cs 175 project in ai. 2 lectures: ics 180 tuesday, thursday hours: 9.30-10.50 am discussion: dbh...

CS 175 Project in AI

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Lectures: ICS 180Tuesday, ThursdayHours: 9.30-10.50 am

Discussion: DBH 1300WednesdayHours: 1.00-1.50 pm

Instructor: Natalia FlerovaDBH 4099e-mail: [email protected]

TA: TBA

Class website:www.ics.uci.edu/~nflerova/cs175/index.html

mailto:[email protected]

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Grading

Project proposal: 5%Midterm report: 30%Demo-presentation: 30%Final report: 35%

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Grading

Project proposal: due April 7thMidterm report: due May 5thDemo-presentation: during 10th and finals weeksFinal report: due June 11th

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Project ideasConstraints:

Class scheduling

TA assignment

Sudoku

Bayesian networks:

Advising a first year student

Vision:

Image classification

Image recognition (e.g. handwritten digits recognition)

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Project ideasGames:

Othello (Reversi)

Checkers

Go

Japanese crossword

Etc.

Other:

RoboSoccer simulation

Email management project

Estimating space explored by heuristic search

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Project proposal

Single page report describing the 2 ideas for your project (indicate priorities). Be sure to address the following issues: What is the main purpose of your project? What tasks will your end product solve? What AI techniques and approaches are you planning to use (it's OK to provide only high level description) How are you planning to split the responsibilities between group members?

Constraint NetworksOverview

Suggested reading

Russell and Norvig. Artificial Intelligence: Modern Approach. Chapter 5.

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Good source of advanced information

Rina Dechter,

Constraint Processing,Morgan Kaufmann

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Outline CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:

– Consistency algorithms– Look-ahead methods– Look-back methods

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A Bred greenred blackgreen redgreen blackblack greenblack red

Constraint Satisfaction

Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, black) Constraints: A ≠ B, A ≠ D , D ≠ E, etc .

C

A

B

DE

F

G

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Constraint Network; DefinitionA constraint network is: R=(X,D,C)X variables

D domain

C constraints

R expresses allowed tuples over scopes

A solution is an assignment to all variables that satisfies all constraints (join of all relations).

Tasks: consistency?, one or all solutions, counting, optimization

X= { X 1 , .. . ,X n }D= {D1 , .. . ,Dn } ,Di= {v1 , . .. v k }

C= {C1 ,. ..C t },,,C i=( S i ,Ri )

Example The 4-queen problem

Q

QQ

Q QQ

QQ

Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.

Standard CSP formulation of the problem:• Variables: each row is a variable.

QQ

QQ

X 1

X 4

X 3

X 2

1 2 3 4

• Domains: D i= {1,2,3,4 } .

Example The 4-queen problem

Q

QQ

Q QQ

QQ

Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.

Standard CSP formulation of the problem:• Variables: each row is a variable.

QQ

QQ

X 1

X 4

X 3

X 2

1 2 3 4

• Domains: D i= {1,2,3,4 } .

• Constraints: There are = 6 constraints involved:42( )

R12= {(1,3 )(1,4 )(2,4 )(3,1)( 4,1) (4,2 )}R13= {(1,2 )(1,4)(2,1)( 2,3)(3,2 )(3,4 )( 4,1)( 4,3)}R14= {(1,2 )(1,3)( 2,1)( 2,3)( 2,4 )(3,1)(3,2 )(3,4 )(4,2 )( 4,3)}R23={(1,3 )(1,4 )(2,4 )(3,1 )(4,1 )(4,2 )}R24={(1,2)(1,4 )( 2,1)( 2,3)(3,2 )(3,4 )(4,1 )(4,3 )}R34={(1,3)(1,4 )(2,4 )(3,1 )(4,1 )(4,2 )}

• Constraint Graph :X 1

X 2 X 4

X 3

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Constraint’s representations

Relation: allowed tuples

Algebraic expression:

Propositional formula:

Constraint graph

X+Y 2≤10 ,X ≠Y

(a∨b)→¬c

X Y Z1 3 22 1 3

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Figure 1.8: Example of set operations intersection, union, and difference applied to relations.

