Artificial IntelligenceArtificial Intelligence

University Politehnica of Bucharest

2007-2008

Adina Magda Florea

http://turing.cs.pub.ro/ai_07

Lecture No. 2

Problem solving strategies Representing problem solution Basic search strategies Informed search strategies

1. Representing problem solution

Symbolic structure Computational instruments Planning method / approach

1.1. State space representation

state, state space, initial state, final state(s), operators

(Si, O, Sf) Problem solution

2 3

1

7 6

8 4

5

S i

2 31

7 6

8 4

5

S f

(a) Stare initiala (b) Stare finala

SUS - Mutare patrat liber in susSTINGA - Mutare patrat liber la stingaJOS - Mutare patrat liber in josDREAPTA - Mutare patrat liber la dreapta

(c) Operatori

2 3

1

7 6

8 4

5

S i

2 3

1

7 6

8 4

5

2 3

1

7 6

8

4

5

2 3

1

7 6

8 4

5

STINGA JOS DREAPTA

(d) Tranzitii posibile din starea Si

S1 S2 S3

8-puzzle

1.2 AND/OR graph representation

Problem decomposition into sub-problems (Pi, O, Pe) AND/OR graph Solved node Unsolvable node Problem solution

Nod SAU

Noduri SI

Noduri SAU

AND/OR graph

OR

AND

A B C A B C

(a) Stare initiala (b) Stare finala

n = 3A la C

n = 2 n = 2

n = 1

A la B

A la C

B la C

n = 1A la C

n = 1A la B

n = 1C la B

n = 1B la A

n = 1B la C

n = 1A la C

(c) Arborele SI/SAU de descompunere in subprobleme

O - operator de descompunere

Towers of Hanoi

1.3 Equivalence of representations

(a) Spatiul starilor

S k

S j

Tranzitie din in S j S f

S , - stari intermediarej S k S - stare finala f

(b) Descompunerea problemei in subprobleme

Tranzitie din in S j S k Tranzitie din in S k S f

State spaceProblem decomposition

2. Basic search strategies

Criteria Completeness Optimality Complexity Possibility to backtrack Informedness

Conventions: unknown node, evaluated node, expanded node, OPEN, CLOSED

Grad deinformare

CostComputational

Cost total

Cost control(cost evaluare stari)Cost parcurgere

(cost aplicareoperatori)

Neinformat Informat

Computational costs of search

2.1. Uninformed search in state space Algorithm BREADTH: Breadth first search in state space1. Init lists OPEN {Si}, CLOSED {}2. if OPEN = {}

then return FAIL3. Remove first node S from OPEN and insert it in CLOSED4. Expand node S

4.1. Generate all direct successors Sj of node S4.2. for each successor Sj of S do

4.2.1. Make link Sj S4.2.2. if Sj is final state

theni. Solution is (Sj, S, .., Si)ii. return SUCCESS

4.2.3. Insert Sj in OPEN, at the end5. repeat from 2end.

Features of breadth first search Previous algorithm for tree space not graph For graphs - Insert step 3’

3’. if S OPEN CLOSED then repeat from 2

Depth first search in state space Depth of a node Ad(Si) = 0, where Si is the initial state, Ad(S) = Ad(Sp)+1, where Sp is the predecessor

node of S.

Algorithm DEPTH(AdMax): Depth first search in state space1. Init lists OPEN {Si}, CLOSED {}2. if OPEN = {}

then return FAIL3. Remove first node S from OPEN and insert it into CLOSED3’. if Ad(S) = AdMax then repeat from 24. Expand node S

4.1. Generate all successors Sj of node S4.2. for each succesor Sj of S do

4.2.1. Set link Sj S4.2.2. if Sj is final state

theni. Solution is (Sj,.., Si)ii. return SUCCESS

4.2.3. Insert Sj in OPEN, at the beginning5. repeat from 2end.

Features of depth first search

Iterative deepeningfor AdMax=1, Val do

DEPTH(AdMax)

Features of iterative deepening

Bidirectional search

Which strategy to choose ?

2.2. Uninformed search in AND/OR graphs

Depth of a node Ad(Si) = 0, where , Si is the node of the

initial problem Ad(S) = Ad(Sp) + 1 if Sp is node OR

predecesor of node S, Ad(S) = Ad(Sp) if Sp is node AND

predecesor of node S.

