Solving problems by searching

Chapter 3

Outline

Problem-solving agents

Problem types

Problem formulation

Example problems

Basic search algorithms

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

Example: Romania

Problem types

Deterministic, fully observable à single-state problem

Agent knows exactly which state it will be in; solution is a sequence

Non-observable à sensorless problem (conformant problem)

Agent may have no idea where it is; solution is a sequence

Nondeterministic and/or partially observable à contingency problem

percepts provide new information about current state

often interleave} search, execution

Unknown state space à exploration problem

Example: vacuum world

Single-state, start in #5. Solution?

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?

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?

Single-state problem formulation

A problem is defined by four items:

initial state e.g., "at Arad"

actions or successor function S(x) = set of action–state pairs

e.g., S(Arad) = {<Arad à Zerind, Zerind>, … }

goal test, can be

explicit, e.g., x = "at Bucharest"

implicit, e.g., Checkmate(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

Selecting a state space

Real world is absurdly complex

à state space must be abstracted for problem solving

(Abstract) state = set of real states

(Abstract) action = complex combination of real actions

e.g., "Arad à 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

Vacuum world state space graph

states?

actions?

goal test?

path cost?

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

Example: The 8-puzzle

states?

actions?

goal test?

path cost?

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]

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

Tree search algorithms

Basic idea:

offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)

Tree search example

Implementation: general tree search

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.

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 ∞)

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

Breadth-first search

Expand shallowest unexpanded node

Implementation:

fringe is a FIFO queue, i.e., new successors go at end

Properties of breadth-first search

Complete? Yes (if b is finite)

Time? 1+b+b²+b³+… +b^d + b(b^d-1) = O(b^d+1)

Space? O(b^d+1) (keeps every node in memory)

Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

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(b^ceiling(C*/^ε⁾) where C^* is the cost of the optimal solution

Space? # of nodes with g ≤ cost of optimal solution, O(b^ceiling(C*/^ε⁾)

Optimal? Yes – nodes expanded in increasing order of g(n)

Depth-first search

Expand deepest unexpanded node

Implementation:

fringe = LIFO queue, i.e., put successors at front

Properties of depth-first search

Complete? No: fails in infinite-depth spaces, spaces with loops

Modify to avoid repeated states along path

à complete in finite spaces

Time? O(b^m): 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? No

Depth-limited search

= depth-first search with depth limit l,

i.e., nodes at depth l have no successors

Recursive implementation:

Iterative deepening search

Iterative deepening search l =0

Iterative deepening search l =1

Iterative deepening search l =2

Iterative deepening search l =3

Iterative deepening search

Number of nodes generated in a depth-limited search to depth d with branching factor b:

N_DLS = b⁰ + b¹ + b² + … + b^d-2 + b^d-1 + b^d

Number of nodes generated in an iterative deepening search to depth d with branching factor b:

N_IDS = (d+1)b⁰ + d b^¹ + (d-1)b^² + … + 3b^d-2 +2b^d-1 + 1b^d

For b = 10, d = 5,

N_DLS= 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111

N_IDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

Overhead = (123,456 - 111,111)/111,111 = 11%

Properties of iterative deepening search

Complete? Yes

Time? (d+1)b⁰ + d b¹ + (d-1)b² + … + b^d = O(b^d)

Space? O(bd)

Optimal? Yes, if step cost = 1

Summary of algorithms

Repeated states

Failure to detect repeated states can turn a linear problem into an exponential one!

Graph search

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


	Problem-solving agents
	Problem types
	Problem formulation
	Example problems
	Basic search algorithms


	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


	Deterministic, fully observable à single-state problem
		Agent knows exactly which state it will be in; solution is a sequence
	Non-observable à sensorless problem (conformant problem)
		Agent may have no idea where it is; solution is a sequence
	Nondeterministic and/or partially observable à contingency problem
		percepts provide new information about current state
		often interleave} search, execution
	Unknown state space à exploration problem


	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?


	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?


	A problem is defined by four items:

	initial state e.g., "at Arad"
	actions or successor function S(x) = set of action–state pairs
		e.g., S(Arad) = {<Arad à Zerind, Zerind>, … }
	goal test, can be
		explicit, e.g., x = "at Bucharest"
		implicit, e.g., Checkmate(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


	Real world is absurdly complex
		à state space must be abstracted for problem solving
	(Abstract) state = set of real states
	(Abstract) action = complex combination of real actions
		e.g., "Arad à 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







	states? integer dirt and robot location
	actions? Left, Right, Suck
	goal test? no dirt at all locations
	path cost? 1 per action







	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]







	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


	Basic idea:
		offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)


	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.


	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 ∞)


	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


	Expand shallowest unexpanded node
	Implementation:
		fringe is a FIFO queue, i.e., new successors go at end


	Complete? Yes (if b is finite)
	Time? 1+b+b²+b³+… +b^d + b(b^d-1) = O(b^d+1)
	Space? O(b^d+1) (keeps every node in memory)
	Optimal? Yes (if cost = 1 per step)

	Space is the bigger problem (more than time)


	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(b^ceiling(C/^ε⁾) where C^ is the cost of the optimal solution
	Space? # of nodes with g ≤ cost of optimal solution, O(b^ceiling(C*/^ε⁾)
	Optimal? Yes – nodes expanded in increasing order of g(n)


	Expand deepest unexpanded node
	Implementation:
		fringe = LIFO queue, i.e., put successors at front


Complete? No: fails in infinite-depth spaces, spaces with loops
	Modify to avoid repeated states along path
		à complete in finite spaces
Time? O(b^m): 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? No


	= depth-first search with depth limit l,
	i.e., nodes at depth l have no successors

	Recursive implementation:


	Number of nodes generated in a depth-limited search to depth d with branching factor b:
	N_DLS = b⁰ + b¹ + b² + … + b^d-2 + b^d-1 + b^d

	Number of nodes generated in an iterative deepening search to depth d with branching factor b:
	N_IDS = (d+1)b⁰ + d b^¹ + (d-1)b^² + … + 3b^d-2 +2b^d-1 + 1b^d

	For b = 10, d = 5,
		N_DLS= 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111
		N_IDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

	Overhead = (123,456 - 111,111)/111,111 = 11%


	Complete? Yes
	Time? (d+1)b⁰ + d b¹ + (d-1)b² + … + b^d = O(b^d)
	Space? O(bd)
	Optimal? Yes, if step cost = 1


	Failure to detect repeated states can turn a linear problem into an exponential one!


	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