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Problem solving
Chapter 3, Sections 1–3
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
1
Problem types
Deterministic, fully observable =⇒ single-state problem
Agent knows exactly which state it will be in; solution is a sequence
Non-observable =⇒ conformant problem
Agent may have no idea where it is; solution (if any) is a sequence
Nondeterministic and/or partially observable =⇒ contingency problem
percepts provide new information about current state
solution is a contingent plan or a policy
often interleave search, execution
Unknown state space =⇒ exploration problem (“online”)
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
2
Example: Romania
We are on holiday in Romania; currently in Arad
Our 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
Problem type:
deterministic, fully observable =⇒ single-state problem
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
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Example: Romania
Oradea
71
75
Neamt
Zerind
87
151
Iasi
Arad
140
Sibiu
92
99
Fagaras
118
Timisoara
111
Vaslui
80
Rimnicu Vilcea
Lugoj
Pitesti
97
142
211
70
98
Mehadia
75
Dobreta
146
85
101
86
138
Bucharest
120
Craiova
Hirsova
Urziceni
90
Giurgiu
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Eforie
Chapter 3, Sections 1–3
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Single-state problem formulation
A problem is defined by five components:
initial state, e.g., In(Arad)
actions Actions(s) = set of actions for state s
e.g., Actions(In(Arad)) = {Go(Sibiu), Go(T imisoara), Go(Zerind)}
transitions Result(s, a) = the successor state
e.g., Result(In(Arad), Go(Zerind)) = In(Zerind)
goal test, can be an explicit set of states, e.g., {In(Bucharest)}
or an implicit property, e.g., checkmate in chess
path cost is the sum of the step costs c(s, a, s′)
e.g., sum of distances, number of actions executed, etc.
in this chapter we assume c to be positive
A solution is a sequence of actions
leading from the initial state to a goal state
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
5
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., Result(In(Arad), Go(Zerind)) represents a complex set
of possible routes, detours, rest stops, etc.
(Abstract) solution =
set of real paths that are solutions in the real world
Each abstract action should be “easier” than the original problem!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
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Example: vacuum world state space graph
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states??: integer dirt and robot locations (ignore dirt amounts etc.)
initial state??: any state
actions??: Left, Right, Suck, N oOp
goal test??: no dirt in any location
path cost??: 1 per action (0 for N oOp)
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
7
Example: The 8-puzzle
states??: a 3 × 3 matrix of integers 0 ≤ n ≤ 8
initial state??: any state
actions??: move the blank space: left, right, up, down
goal test??: equal to goal state (given above)
path cost??: 1 per move
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
8
Example: The 8-queens problem
states??: any arrangement of 0 to 8 queens on the board
initial state??: no queens on the board
actions??: add a queen to any empty square
goal test??: 8 queens on the board, none attacked
path cost??: 1 per move
Using this formulation, there are 64 · 63 · · · · 57 ≈ 1.8 × 1014
possible sequences to explore!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
9
Example: The 8-queens problem (alternative)
states??: one queen per column in the leftmost columns, none attacked
initial state??: no queens on the board
actions??: add a queen to any square in the leftmost empty column,
making sure that no queen is attacked
goal test??: 8 queens on the board, none attacked
path cost??: 1 per move
Using this formulation, we have only 2,057 sequences!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
10
Example: robotic assembly
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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 of the object
path cost??: time to execute
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Sections 1–3
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