course: engineering artificial intelligence

University College Cork (Ireland) Department of Civil and Environmental Engineering

Course:Engineering Artificial Intelligence

Dr. Radu Marinescu

Lecture 2


Fundamental issues for most AI problems

o Representationo Searcho Inferenceo Planningo Learning

R. Marinescu October-2009 page 2


Representation

o Facts about the world have to be represented in some way, e.g., mathematical logic

o Deals with:• What to represent and how to represent?• How to structure knowledge?• What is explicit and what must be inferred?• How to encode “rules”?• How to deal with incomplete, inconsistent and probabilistic

knowledge?• What kinds of knowledge are required to solve problems?



Search

o Many tasks can be viewed as searching a very large problem space for a solution

• For example, Tic-Tac-Toe has 765 states, Chess has about 250 states, while Go has about 2100 states



Inference

o From some facts others can be inferred (related to search)

• For example, knowing "All elephants have trunks" and "Clyde is an elephant," can we answer the question "Does Clyde have a trunk?"

• What about "Peanuts has a trunk, is it an elephant?" Or "Peanuts lives in a tree and has a trunk, is it an elephant?”



Learning and planning

o Learning• Learn new facts about the world: e.g., machine learning

o Planning• Starting with general facts about the world, facts about the effects

of basic actions, facts about a particular situation, and a statement of a goal generate a strategy for achieving that goal in terms of a sequence of primitive steps or actions



Fundamental issues for most AI problems

o Representationo Searcho Inferenceo Planningo Learning



Search

o Main idea: Search allows exploring alternatives

o Backgroundo State space representationo Uninformed vs. informedo Any path vs. optimal patho Implementation and performance



Trees and graphs


C

B

ATree

root

Terminal(leaf)

Link(edge)

B is parent of CC is child of BA is ancestor of CC is descendant of A


Trees and graphs


C

B

ATree

root

Terminal(leaf)

Link(edge)


Directed Graph(one way streets)


Trees and graphs


C

B

ATree

root

Terminal(leaf)

Link(edge)


Directed Graph(one way streets)

Undirected Graph(two way streets)


Examples of graphs


San FranBoston

Wash DC

LA Dallas

Airline routes


Examples of graphs


San FranBoston

Wash DC

LA Dallas

Airline routes

Planning actions(graph of possible statesof the world)

A B C

A BC

A

BC

ACB B

AC

Put C on A

Put C on B

Put B on C

Put A on C

Put C on A


Problem solving paradigm

o What are the states? (All relevant aspects of the problem)

• Arrangement of parts (to plan an assembly)• Positions of trucks (to plan package distribution)• Cities (to plan a trip)• Set of facts (e.g., to prove a mathematical theorem)

o What are the actions (operators)? (deterministic, discrete)

• Assemble two parts• Move a truck to a new position• Fly to a new city• Apply a theorem to derive a new fact

o What is the goal test? (Condition for success)

• All parts in place• All packages delivered• Reached destination city• Derived goal fact



Example: holiday in Romania

o On vacation in Romania, currently in Arado Flight home leaves tomorrow from Bucharest



Example: holiday in Romania

o Goal• Be in Bucharest

o State-space• States: various cities• Actions: drive between cities

o Solution• Sequence of actions to destination



Solution to the holiday problem


Solution: go(Sibiu), go(Fagaras), go(Bucharest)

Cost: 140 + 99 + 211 = 450


State-space problem formulation

o A problem is defined by 4 items:• Initial state: e.g., in(Arad)• Actions or successor function: S(X) = set of action-state pairs

e.g., S(Arad) = {<go(Sibiu), in(Sibiu)>, <go(Zerind), in(Zerind)>, <go(Timisoara), in(Timisoara)>}

• Goal test: e.g., in(Bucharest)• Path cost (additive)

e.g., sum of distances to drive

o A solution is a sequence of actions leading from the initial state to a goal state



Vacuum cleaner state space

o States: location of dirt and roboto Initial state: anyo Actions: move robot left, right and sucko Goal state: no dirt at all locationso Path cost: 1 per action



8-queen puzzle state space

o States: arrangements of n ≤ 8 queens in the leftmost n columns, 1 per column, such that no queen attacks another

o Initial state: no queens on the board

o Actions: add queen to the leftmost empty column such that it is not attacked by any other queen

o Goal state: 8 queens on the board, none attacked

o Path cost: 1 per action



Sliding tile puzzle state space

o Stateso Initial stateo Actionso Goal stateo Path cost


Try yourselves



o States: locations of tileso Initial state: given (left)o Actions: move blank left, right, up, downo Goal state: given (right)o Path cost: 1 per action



Search algorithms

o Basic idea• Exploration of state space graph by generating successors of already-

explored states (a.k.a. expanding states)

• Every states is evaluated: is it a goal state?



