(n) (heuristic)= estimate of cost from n to goale.g., hSLD(n) = straight-line distance from n to BucharestGreedy best-first search expands the node that appears to be closest to goal
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Properties of greedy best-first search• Complete? No – can get stuck in
loops, e.g., Iasi Neamt Iasi Neamt
• Time? O(bm), but a good heuristic can give dramatic improvement
• Space? O(bm) -- keeps all nodes in memory
• Optimal? No
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A* search
Idea: avoid expanding paths that are already expensiveEvaluation function f(n) = g(n) + h(n)g(n) = cost so far to reach nh(n) = estimated cost from n to goalf(n) = estimated total cost of path through n to goal
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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State Spaces
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Example: vacuum world
Single-state, start in #5. Solution?
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Example: vacuum world
Single-state, start in #5. Solution? [Right, Suck]
Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution?
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Example: vacuum world
Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution? [Right,Suck,Left,Suck]
Contingency Nondeterministic: Suck may
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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Example: vacuum world
Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution? [Right,Suck,Left,Suck]
Contingency Nondeterministic: Suck may
dirty a clean carpet Partially observable: location, dirt at current location. Percept: [L, Clean], i.e., start in #5 or #7
Solution? [Right, if dirt then Suck]
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Selecting a state space
Real 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 e.g., "Arad 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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Vacuum world state space graph
states? actions? goal test? path cost?
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Vacuum world state space graph
states? integer dirt and robot location actions? Left, Right, Suck goal test? no dirt at all locations path cost? 1 per action
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Modified vacuum world?
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Example: The 8-puzzle
states? actions? goal test? path cost?
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Example: The 8-puzzle
states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given) path cost? 1 per move
[Note: optimal solution of n-Puzzle family is NP-hard]
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Example: robotic assembly
states?: real-valued coordinates of robot joint angles parts of the object to be assembled
actions?: continuous motions of robot joints goal test?: complete assembly path cost?: time to execute
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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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Depth-limited search
= depth-first search with depth limit l,i.e., nodes at depth l have no successors
Recursive implementation:
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Iterative deepening search
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Iterative deepening search l =0
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Iterative deepening search l =1
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Iterative deepening search l =2
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Iterative deepening search l =3
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Iterative deepening search
Number of nodes generated in a depth-limited search to depth d with branching factor b:
NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd
Number of nodes generated in an iterative deepening search to depth d with branching factor b:
NIDS = (d+1)b0 + d b1 + (d-1)b2 + … + 3bd-2 +2bd-1 + 1bd
For b = 10, d = 5, NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111 NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456
Overhead = (123,456 - 111,111)/111,111 = 11%
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Properties of iterative deepening search
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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Admissible heuristics
A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimisticExample: hSLD(n) (never overestimates the actual road distance)• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
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Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
f(G2) = g(G2) since h(G2) = 0
g(G2) > g(G) since G2 is suboptimal
f(G) = g(G) since h(G) = 0 f(G2) > f(G) from above
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Consistent heuristicsA heuristic is consistent if for every node n, every successor n' of n generated by any action a,
h(n) ≤ c(n,a,n') + h(n')
If h is consistent, we havef(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ≥ g(n) + h(n) = f(n)i.e., f(n) is non-decreasing along any path.• Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is
optimal
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Optimality of A*
A* expands nodes in order of increasing f value
Gradually adds "f-contours" of nodes Contour i has all nodes with f=fi, where fi < fi+1
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Properties of A$^*$
• Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
• Time? Exponential• Space? Keeps all nodes in memory• Optimal? Yes
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Admissible heuristicsE.g., for the 8-puzzle:h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
• h1(S) = ?
• h2(S) = ?
•
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Admissible heuristicsE.g., for the 8-puzzle:h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
• h1(S) = ? 8
• h2(S) = ? 3+1+2+2+2+3+3+2 = 18
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Repeated states
Failure to detect repeated states can turn a linear problem into an exponential one!
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Graph search
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Summary
Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored
Variety of uninformed search strategies
Iterative deepening search uses only linear space and not much more time than other uninformed algorithms