Algorithm BREADTH-ANDOR: Breadth first search in AND/OR trees

1. Init lists OPEN {Si}, CLOSED {}

2. Remove first node S from OPEN and insert it into CLOSED

3. Expand node S

3.1. Generate all direct successors Sj of node S

3.2. for each successor Sj of S do

3.2.1. Make link Sj S

3.2.2. if Sj is a set of at least 2 sub-problems

then /* is an AND node*/

i. Generate all successors sub-problems Skj of Sj

ii. Set link Skj Sj

iii. Insert nodes Skj in OPEN, at the end

3.2.3. else Insert Sj in OPEN, at the end

4. if no successor of S was generated in previous step (3)then4.1. if S is terminal node labeled with a non-elementary problem

then4.1.1. Label S unsolvable4.1.2. Label with unsolvable all nodes predecessor of S which become

unsolvable because of S4.1.3. if node AND is unsolvable

then return FAIL /* no solution */4.1.4. Remove from OPEN all nodes which have unsolvable

predecessors4.2. else /* S is terminal node labeled with a non-solvable problem */

4.2.1. Label S solved4.2.2. Label with solved all nodes predecessor of S which become

unsolvable because of S4.2.3. if node AND is solved

theni. Build solution tree following the linksii. return SUCCESS /* Solution found */

4.2.4. Remove from OPEN all solved nodes and all nodes which have solved predecessors

5. repeat from 2end.

2.3. Complexity of search strategies B – branching factor of the search space8-puzzleNumber of moves: 2 m for corners = 8 3 m for laterals = 12 4m for center 24 moves B = no. moves / no. pozitions of free square = 2.67Number of moves : 1 m for corners = 4 2 m for laterals = 8 3m for center 15 moves B = 1.67

Complexity of search strategies

B - branching factor

Root – B nodes, B2 on level 2, etc. Number of states possible to be generated on a

search level d is Bd

T – total number of states generated during a search, d – depth of solution node

T = B + B2 + … + Bd = O(Bd)

Number of generated nodes

Breadth first search

B + B2 + … + Bd + (Bd+1-B) = O(Bd+1)B – branching factor, d – depth of solution

Depth first searchB - branching factor, m – maximum depth

B*m+1 Iterative deepening search

d*B+(d-1)*B2+ … + (1)*Bd = O(Bd)

Complexity of search strategiesCriterion Level Depth Limited

depthIterative deepening

Bidirectional

Time Bd Bm Bl Bd Bd/2

Space Bd B*m B*l Bd Bd/2

Optimality?

Yes No No Yes Yes

Completeness

Yes No Yes if ld Yes Yes

B – branching factor, d – solution depth,m – maximum depth of the tree, l –limit of search (AdMax)

3. Informed search strategies

Use heuristic knowledge to incraese efficiency of search:

Select which node to expand nex during search

While expanding a node decide which successors to generate and which to ignore

Remove from the search space some nodes that have previously been generated – prune the search space

3.1 Best-first search

Evaluate the information that can be obtained by expanding a node and ist importance in guiding the search

The quality of a node is estimated by the heuristic search function w(n) for node n

hill climbing strategy best-first strategy

Algorithm BFS: Best-first in state space

1. Init lists OPEN {Si}, CLOSED {}

2. Compute w(Si) and associate this value to Si

3. if OPEN = {}

then return FAIL

4. Remove node S with minimum w(S) from OPEN and insert it in CLOSED

5. if S is final state

then

i. Solution is (S,.., Si)

ii. return SUCCESS

6. Expand node S

6.1. Generate all successors Sj of node S

6.2. for each succesor Sj of S do

6.2.1 Compute w(Sj) and associate it to Sj

6.2.2. Set link Sj S

6.2.3. if Sj OPEN CLOSED

then insert Sj in OPEN with associated w(Sj) 6.2.5. else

i. Be S’j the copy of Sj from OPEN or CLOSED

ii. if w(Sj) < w(S’j) then

- Remove link S’j Sp, with Sp pred. of S’j

- Set link S’j S, and update cost of S’j to w(Sj)

- if S’j is in CLOSED

- then remove S’j from CLOSED and insert S’j in OPEN

iii. else ignore node Sj

7. repeat from 3

end.

Cases

Best-first strategy is a generalization of uninformed search strategies- Breadth first search w(S) = Ad(S)- Depth first search w(S) = -Ad(S)

Uniform cost strategy

Minimize the search effort – heuristic searchw(S) = heuristic function

w(S ) = cost_ arc(S ,S )j k k 1k i

j 1

3.2 Optimal solutions: A*Algorithm

w(S) becomes f(S) with 2 components: g(S), estimates the real cost g*(S) of the

search path from Si to S, h(S), estimates the real cost h*(S) of the

search path from S to Sf. f(S) = g(S) + h(S) f*(S) = g*(S) + h*(S)

S i

S

S f

g(S)

h(S)

f(S)

Components of A*

Computing f(S) Computation of g(S)

Computation of h(S) Must be admissible A heuristic function is called admissible if for any

state S, h(S) h*(S). This definition gives the admissibility condition

of function h and is used to define the property of admissibility of an algorithm A*.

g(S) = cost_ arc(S ,S )k k 1k i

n

A* admissibility

Consider an algorithm A* which uses g and h compenents of f if

(1) h satisfies the admissibility condition (2) for any 2 states S, S', where c > 0 is a constant and

the cost is finite then A* algorithm is admissible, i.e., is

guaranteed to find the path of minimal cost to solution.