Terminology

o State – Used to refer to the vertices in the underlying graph that is being searched, that is states in the problem domain, for example, a city, an arrangement of blocks or the arrangement of parts in a puzzle

o Search node – Refers to the vertices in the search tree that is being generated by the search algorithm. Each node refers to a state of the world; many nodes may refer to the same state.

• Importantly, a node implicitly represents a path (from the start state of the search tree to the state associated with the node). Because search nodes are part of the search tree, they have a unique ancestor node (except for the root node)



Terminology: more details

o A state is a (representation of) a physical configurationo A node is a data structure constituting part of a search

tree contains info such as: state, parent node, action, path cost g(x), depth



Search strategies

o A search strategy is defined by picking the order of node expansion

o Search 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?

o Time and space complexity are measured in terms of

• b: maximum branching factor of the search tree• d: depth of the search tree



Classes of search

o Any path search• Uninformed search• Informed search

o Optimal path search• Uninformed search• Informed search



Classes of search


Class Name Operation

Any Path Depth-First Systematic exploration of whole tree

Uninformed Breadth-First until a goal node is found.

Any Path Best-First Uses heuristic measure of goodness

Informed of a state, e.g. estimated distance to goal.

Optimal Uniform-Cost Uses path “length” measure.

Uninformed Finds shortest path.

Optimal A* Uses path “length” measure and heuristic

Informed Finds shortest path.


Simple search algorithm

o A search node is a path from some state X to the start state, e.g., (X B A S)o The state of a search node is the most recent state of the path, e.g., Xo Let Q be a list of search nodes, e.g., ((X B A S) (C B A S) …)o Let S be the start state

1. Initialize Q with search node (S) as only entry, set Visited = (S)

2. If Q is empty, fail. Else, pick some node N from Q

3. If state(N) is goal, return N (we’ve reached the goal)

4. (Otherwise) Remove N from Q

5. Find all descendants of state(N) not in Visited and create all the one-step extensions of N to each descendant

6. Add the extended paths to Q; add children of state(N) to Visited

7. Go to step 2.



Simple search algorithm

o A search node is a path from some state X to the start state, e.g., (X B A S)o The state of a search node is the most recent state of the path, e.g., Xo Let Q be a list of search nodes, e.g., ((X B A S) (C B A S) …)o Let S be the start state

1. Initialize Q with search node (S) as only entry, set Visited = (S)

2. If Q is empty, fail. Else, pick some node N from Q

3. If state(N) is goal, return N (we’ve reached the goal)

4. (Otherwise) Remove N from Q

5. Find all descendants of state(N) not in Visited and create all the one-step extensions of N to each descendant

6. Add the extended paths to Q; add children of state(N) to Visited

7. Go to step 2.


Critical decisions:Step 2: picking N from Q

Step 6: adding extensions of N to Q


Implementing the search strategies

o Depth-first• Pick first element of Q• Add path extensions to front of Q

o Breadth-first• Pick first element of Q• Add path extensions to end of Q



Terminology

o Visited – a state M is first visited when a path to M first gets added to Q. In general, a state is said to have been visited if it has ever shown up in a search node in Q. The intuition is that we have briefly “visited” them to place them in Q, but we have not yet examined them carefully.

o Expanded – a state M is expanded when it is the state of a search node that is pulled off of Q. At that point, the descendants of M are visited and the path that led to M is extended to the eligible descendants. We sometimes refer to the search node that led to M (instead of M itself) as being expanded. However, once a node is expanded we are done with it; we will not need to expand it again. In fact, we discard it from Q



Depth-First


S

A

B

C

D

G

Pick first element of Q; Add path extensions to front of Q

Q Visited

1

2

3

4

5

Added paths in blueWe show the paths in reversed order; the node’s state is the first entry


Depth-First


S

A

B

C

D

G


Q Visited

1 (S) S

2

3

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5


1


Depth-First


S

A

B

C

D

G


Q Visited

1 (S) S

2 (A S) (B S) A, B, S

3

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5


1

2


Depth-First


S

A

B

C

D

G


Q Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (C A S) (D A S) (B S) C, B, A, S

