Planning and search
Lecture 1: Introduction and Revision of Search
Lecture 1: Introduction and Revision of Search
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Contact and web page
♦ Lecturer: Natasha Alechina
♦ email: nza@cs.nott.ac.uk
♦ web page: http://www.cs.nott.ac.uk/∼nza/G52PAS
♦ Textbook: Stuart Russell and Peter Norvig. Artificial Intelligence: A
Modern Approach, 3rd edition
♦ Slides: mostly by Stuart Russell (big thanks!)
Lecture 1: Introduction and Revision of Search
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Outline
♦ What this module is about
♦ Where search and planning fit in AI
♦ Reminder of uninformed search algorithms
♦ Definition of planning
♦ Plan of the module
♦ What to read for next week
Lecture 1: Introduction and Revision of Search
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Problem-solving agents
Simple problem-solving agent:
function Simple-Problem-Solving-Agent( percept) returns an action
static: seq, an action sequence, initially empty
state, some description of the current world state
goal, a goal, initially null
problem, a problem formulation
state ← Update-State(state, percept)
if seq is empty then
goal ← Formulate-Goal(state)
problem ← Formulate-Problem(state, goal)
seq ← Search( problem)
if seq=failure then return a null action
action ← First(seq)
seq ← Rest(seq)
return action
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Example of a search problem: 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
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
Eforie
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Single-state problem formulation
A problem is defined by four items:
initial state
e.g., “at Arad”
successor function S(x) = set of action–state pairs
e.g., S(Arad) = {hDrive(Arad, Zerind), Zerindi, . . .}
goal test, can be
explicit, e.g., x = “at Bucharest”
implicit, e.g., HasAirport(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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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., “Drive(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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Example: The 8-puzzle
7
2
5
8
3
Start State
4
5
1
2
3
6
4
5
6
1
7
8
Goal State
states??
actions??
goal test??
path cost??
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Example: The 8-puzzle
7
2
5
8
3
Start State
4
5
1
2
3
6
4
5
6
1
7
8
Goal State
states??: integer locations of tiles (ignore intermediate positions)
actions??
goal test??
path cost??
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Example: The 8-puzzle
7
2
5
8
3
Start State
4
5
1
2
3
6
4
5
6
1
7
8
Goal State
states??: integer locations of tiles (ignore intermediate positions)
actions??: move blank left, right, up, down (ignore unjamming etc.)
goal test??
path cost??
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Example: The 8-puzzle
7
2
5
8
3
Start State
4
5
1
2
3
6
4
5
6
1
7
8
Goal State
states??: integer locations of tiles (ignore intermediate positions)
actions??: move blank left, right, up, down (ignore unjamming etc.)
goal test??: = goal state (given)
path cost??
Lecture 1: Introduction and Revision of Search
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Example: The 8-puzzle
7
2
5
8
3
Start State
4
5
1
2
3
6
4
5
6
1
7
8
Goal State
states??: integer locations of tiles (ignore intermediate positions)
actions??: move blank left, right, up, down (ignore unjamming etc.)
goal test??: = goal state (given)
path cost??: 1 per move
Lecture 1: Introduction and Revision of Search
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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)
function Tree-Search( problem, strategy) returns a solution, or failure
initialize the search tree using the initial state of problem
loop do
if there are no candidates for expansion then return failure
choose a leaf node for expansion according to strategy
if the node contains a goal state then return the corresponding solution
else expand the node and add the resulting nodes to the search tree
end
Lecture 1: Introduction and Revision of Search
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Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
Arad
Oradea
Lecture 1: Introduction and Revision of Search
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Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
Arad
Oradea
Lecture 1: Introduction and Revision of Search
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Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
Arad
Oradea
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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 parent, children, depth, path cost g(x)
States do not have parents, children, depth, or path cost!
