1. Blay Whitby 2003 blayw@sussex.ac.uk
Search
Blay Whitby 2003

2. Knowledge Representation
Spot the Connection…
Chippenham, Wiltshire April 17th 1960.
Finding routes, making plans, Deep Blue.
One greater than two (if you believe in integer theory).
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3. Knowledge Representation
Jargon
State Space or Problem Space
ALL the steps achievable in a particular problem
Search Space
Those steps achievable GIVEN an initial start state and operators for moving between states [usually a directed graph]
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4. Knowledge Representation
Jargon…
Directed Graph
A graph in which the ARCS are directional and may rejoin as opposed to a:
Tree
A Special case of a directed graph in which each NODE may have several descendants, but only one ancestor.
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5. Knowledge Representation
State Space Search
A general problem solving technique, the basis of:
Game playing
Planning
Natural Language Processing
Theorem proving
The main step forward in 50’s – 60’s AI
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6. Knowledge Representation
State Space Search
Represent the problem as a space of possibilities
Solving the problem is then a matter of ‘finding’ if one of these possibilities is the solution.
There are many techniques for doing this.
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7. Knowledge Representation
Trees
Root
Root node
Arcs
Goal
‘Daughter’ nodes
Nodes
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8. Knowledge Representation
How to Grow a Tree
Most problems have:
A start state
A possible solution (or goal)
Possible transitions (or legal moves)
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9. Knowledge Representation
How to Grow a Tree
Problem:
Build a tower of bricks 5cm high from any of 4 bricks: 1cm, 2cm, 3cm & 4cm high.
Start state: No bricks at all
Possible transitions: Add any one brick, provided it is not already on the pile.
Goal: 5cm pile of bricks
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10. Tower of Bricks Search space tree
[ ]
[1]
[2]
[3]
[4]
[2,1] [2,3] [2,4]
[4,1] [4,2] [4,3]
[1,2]
[1,3]
[1,4]
[3,1] [3,2] [3,4]
[1,2,3]
[1,2,4]
Note 4 paths to goal
[1,2,3,4]
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11. State Space Representation of Predicate Calculusq r u v t s
qp t r
r p s u
v q s
s r t p q r p a b
Q Λ r p
a V b p
Luger & Stubblefield pp
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12. Knowledge Representation
Implementation
Most search programs use lists: [a,b,c,d,e]
It is important to know if a state has already been visited, so two lists are needed:
Considered – states already looked at – closed
Alternatives – generated but not yet examined – open
Generating and examining these lists is an algorithmic procedure
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13. Generate & Test Algorithm
Generate a possible solution
(legal move)
Test to see if this is the goal
(or 1 of a set of end states)
If goal then STOP else return to 1
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14. Knowledge Representation
Search Strategies
Depth First: Explore one path at a time to final node or goal.
Breadth First: Explore all nodes that are one step away until all visited or goal found.
Exhaustive search strategies. They will find a solution if one exists.
As such they may be inefficient and need to be limited
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15. More efficient Strategies
For most non-trivial problems exhaustive search is not practicable e.g., Chess (2.5x10154)
Many techniques exist to reduce the need to explore unpromising parts of the search space
Take into account how much effort has been made in getting this far or how near we think we are to the goal: or, best, both.
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16. Best First and Hill Climbing
Best First: Explore the most promising node first.
Hill Climbing:Explore the daughter of the most promising node first.
To do this we need a static evaluation function: measure of how good this node is, without looking ahead or:
An heuristic* evaluation function?
* rule of thumb
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17. Knowledge Representation
Branch and Bound
Consider the ‘cost’ of getting to a node
Euston Vic.
Charing X Northern
Oxford Circus Vic.
Warren St. Northern
Warren St. Vic.
Vic. Vic.
Green Pk. 2 5 1 4
Here ‘cost’ = travelling time between nodes
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18. Knowledge Representation
Branch and Bound
Good in that it guarantees an ‘optimal path’ but:
Poor in that it will explore ‘pointless’ options because it is unguided.
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19. Knowledge Representation
A* Algorithm
Requires an evaluation of the cost so far (K) and a heuristic (guess) of how close we are to the goal (>K)
Where: g(n)=cost of best path from start to n
h(n)=estimate of cost of best path to goal from n.
Evaluate nodes and always choose lowest f value
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20. Knowledge Representation
A* Algorithm
If will find the lowest cost path.
Where: h*(n) is the actual cost to goal.
In other words: The heuristic must underestimate the cost to goal
The closer the heuristic is to the real costs, the more effective is the search.
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21. Knowledge Representation
The 8 Puzzle
Tiles 1 3 4 6 8 7 5 2
Represent as a list
[1,3,4,6,8,7,hole,5,2]
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22. Knowledge Representation
State Representation 1 4 3 2 8 7 5 6
= [1,4,3,2,8,hole,7,5,6] 1 4 2 8 3 7 5 6
= [1,4,hole,2,8,3,7,5,6]
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23. Knowledge Representation
Program run:
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24. First 4 layers of search tree
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25. Comparison of search methods
Source Thornton & DuBoulay (1992)
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26. Comparison of search methods
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