4

5


1

2

3


Depth-First


S

A

B

C

D

G


Q Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (C A S) (D A S) (B S) C, B, A, S

4 (D A S) (B S) C, D, B, A, S

5


1

2

3

4


Depth-First


S

A

B

C

D

G


Q Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (C A S) (D A S) (B S) C, B, A, S

4 (D A S) (B S) C, D, B, A, S

5 (G D A S) (B S) G, C, D, B, A, S


1

2

3

4

5


Depth-First: another (easier?) way to see it


S

A

B

C

D

G

S

A B

1

1

Numbers indicate order pulled off of Q (expanded)

Blue fill = Visited & ExpandedGray fill = Visited




S

A

B

C

D

G

S

A B

1

1



C D

2

2




S

A

B

C

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S

A B

1

1



C D

2

23

3




S

A

B

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S

A B

1

1



C D

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23

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C G

4

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NB: C is notvisited again




S

A

B

C

D

G

S

A B

1

1



C D

2

23

3

C G

4

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5

5


Implementing the search strategies

o Depth-first• Pick first element of Q• Add path extensions to front of Q

o Breadth-first• Pick first element of Q• Add path extensions to end of Q



Breadth-First


Pick first element of Q; Add path extensions to end of Q

S

A

B

C

D

GQ Visited

1 (S) S

2

3

4

5

6



Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3

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6


1


Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (B S) (C A S) (D A S) C, D, B, A, S

4

5

6


1

2


Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (B S) (C A S) (D A S) C, D, B, A, S

4 (C A S) (D A S) (G B S)* G, C, D, B, A, S

5

6

Added paths in blueWe show the paths in reversed order; the node’s state is the first entry* We could have stopped here, when the first path to the goal was generated

1

2

3


Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (B S) (C A S) (D A S) C, D, B, A, S


5 (D A S) (G B S) G, C, D, B, A, S

6


1

2

3

4


Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (B S) (C A S) (D A S) C, D, B, A, S


5 (D A S) (G B S) G, C, D, B, A, S

6


1

2

3

4

5


Breadth-First



S

A

B

C

D

GQ Visited

1 (S) S

2 (A S) (B S) A, B, S

3 (B S) (C A S) (D A S) C, D, B, A, S


5 (D A S) (G B S) G, C, D, B, A, S

6 (G B S) G, C, D, B, A, S


1

2

3

4

5

6


Breadth-First: another (easier?) way to see it


1

2

3

4

5

6

S

A B

1

C D

2

45

3

D G 6



NB: D is notvisited again


Worst case running time

o The number of states in the search space may be exponential in the some “depth” parameter, e.g. number of actions in a plan, number of moves in a game

o All searches may have to visit each state at least one, in the worst case

o So, all searches will have worst case running times that are at least proportional to the total number of states and therefore exponential in the “depth” parameter


b=2d=0

d=1

d=2

d is depthb is branching factor

bd < (bd+1 – 1) / (b - 1) < bd+1 states in tree

Max Time α Max # Visited


Worst case space


Depth-first max Q size(b – 1)d ≈ bd

visitedexpanded

Max Q size = Max(#Visited - #Expanded)

Breadth-first max Q sizebd


Cost performance of any-path methods


Search Worst Worst Fewest Guaranteed to

Method Time Space States find path?

Depth-First bd+1 bd No Yes*

Breadth-First bd+1 bd Yes Yes

Searching a tree with branching factor b and depth d(without a Visited list)

* If there are no infinitely long paths in the search space

Worst case time is proportional to the number of nodes added to QWorst case space is proportional to the maximal length of Q


Space (the final frontier)

o In large search problems, memory is often the limiting factor

o Imagine searching a tree with branching factor 8 and depth 10. Assume a node requires just 8 bytes of storage. The, breadth-first search might require up to

(23)10 x 23 = 233 bytes = 8,000 Mbytes = 8 Gbyteso One strategy is to trade time for memory. For example, we can

emulate breadth-first search by repeated application of depth-first search, each up to a preset depth limit. This is called iterative deepening search (IDS)

1. C = 12. Do DFS to max depth C. If path found, return it3. Otherwise, increment C and go to 2.



Summary

o State-space formulation• Abstract representation of the problem

o Search for a solution• Depth-first search: exponential time, linear space• Breadth-first search: exponential time, exponential space, but

finds the “shallowest” goal• Iterative deepening search: trades off time for space (simulates

breadth-first search) o Readings

• Chapter 3 in R&N book


course: engineering artificial intelligence

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