parent, action
State
Node
5
4
6
1
88
7
3
22
depth = 6
g=6
state
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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Implementation: general tree search
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function Tree-Search( problem, fringe) returns a solution, or failure
fringe ← Insert(Make-Node(Initial-State[problem]), fringe)
loop do
if fringe is empty then return failure
node ← Remove-Front(fringe)
if Goal-Test(problem, State(node)) then return node
fringe ← InsertAll(Expand(node, problem), fringe)
function Expand( node, problem) returns a set of nodes
successors ← the empty set
for each action, result in Successor-Fn(problem, State[node]) do
s ← a new Node
Parent-Node[s] ← node; Action[s] ← action; State[s] ← result
Path-Cost[s] ← Path-Cost[node] + Step-Cost(State[node], action,
result)
Depth[s] ← Depth[node] + 1
add s to successors
return successors
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Search strategies
A 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/expanded
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 ∞)
Lecture 1: Introduction and Revision of Search
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Uninformed search strategies
Uninformed 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
Lecture 1: Introduction and Revision of 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
A
B
D
C
E
F
G
Lecture 1: Introduction and Revision of 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
A
B
D
C
E
F
G
Lecture 1: Introduction and Revision of 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
A
B
D
C
E
F
G
Lecture 1: Introduction and Revision of 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
A
B
D
C
E
F
G
Lecture 1: Introduction and Revision of Search
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Uniform-cost search
Expand least-cost unexpanded node
Implementation:
fringe = queue ordered by path cost, lowest first
Equivalent to breadth-first if step costs all equal
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-first search
Expand deepest unexpanded node
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
B
C
D
H
E
I
J
F
K
L
G
M
N
O
Lecture 1: Introduction and Revision of Search
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Depth-limited search
Sometimes DFS does not terminate (infinite branch)
Fix: introduce a depth limit l
Backtrack when reach l (as if found a leaf node)
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Iterative deepening search
Do depth-limited search with l = 1, 2, 3, . . .
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Iterative deepening search l = 0
Limit = 0
A
A
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Iterative deepening search l = 1
Limit = 1
A
B
A
C
B
A
C
B
A
C
B
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Iterative deepening search l = 2
Limit = 2
A
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B
D
C
E
F
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G
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Lecture 1: Introduction and Revision of Search
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Iterative deepening search l = 3
Limit = 3
A
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Lecture 1: Introduction and Revision of Search
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O
Summary of basic search algorithms
Criterion
Complete?
Time
Space
Optimal?
Breadth- Uniform- DepthFirst
Cost
First
Yesa
O(bd)
O(bd)
Yesc
DepthLimited
Yesa,b
No No (Yes, if l ≥ d)
∗
O(b⌈C /ǫ⌉) O(bm)
O(bl )
∗
O(b⌈C /ǫ⌉) O(bm)
O(bl)
Yes
No
No
Iterative
Deepening
Yesa
O(bd)
O(bd)
Yesc
C ∗ is the cost of the optimal solution, ǫ is the minimal cost of a step, b the
branching factor, d the depth of the shallowest solution, m the maximum
depth of the search tree, l the depth limit.
a
complete if b is finite; b complete if step costs ≥ ǫ, c optimal if step costs
are identical
Lecture 1: Introduction and Revision of Search
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Planning
Planning: devising a plan of action to achieve the goal (for example: buy
milk, bananas, and a cordless drill)
Also talking about states of the world and actions, but more sophisticated
representation
States have structure (properties); actions have pre- and post-conditions.
Action: Buy(x)
Precondition: At(p), Sells(p, x)
Effect: Have(x)
At(p) Sells(p,x)
Buy(x)
Have(x)
Lecture 1: Introduction and Revision of Search
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Search vs. planning contd.
Planning systems do the following:
1) open up action and goal representation to allow selection
2) divide-and-conquer by subgoaling
3) relax requirement for sequential construction of solutions
Search
States Data structures
Actions Code
Code
Goal
Plan
Sequence from S0
Planning
Logical sentences
Preconditions/outcomes
Logical sentence (conjunction)
Constraints on actions
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Plan of the module: search topics
♦ Another revision lecture on properties of uninformed search algorithms,
heuristic search (A∗)
♦ Graph search, direction of search
♦ Local search (annealing, tabu, ...)
♦ Population-based methods (genetic algorithms...)
♦ Reducing search to SAT
♦ Search with non-determinism and partial observability
♦ Logical agents; first-order logic
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Plan of the module: planning topics
♦ Situation calculus
♦ What is classical planning. Forward planning.
♦ Classical planning continued. Regression Planning
♦ Classical planning continued. Partial-Order Planning
♦ Classical planning continued. GraphPlan.
♦ Classical planning continued. SatPlan.
♦ Planning with time and resources
♦ HTN planning.
♦ Planning and acting in non-deterministic domains.
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What to read for the next lecture
Chapter 3 in Russell and Norvig (this is revision)
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