artificial intelligence [intelligent agents paradigm]

ARTIFICIAL INTELLIGENCE[INTELLIGENT AGENTS PARADIGM]

Professor Janis Grundspenkis

Riga Technical University

Faculty of Computer Science and Information Technology

Institute of Applied Computer Systems

Department of Systems Theory and Design

E-mail: [email protected]

PROBLEM SOLVING BY SEARCHING

Problem Solving by Searching (1)

• Problem Solving Agent(one kind of a Goal-Based Agent)

–Relevant feature:Decision making what to do by finding sequence of actions that leads to desired states

Problem Solving by Searching (2)

• How problem is solved?Step 1. Goal formulationStep 2. Problem formulation – a process of

deciding what actions and states to considerStep 3. Search – systematic exploration of the

sequence of alternative states that appear in a problem solving process

Step 4. Solution – an action sequenceStep 5. Execution – recommended actions can

be accomplished

Simple-Problem-Solving Agent

> PerceptUpdate-State(state, percept)

> GoalFormulate-Goal(state)

> ProblemsFormulate-Problem(state, goal)

> Current stateSearch(problem)

> ActionRecommendation(state, action)

Return: action

Problem Types (1)

• Single-state problem– Percepts give enough information to tell

exactly which state is it

– Agent knows exactly what each of its actions does

– Agent can calculate exactly which state it will be in after any sequence of actions

Problem Types (2)

• Multiple-state problem– Percepts give not enough information to

tell which state is it

– Agent knows all the effects of its actions

– Agent must reason about sets of states that it might get to

Basic Elements of a Problem Definition

• A problem is a collection of information that the agent can use to decide what to do

• Information specification for single-state problem definition

Basic elements:– States– Actions

State Space of the Problem

• State space is the set of all states reachable from the initial state by any sequence of actions– The initial state is the state that the agent

knows itself to be in– The goal state corresponds to the solution of

the problem– The operator describes an action in terms of

which state will be reached by carrying out the action in a particular state

State Space Representation of the Problem Using Graph

• The nodes of a graph corresponds to particular problem solution states

• The arcs correspond to steps in a problem-solving process

• Initial states form root of the graph and correspond to the given information in a problem instance

• Goal states of the problem corresponds to the solutions to a problem instance

Formal State Space Representation of the Problem (1)

• A state space is represented by a four-tuple

[N, A, IS, GS], where

N is the set of nodes or states of the graph

A is the set of arcs between nodes

Formal State Space Representation of the Problem (2)

• A state space is represented by a four-tuple

[N, A, IS, GS], where

IS is the non-empty set that contains the initial states of the problem

(IS N)

GS is the non-empty set that contains the goal states of the problem

(GS N)

Testing the Goal States

• Agent can apply the goal test to a single-state description to determine if it is a goal state

• Description of the goal states:– Explicit set of possible goal states– A measurable property of the states

encountered in the search– A property of the path developed in the

search

Paths in the State Space

• Each path that does reach the goal is called a complete path

• Each path that does no reach the goal is called a partial path

• A solution path is a complete path through the graph from a node in IS to a node in GS

• A path cost function g is a function that assigns a cost to a path

• The cost of a path is the sum of the costs of the individual actions along the path

Measuring Problem-Solving Performance

• Three ways to measure the effectiveness of a search:– Does a search find a solution at all?– Is a solution good (one with a low path cost)?– What is the search cost associated with the

time and memory required to find a solution?

• The total cost of the search:Total cost = Path cost + Search

cost

Searching for Solutions (1)

• Search process builds up a search tree (subgraph of the state space)

• Search starts from the search node that corresponds to the initial state (the root of the search tree)

• Determining the children of a state is called expanding the state

Searching for Solutions (2)

• At each step, the search algorithm chooses one leaf node to expand– Leaf nodes have not successors

(children)Two cases:• They have not been expanded• They were expanded, but generated the

empty set

• Nodes that are waiting to be expanded are called a frontier

Search Algorithms for Agents

• INPUT: instances of data type PROBLEM

• OUTPUT: solution path• Data type PROBLEM has four

components:– Initial state– Operators– Goal test– Path cost function

Distinction Between Nodes and States

• A node is a data structure used to represent the search tree for a particular problem instance as generated by a particular problem

• A state represents a configuration of the world and is characterized by a set of features

Data Structure for Search Trees

• Node v is data structure with five components:– The corresponding state in the state space– The node that generates node v (the parent

node)– The operator that was applied to generate v– The depth of the node v (the number of

nodes on the path from the search node to v)– The path cost of the path from the search

node to v

Classification of Search Strategies (1)