Completeness

cost_ arc(S,S') c

Implementation of A*Modify the algorithm correspnding to the "best-first" strategy

in state space as such:…2. Compute w(Si)=g(Si) + h(Si) and associate this value to Si

3. if OPEN = {}then return FAIL - unmodified

4. Remove node S with w(S) minimum from OPEN and insert it into CLOSED - unmodified

…..6.2.5. else

i. Be S’j the copy of Sj from OPEN or CLOSED

ii. if g(Sj) < g(S’j) then …

The heuristic of A*

Consider 2 algorithms A*, A1 and A2, with evaluation functions h1 and h2 admissibile, g1=g2

It is said that A2 is more informed than A1 if for any state S

with SSf

We say that h2 dominates h1 monotony

h (S) > h (S)2 1

f (S) g (S) h (S)1 1 1 f (S) g (S) h (S)2 2 2

h(S) h(S') + cost_ arc(S,S')

How we compute f 8-puzzle

Traveling salesman

h2(S) = the cost of the minimum spanning tree of unvisited cities until S

h (S) = t (S)1 ii=1

8

h (S) = Distanta(t2 ii=1

8)

h (S) = cost_ arc(S ,S)1 i

Missionaries and cannibals

ESTVEST

n

n

n

n

Em

Vm

Vc

Ec

(S

(S

(S

(Si

i i

i

)=3

)=3)=0

)=0

ESTVEST

n

n

n

n

Em

Vm

Vc

Ec

(S

(S

(S

(Sf

f f

f

)=0

)=0)=3

)=3

(b) Stare finala(a) Stare initiala

Missionaries and cannibals

f (S) g(S) h (S)1 1

f (S) g(S) h (S)2 2

f (S) g(S) h (S)3 3

h (S) = n (S)1E

h (S) n (S) / 22E

0(S)n daca0

0(S)n si EST de malul pe este barca daca1(S)n

0(S)n si VEST de malul pe este barca daca1(S)n

(S)hE

EE

EE

3

Relaxing the admissibility condition of A* An heuristic function h is called -admissible if

with > 0 An A* algorithm which uses an evaluation

function f with a component h -admissible will find a solution which has a cost greater than the cost of the optimal solution with at most .

such an algorithm - an -admissible algorithm and the solutions is called -optimal solution.

h(S) h (S)*

Relaxing the admisibility condition of A*

8-puzzle

f (S) g(S) h (S)3 3 h (S) h (S) 3 T(S)3 2

T(S) = Scor[t (S)]ii 1

8

mozaicului centrul laaflat pentru t 1

centru de diferita tlui a pozitie oricepentru 0

finala stareadin corect succesorul

deurmat estenu S stareain tpatratul daca 2

=(S)]Scor[t

i

i

i

i

Recursive Best First

Best first with linear space Recursive implementation Remembers the value f of the best alternate

path which starts from any precedent node of the current node

Finds the minimum cost solution if h is admissible but has a space complexity of O(B*d)

S1

S3

S8S5

S2

S6 S7

S4

S9 S10 S11

447447 449

inf

393

646 415 526

413

526

417

553

415

S1

S3

S8S5

S2

S6 S7

S4

S12 S13

447447 449

inf

393

646 415 526

417

450591

417

S1

S3

S8S5

S2

S6 S7

S4

S9 S10 S11

447447 449

inf

393

646 450 526

417

526 417 553

447

S14 S15

615418

S16

607

447

BestFR(S) Recursive Best First strategy/* returns the solution or FAIL */

BFR(Si, inf)

Algorithm BFR(S, f_lim): Recursive Best First strategy

/* Returns a solution or FAIL and a new limit f_limit */

1. if S final state then return S, f_lim

2. Generate all successors Sj of S

3. if there are no successors

then return FAIL, inf

4. for each successor Sj do

f(Sj) max(g(Sj) + h(Sj), f(S))

5. Best Sjmin, node with minimal value f(Sj) among successors

6. if f(Best) > f_lim

then return FAIL, f(Best)

7. Alternat f(Sjmin2), the 2nd smallest value f(Sj)

8. Rez, f(Best) BFR(Best, min(f_lim, Alternat)

9. if Rez FAIL then return Rez, f(Best)

10. repeat from step 5

end.