程序代写代做代考 data structure assembly algorithm Problem solving and search

Problem solving and search
Chapter 3
Chapter 3 1

♦ Problem-solving agents ♦ Problem types
♦ Problem formulation ♦ Example problems
♦ Basic search algorithms
Outline
Chapter 3 2

Restricted form of general agent:
Problem-solving agents
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)
action ← Recommendation(seq, state) seq ← Remainder(seq, state)
return action
Note: this is offline problem solving; solution executed “eyes closed.” Online problem solving involves acting without complete knowledge.
Chapter 3 3

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
Example: Romania
Find solution:
sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest
Chapter 3 4

Arad
140
118
99 80
Vaslui
75
151
71
Neamt
Zerind
87
Timisoara
Rimnicu Vilcea
111
70
Lugoj
97
Pitesti
Dobreta
120
Oradea
75
Urziceni Bucharest
86
Mehadia
146 101 138
85
Example: Romania
Sibiu
Fagaras
Craiova
Eforie
211
142
90
Giurgiu
Iasi
92
98
Hirsova
Chapter 3 5

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”)
Chapter 3 6

Example: vacuum world
Single-state, start in #5. Solution??
12
34
56
78
Chapter 3 7

Example: vacuum world
Single-state, start in #5. Solution?? [Right, Suck]
12
Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8} e.g., Right goes to {2, 4, 6, 8}. Solution??
34
56
78
Chapter 3 8

Example: vacuum world
Single-state, start in #5. Solution?? [Right, Suck]
12
Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8} e.g., Right goes to {2, 4, 6, 8}. Solution?? [Right, Suck, Left, Suck]
34
Contingency, start in #5
Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only.
Solution??
78
56
Chapter 3 9

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) = {⟨Arad → Zerind, Zerind⟩, . . .}
goal test, can be
explicit, e.g., x = “at Bucharest” implicit, e.g., NoDirt(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
Chapter 3 11

(Abstract) state = set of real states
Selecting a state space
Real world is absurdly complex
⇒ state space must be abstracted for problem solving
(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!
Chapter 3 12

Example: vacuum world state space graph
states?? actions?? goal test?? path cost??
R LR
L SS
RR LRLR
LL
SS SS
R LR
L SS
Chapter 3 13

Example: vacuum world state space graph
R LR
L SS
RR LRLR
LL
SS SS
R LR
L SS
states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??
goal test??
path cost??
Chapter 3 14

Example: vacuum world state space graph
R LR
L SS
RR LRLR
LL
SS SS
R LR
L SS
states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Left, Right, Suck, NoOp
goal test??
path cost??
Chapter 3 15

states?? actions?? goal test?? path cost??
Example: The 8-puzzle
7 2 4 51 2 3
56456
831 78
Start State
Goal State
Chapter 3 18

Example: The 8-puzzle
7 2 4 51 2 3
56456
831 78
Start State Goal State
states??: integer locations of tiles (ignore intermediate positions) actions??
goal test??
path cost??
Chapter 3 19

Example: The 8-puzzle
7 2 4 51 2 3
56456
831 78
Start State Goal State
states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??
path cost??
Chapter 3 20

Example: robotic assembly
states??: real-valued coordinates of robot joint angles parts of the object to be assembled
P
R
R
R
actions??: continuous motions of robot joints
goal test??: complete assembly with no robot included! path cost??: time to execute
RR
Chapter 3 23

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
end
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
Chapter 3 24

Arad Fagaras
Oradea Rimnicu Vilcea Arad
Lugoj Arad Oradea
Sibiu
Timisoara Zerind
Tree search example
Arad
Chapter 3 25

Arad Fagaras
Oradea Rimnicu Vilcea Arad
Lugoj Arad Oradea
Sibiu
Timisoara Zerind
Tree search example
Arad
Chapter 3 26

Arad Fagaras
Oradea Rimnicu Vilcea Arad
Lugoj Arad Oradea
Sibiu
Timisoara Zerind
Tree search example
Arad
Chapter 3 27

state
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!
State
5 4 Node 618
depth = 6 g =6
618 7 3 2
The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.
parent, action
Chapter 3 28

Implementation: general tree search
Chapter 3 29

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
Chapter 3 30

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 ∞)
Chapter 3 31

Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search
Uninformed search strategies
Uninformed strategies use only the information available in the problem definition
Chapter 3 32

Expand shallowest unexpanded node Implementation:
Breadth-first search
fringe is a FIFO queue, i.e., new successors go at end A
BC
DEFG
Chapter 3 33

Expand shallowest unexpanded node Implementation:
Breadth-first search
fringe is a FIFO queue, i.e., new successors go at end A
BC
DEFG
Chapter 3 34

Expand shallowest unexpanded node Implementation:
Breadth-first search
fringe is a FIFO queue, i.e., new successors go at end A
BC
DEFG
Chapter 3 35

Expand shallowest unexpanded node Implementation:
Breadth-first search
fringe is a FIFO queue, i.e., new successors go at end A
BC
DEFG
Chapter 3 36

Complete??
Properties of breadth-first search
Chapter 3 37

Properties of breadth-first search
Complete?? Yes (if b is finite) Time??
Chapter 3 38

Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1), i.e., exp. in d Space??
Chapter 3 39

Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1), i.e., exp. in d Space?? O(bd+1) (keeps every node in memory)
Optimal?? Yes (if cost = 1 per step); not optimal in general
Space is the big problem; can easily generate nodes at 100MB/sec so 24hrs = 8640GB.
Chapter 3 41

Expand least-cost unexpanded node Implementation:
Complete?? Yes, if step cost ≥ ǫ
Uniform-cost search
fringe = queue ordered by path cost, lowest first
Equivalent to breadth-first if step costs all equal
Time?? # of nodes with g ≤ cost of optimal solution, O(b⌈C∗/ǫ⌉) where C∗ is the cost of the optimal solution
Space?? # of nodes with g ≤ cost of optimal solution, O(b⌈C∗/ǫ⌉) Optimal?? Yes—nodes expanded in increasing order of g(n)
Chapter 3 42

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 43

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 44

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 45

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 46

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 47

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 48

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 49

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 50

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 51

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 52

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 53

Expand deepest unexpanded node
Depth-first search
Implementation:
fringe = LIFO queue, i.e., put successors at front
A
BC
DEFG
HIJKLMNO
Chapter 3 54

Complete??
Properties of depth-first search
Chapter 3 55

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??
Chapter 3 56

Space??
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 is much larger than d
but if solutions are dense, may be much faster than breadth-first
Chapter 3 57

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 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??
Chapter 3 58

= depth-first search with depth limit l, i.e., nodes at depth l have no successors
Depth-limited search
Recursive implementation:
function Depth-Limited-Search( problem, limit) returns soln/fail/cutoff
Recursive-DLS(Make-Node(Initial-State[problem]), problem, limit)
function Recursive-DLS(node, problem, limit) returns soln/fail/cutoff cutoff-occurred? ← false
if Goal-Test(problem,State[node]) then return node
else if Depth[node] = limit then return cutoff
else for each successor in Expand(node, problem) do result ← Recursive-DLS(successor, problem, limit) if result = cutoff then cutoff-occurred? ← true
else if result ̸= failure then return result
if cutoff-occurred? then return cutoff else return failure
Chapter 3 60

Iterative deepening search
function Iterative-Deepening-Search( problem) returns a solution inputs: problem, a problem
for depth ← 0 to ∞ do
result ← Depth-Limited-Search( problem, depth) if result ̸= cutoff then return result
end
Chapter 3 61

Limit = 0
Iterative deepening search l = 0 AA
Chapter 3 62

Limit = 1
Iterative deepening search l = 1 AAAA
BCBCBCBC
Chapter 3 63

Limit = 2
Iterative deepening search l = 2 AAAA
BCBCBCBC
DEFGDEFGDEFGDEFG
AAAA
BCBCBCBC
DEFGDEFGDEFGDEFG
Chapter 3 64

Limit = 3
Iterative deepening search l = 3 AAAA
BCBCBCBC
DEFGDEFGDEFGDEFG
HIJKLMNOHIJKLMNOHIJKLMNOHIJKLMNO
AAAA
BCBCBCBC
DEFGDEFGDEFGDEFG
HIJKLMNOHIJKLMNOHIJKLMNOHIJKLMNO
AAAA
BCBCBCBC
DEFGDEFGDEFGDEFG
HIJKLMNOHIJKLMNOHIJKLMNOHIJKLMNO
Chapter 3 65

Complete??
Properties of iterative deepening search
Chapter 3 66

Properties of iterative deepening search
Complete?? Yes Time??
Chapter 3 67

Properties of iterative deepening search
Complete?? Yes
Time?? (d+1)b0 +db1 +(d−1)b2 +…+bd =O(bd) Space??
Chapter 3 68

Properties of iterative deepening search
Complete?? Yes
Time?? (d+1)b0 +db1 +(d−1)b2 +…+bd =O(bd) Space?? O(bd)
Optimal??
Chapter 3 69

Properties of iterative deepening search
Complete?? Yes
Time?? (d+1)b0 +db1 +(d−1)b2 +…+bd =O(bd)
Space?? O(bd)
Optimal?? Yes, if step cost = 1
Can be modified to explore uniform-cost tree
Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(IDS) = 50+400+3,000+20,000+100,000=123,450
N(BFS) = 10+100+1,000+10,000+100,000+999,990=1,111,100 IDS does better because other nodes at depth d are not expanded
BFS can be modified to apply goal test when a node is generated
Chapter 3 70

Criterion Breadth- First
Uniform- Depth- Cost First
Depth- Limited
Iterative Deepening
Complete? Yes∗ Time bd+1 Space bd+1 Optimal? Yes∗
Yes∗ No b⌈C∗/ǫ⌉ bm
Yes, if l ≥ d Yes bl bd bl bd
Summary of algorithms
b⌈C∗/ǫ⌉ bm Yes No
No Yes∗
Chapter 3 71

B
BB
Repeated states
Failure to detect repeated states can turn a linear problem into an exponential one!
AA
CCCCC
D
Chapter 3 72

end
Graph search
function Graph-Search( problem, fringe) returns a solution, or failure
closed ← an empty set
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 if State[node] is not in closed then
add State[node] to closed
fringe ← InsertAll(Expand(node, problem), fringe)
Chapter 3 73

Iterative deepening search uses only linear space
and not much more time than other uninformed algorithms
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
Graph search can be exponentially more efficient than tree search
Chapter 3 74

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