If available information is specified:• Uninformed Search (Blind Search)

No information is available about:• the number of steps• the path cost

from the current state to the goal

• Informed Search (Heuristic Search)Problem-specific knowledge is used


If the search direction is specified:• Data-Driven Search (Forward

Chaining)– Search starts from the node(s) representing

the given facts of the problem, applies operators to produce new nodes that lead to a goal

– Searching forwards means generating successors successively starting from the initial (root) node


If the search direction is specified:• Goal-Driven Search (Backward

Chaining)– Search starts from the goal, applies operators

that could produce the goal, and chains backward to the given facts of the problem

– Searching backwards means generating predecessors successively starting from the goal node


If the order in which nodes are expanded is specified:

• Breadth-First Search• Uniform Cost Search• Depth-First Search• Depth-Limited Search• Iterative Deepening

Search• Bidirectional Search

UninformedSearch



• Best-First Search

– Greedy Search (minimizing theestimate cost to reach the goal)

– A* Search (minimizing the totalpath cost)

InformedSearch



• Memory Bounded Search

– IDA* Search (IterativeDeepening A* search)

– SMA* Search (SimplifiedMemory-Bounded A*)

InformedSearch



• Iterative ImprovementAlgorithms

– Hill-Climbing Search

– Random-Restart Hill-Climbing

– Simulated Annealing

– Beam Search

InformedSearch

UninformedSearch Strategies (1)

• Breadth-First Search–This search expands the shallowest

node in the search tree first

–This search explores the space in a level-by-level fashion

E

A

F

CB

IH

D G

JB- the initial node H – the goal

Iteration

0

1

2

3

4

5

6

OPEN

B

D F G

F G I J

G I J H

I J H

J H

H

CLOSED

B

B D

B D F

B D F G

B D F G I

B D F G I J

Breadth-first search

Data driven search

Goal


• Uniform-Cost Search–This search expands the lowest-

cost node on the frontier first

–The path cost must never decrease

g(successor(n)) g(n)for every node n


• Depth-First Search– This search expands the deepest node in

the search tree first– This search goes deeper into the search

space whenever this is possible

• Depth-Limited Search– This search places a limit on how deep a

depth-first search can go (imposes a cutoff on the maximum depth of a path)

Iteration

0

1

2

3

4

5

OPEN

B

D F G

I J F G

J F G

F G

H G

CLOSED

B

B D

B D I

B D I J

B D I J F

Depth-first search

E

A

F

CB

IH

D G

J

Goal

B- the initial node H – the goal

Data driven search

B- the initial node

F- the goal

Depth limit - 2

Iteration

0

1

2

OPEN

B0

D1 G1 E1

F2 H2 G1 E1

CLOSED

B

B D

Depth limited search

Goal

Node depth

E

F

C

I

H

D

J

A B

G

Data driven search


• Iterative Deepening Search– This search call depth-limited search with

increasing limits until a goal is found

– This search expands many states multiple times

• Bidirectional Search– This search goes both forwards from the initial

state and backward from the goal, and stop when the two searches meet in the middle

Iteration

0

1

2

OPEN

A

C D

D F G

CLOSED

A

A C

Solution is found because the node D is found in both OPEN lists

Bidirectional search

E

F

C

I

H

D

J

A B

G

A- the initial node

J- the goal

Iteration

0

1

2

OPEN

J

G H

H C D

CLOSED

J

J G

Data-driven search Goal-driven search

Evaluation of Uninformed Search Strategies

• Four Criteria– Completeness (is a solution guaranteed?)– Time Complexity (time needed for search)– Space Complexity (memory needed for

search)– Optimality (is highest-quality solution

guaranteed when there are several different solutions?)

Comparing Uninformed Search Strategies (1)

Criterion Breadth-First

Uniformcost

Depth-First

Complete? + + −Time Complexity bd bd bm

Space Complexity bd bd

bm

Optimal? + + −b is the branching factor (every state can be expanded to yield b new states),d is the depth of solution, m is the maximum depth of the search tree, and l is the depth limit

Comparing Uninformed Search Strategies (2)

Criterion Depth-Limited

Iterative Deepening

Bidirectional

Complete? +, if l d + +Time Complexity bl bd bd/2

Space Complexity bl bd bd/2

Optimal? − + +

b is the branching factor (every state can be expanded to yield b new states)d is the depth of solutionm is the maximum depth of the search treel is the depth limit

Strengths of Uninformed Search Strategies (1)