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Constraint Graphs: Primal, Dual and Hypergraphs

A (primal) constraint graph: a node per variablearcs connect constrained variables.A dual constraint graph: a node per

constraint’s scope, an arc connect nodes sharing variables =hypergraph

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Search space

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Backtracking search

Backtracking

The search space A tree of all partial solutions A partial solution: (a1,…,aj) satisfying all

relevant constraints The size of the underlying search space depends

on: Variable ordering Level of consistency possessed by the problem

Search space and the effect of ordering

Backtracking

Complexity of extending a partial solution: Complexity of consistent O(e log

t), t bounds tuples, e constraints Complexity of selectvalue O(e k

log t)

A coloring problem example

Backtracking search for a solution

Backtracking Search for a Solution

Backtracking Search for All Solutions

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• Before search: (reducing the search space)– Arc-consistency, path-consistency, i-consistency– Variable ordering (fixed)

• During search:– Look-ahead schemes:

• Value ordering/pruning (choose a least restricting value), • Variable ordering (Choose the most constraining variable)

– Look-back schemes:• Backjumping• Constraint recording• Dependency-directed backtracking

Improving Backtracking

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Consistency methods Constraint propagation – inferring new

constraints Can get such an explicit network that the search

will find the solution without dead-ends. Approximation of inference:

Arc, path and i-consistency Methods that transform the original network

into a tighter and tighter representations

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Arc-consistency

32,1,

32,1, 32,1,

1 X, Y, Z, T 3X YY = ZT ZX T

X Y

T Z

32,1,

=

- infer constraints based on pairs of variables

Insures that every legal value in the domain of a single variable hasa legal match In the domain of any other selected variable

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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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Arc-consistency

32,1,

32,1, 32,1,


X Y

T Z

32,1,

=



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X Y

T Z

=

1 3

2 3

Arc-consistency

Arc-consistency algorithm domain of x domain of y

Arc is arc-consistent if for any value of there exist a matching value of

Algorithm Revise makes an arc consistentBegin

1. For each a in Di if there is no value b in Dj that matches a then delete a from the Dj.

End.Revise is , k is the number of value in each domain.

( X i ,X j ) X i X i

( X i ,X j )

O ( k 2 )

Algorithm AC-3• Begin

– 1. Q <--- put all arcs in the queue in both directions– 2. While Q is not empty do,– 3. Select and delete an arc from the queue Q

• 4. Revise• 5. If Revise cause a change then add to the queue all arcs

that touch Xi (namely (Xi,Xm) and (Xl,Xi)).– 6. end-while

• End• Complexity:

– Processing an arc requires O(k^2) steps– The number of times each arc can be processed is 2·k– Total complexity is

( X i ,X j )( X i ,X j )

O ( ek 3 )

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Sudoku –Constraint Satisfaction

Each row, column and major block must be alldifferent“Well posed” if it has unique solution: 27 constraints

2 34 62

•Variables: empty slots

•Domains = {1,2,3,4,5,6,7,8,9}

•Constraints: 27 all-different

•Constraint •Propagation

•Inference

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Path-consistency

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I-consistency

The Effect of Consistency Level

• After arc-consistency z=5 and l=5 are removed

• After path-consistency– R’_zx– R’_zy– R’_zl– R’_xy– R’_xl– R’_yl

Tighter networks yield smaller search spaces

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Look-back: Backjumping / Learning

• Backjumping: – In deadends, go back to the most recent culprit.

• Learning: – constraint-recording, no-good recording.– good-recording

Backjumping

• (X1=r,x2=b,x3=b,x4=b,x5=g,x6=r,x7={r,b})• (r,b,b,b,g,r) conflict set of x7• (r,-,b,b,g,-) c.s. of x7• (r,-,b,-,-,-,-) minimal conflict-set• Leaf deadend: (r,b,b,b,g,r)• Every conflict-set is a no-good

The cycle-cutset method• An instantiation can be viewed as blocking

cycles in the graph• Given an instantiation to a set of variables that

cut all cycles (a cycle-cutset) the rest of the problem can be solved in linear time by a tree algorithm.

• Complexity (n number of variables, k the domain size and C the cycle-cutset size):

O ( nkC k 2 )

Tree Decomposition

GSAT – local search for SAT(Selman, Levesque and Mitchell, 1992)

1. For i=1 to MaxTries2. Select a random assignment A3. For j=1 to MaxFlips4. if A satisfies all constraint, return A5. else flip a variable to maximize the score 6. (number of satisfied constraints; if no variable 7. assignment increases the score, flip at random)8. end9. end

Greatly improves hill-climbing by adding restarts and sideway moves

WalkSAT (Selman, Kautz and Cohen, 1994)

With probability p random walk – flip a variable in some unsatisfied constraintWith probability 1-p perform a hill-climbing step

Adds random walk to GSAT:

More Stochastic Search: Simulated Annealing, reweighting

• Simulated annealing:– A method for overcoming local minimas– Allows bad moves with some probability:

• With some probability related to a temperature parameter T the next move is picked randomly.

– Theoretically, with a slow enough cooling schedule, this algorithm will find the optimal solution. But so will searching randomly.

• Breakout method (Morris, 1990): adjust the weights of the violated constraints

cs 175 project in ai. 2 lectures: ics 180 tuesday, thursday hours: 9.30-10.50 am discussion: dbh...

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