• Breadth-First Search–This search always finds the

shortest solution path

• Uniform-Cost Search–When the cost of the path never

decrease, the first solution that is found is guaranteed to be the cheapest solution


• Depth-First Search–For problems that have very many

solutions, depth-first may actually be faster than breadth-first search, because chances to find a solution after searching only a small part of the whole space is rather good


• Depth-Limited Search– It avoids the pitfalls of depth-first search

• Iterative Deepening Search– It is the preferred search method when

there is a large search space and the depth of the solution is not known

• Bidirectional Search– Can enormously reduce time complexity

Weaknesses of Uninformed Search Strategies (1)

• Breadth-First Search– Impractical in most cases due to the memory

requirements

– It is optimal only when all operators have the same cost

• Uniform-Cost Search– If some operators have negative cost, only an

exhaustive search of all nodes would guaranteed the optimal solution


• Breadth-First Search(assume that 1 million nodes can be generated per second and

that a node requires 1000 bytes of storage)• At depth 2 there are 110 nodes, time needed is 0,11 milliseconds

and 107 kilobytes of memory are needed• At depth 4 there are 11110 nodes, time needed is 11 milliseconds

and 10,6 megabytes of memory are needed • At depths 6, 8, 10, 12, 14, and 16 the number of nodes is10 in

corresponding power, time needed is 1,1 seconds, 2 minutes, 3 hours, 13 days, 3,5 years, and 350 years, respectively, while 1 gigabyte, 103 gigabites, 10 terabytes, 1 petabyte, 99 petabytes, and 10 exabytes of memory are needed

• Exponential-complexity search is a problem!


• Depth-First Search–Impractical in search trees with large

or infinite maximum depths (it can get stuck going down the wrong paths)

• Depth-Limited Search–In cases when depth limit is too small

there is no guarantee that the search will find the goal


• Iterative Deepening Search–It is wasteful, because many states

are expanded multiple times

• Bidirectional Search–It is not always applicable and its

memory requirements may be impractical

InformedSearch Strategies (1)

• Best-First Search– This search expands the node that appears to

be the best according to the evaluation function

* Greedy Search– This is one of the simplest best-first search

strategies that uses minimization of the estimated cost of the cheapest path (heuristic function h(n)) to reach the goal

– This search expands the node whose state is judged to be closest to the goal state

Best first searchA

CB D

FE G

IH

10 = 0+10

6 =1+5 8 =1+7 4 =1+3

5 =2+3 7 =2+5 4 =2+2

5 =3+2 3 =3+0

Iteration

0

1

2

3

4

5

OPEN

A(10)

C(8) B(6) D(4)

B(6) E(5) D(4) G(4)

F(7) E(5) D(4) G(4)

E(5) D(4) G(4) I(3)

H(5) D(4) G(4) I(3)

CLOSED

A(10)

A(10) C(8)

A(10) C(8) B(6)

A(10) C(8) B(6) F(7)

A(10) C(8) B(6) F(7) E(5)

The best measure is the highest value

A- the initial node

H- the goal

6 Node heuristic measures


• A* Search– This search expands the node n with the

lowest value of the estimated cost of the cheapest solution through n:

f(n) = g(n) + h(n)


• Memory-Bounded Search– This search conserves memory

• Iterative Deepening A* Search (IDA*)– In this search each iteration is a depth-first search

that is modified to use an f-cost limit (not depth limit)– This search expands all nodes inside the contour for

the current f-cost, peeping over the contour to find out where the next contour is (contour in the state space is a set of nodes where all nodes have f(n) less than or equal to f-cost of that contour)


• Simplified Memory-Bounded A* Search (SMA*)– This search drops nodes that are

unpromising (nodes with high f-cost) when it needs to generate a successor but has no memory left


• Iterative Improvement Algorithms– This search explores the state space trying to

find the best value of the estimated cost, which is the optimal solution

• Hill-Climbing Search– This search continually expands nodes that

have better estimated costs of the remaining distance to the goal then previous ones

Hill climbing

A

CB D

FE G

IH

10 = 0+10

9= 1+8 5 =1+4 8 =1+7

4 =2+2 2 =2+0 6 =2+4

3 =3+0 8 =3+5

Iteration

0

1

2

3

OPEN

A(10)

C(5)

E(4)

H(3)

CLOSED

A(10) B(9) D(8)

A(10) B(9) D(8) C(5) G(6)

A(10) B(9) D(8) C(5) G(6) E(4) I(8)

The best measure is the lowest value

A- the initial node

H- the goal



• Simulated Annealing– This search picks a randomly selected

successor of the current node. If the move improves the situation, it is always executed. Otherwise, the algorithm makes the move with some probability less than 1

– The probability decreases exponentially with the “badness” of the move calculated as the difference of values between next and current states


• Beam Search–This search proceeds as in

breadth-first search, except that only the best W states are chosen at each level

Beam searchA

CB D

FE G

IH

10 =0+10

6 =1+5 8 =1+7 4 =1+3

5 =2+3 7 =2+5 4 =2+2

4 = 3+1 3 =3+0

Iterācija

0

1

2

3

OPEN

A(10)

C(8) B(6)

F(7) E(5)

H(4) I(3)

The goal is reached

CLOSED

A(10) D(4)

A(10) D(4) C(8) B(6) G(4)

A(10) D(4) C(8) B(6) G(4) F(7) E(5)

The best measure is the highest value

A- the initial node

H- the goal

Beam width - 2


Strengths of Informed Search Strategies (1)

More information leads to reduced time and space complexity. The time complexity depends on the quality of the heuristic function

• Best-First Search

– This search can find solutions more effectively than uninformed search

– It is good when there is natural measure of goal distance and a good partial path may look like a bad option before more promising partial paths are played out


* Greedy Search– This search substantially reduces the time

and space complexity if a good heuristic function is used

– This search cuts the search cost considerably

* A* Search– This search is optimally efficient for any given

heuristic function (no other optimal algorithm is guaranteed to expand fewer nodes)


• Memory-Bounded Search– This search conserves memory

* IDA* Search– IDA* only goes through two or three

iterations since typically f only increases two or three times along any solution path


* SMA Search–SMA* can make use of all available

memory to carry out the search, and avoids repeated states as far as its memory allows

–Given a reasonable amount of memory, SMA* can solve significantly more difficult problems than A*


• Iterative Improvement Algorithms–This search keeps track of only the

current state and do not look ahead beyond the immediate neighbors of that state


* Hill-Climbing Search– This search does not maintain a search

tree, so the node data structure need only record the state and its evaluation

– It is good when there is a natural measure of distance from each place to the goal and a good path is likely to be among the partial paths that appear to be good at each choice point


* Random-Restart Hill-Climbing Search

– This search eventually finds the optimal solution if enough iterations are allowed

– Usually a reasonably good solution can be found after a small number of iterations

* Simulated Annealing

– This search provides a way to escape local maxima by picking a random move instead of the best move

– It is applied widely to large-scale optimization tasks


* Beam Search– This search moves downward only through the

best W nodes at each level, the other nodes are ignored

– The number of nodes explored remains manageable, even if there is a great deal of branching and the search is deep

– It is good when there is natural measure of goal distance and a good path is likely to be among the partial path that appear to be good at all levels

Weaknesses of Informed Search Strategies (1)

Because heuristics use limited information they can lead a search algorithm to a suboptimal solution or fail to find any solution at all

• Best-First Search– In this search evaluation function

sometimes can lead search astray. It is not good when the branching factor is too large


* Greedy Search– This search is susceptible to false starts

– heuristic may cause unnecessary nodes to be expanded. The algorithm is neither complete nor optimal

* A* Search– This search is not an answer to all

searching needs because at the worst case exponential growth will occur


• Memory-Bounded Search

* IDA* Search– If the heuristic value is different for every

state then each contour only included one more state than the previous contour, and IDA* will expand N2 nodes in N iterations (too long to wait in complex domains)


* SMA* Search– On very hard problems this search is

forced to continually switch back and forth between a set of candidate solution paths

– Extra time is required for repeated generation of the same nodes, and as a consequence memory limitations can make a problem intractable from the point of view of computation time (algorithm is not optimal)


• Interactive Improvement Algorithms* Hill-Climbing Search

– This search suffers from local maxima (on it the algorithm will halt even though the solution may be far from satisfactory), plateaux (the search will conduct a random walk because the evaluation function is essentially flat), ridges (if there are not operators that move directly to the top of the ridge, the search may oscillate from side to side, making little progress)


* Random-Restart Hill-Climbing Search– If it is a complex problem the solution is not

guaranteed due to the inherent limitation of heuristic search

* Simulated Annealing– In cases when the values of the parameter

used to determine the probability of allowing moves are not lowered slowly enough the algorithm will not find a global optimum


* Beam Search–The solution is not guaranteed due

to the inherent limitations of heuristic search (the algorithm is not complete)

artificial intelligence [intelligent agents paradigm]

Documents

problem definitiona

problem instancegoal

inthe goal state

problem formulation

singlestate problempercepts

problem gs ntesting

singlestate description

multiplestate problempercepts