PSU CS 370 – Introduction to Artificial Intelligence 1 Constraint Satisfaction Problems Chapter 5.

PSUCS 370 – Introduction to Artificial Intelligence1

Constraint Satisfaction Problems

Chapter 5


Outline

Constraint Satisfaction Problems (CSP) Backtracking search for CSPs Local search for CSPs

PSUCS 370 – Introduction to Artificial Intelligence

Intro Example: 8-QueensIntro Example: 8-Queens

• Purely generate-and-test• The “search” tree is only used to enumerate all possible 648 combinations



Another form of generate-and-test, with noredundancies “only” 88 combinations




What is Needed?What is Needed?

Not just a successor function and goal test But also a means to propagate the constraints

imposed by one queen on the others and an early failure test

Explicit representation of constraints and constraint manipulation algorithms


Constraint Satisfaction Constraint Satisfaction ProblemProblem Set of variables {X1, X2, …, Xn} Each variable Xi has a domain Di of possible

values Usually Di is discrete and finite Set of constraints {C1, C2, …, Cp} Each constraint Ck involves a subset of variables

and specifies the allowable combinations of values of these variables


Constraint Satisfaction Constraint Satisfaction ProblemProblem Set of variables {X1, X2, …, Xn} Each variable Xi has a domain Di of possible

values Usually Di is discrete and finite Set of constraints {C1, C2, …, Cp} Each constraint Ck involves a subset of variables

and specifies the allowable combinations of values of these variables

Assign a value to every variable such that all constraints are satisfied


Example: 8-Queens Example: 8-Queens ProblemProblem 64 variables Xij, i = 1 to 8, j = 1 to 8 Domain for each variable {yes,no} Constraints are of the forms:

Xij = yes Xik = no for all k = 1 to 8, kj Xij = yes Xkj = no for all k = 1 to 8, kI Similar constraints for diagonals


Example: 8-Queens Example: 8-Queens ProblemProblem 8 variables Xi, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms:

Xi = k Xj k for all j = 1 to 8, ji Similar constraints for diagonals


Example: Map ColoringExample: Map Coloring

• 7 variables {WA,NT,SA,Q,NSW,V,T}

• Each variable has the same domain {red, green, blue}• No two adjacent variables have the same value: WANT, WASA, NTSA, NTQ, SAQ, SANSW, SAV,QNSW, NSWV

WA

NT

SA

Q

NSWV

T

WA

NT

SA

Q

NSWV

T


Example: Street PuzzleExample: Street Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, Italian, Norwegian}Ci = {Red, Green, White, Yellow, Blue}Di = {Tea, Coffee, Milk, Fruit-juice, Water}Ji = {Painter, Sculptor, Diplomat, Violonist, Doctor}Ai = {Dog, Snails, Fox, Horse, Zebra}


Example: Street PuzzleExample: Street Puzzle

1 2 3 4 5Ni = {English, Spaniard, Japanese, Italian, Norwegian}Ci = {Red, Green, White, Yellow, Blue}Di = {Tea, Coffee, Milk, Fruit-juice, Water}Ji = {Painter, Sculptor, Diplomat, Violonist, Doctor}Ai = {Dog, Snails, Fox, Horse, Zebra}

The Englishman lives in the Red houseThe Spaniard has a DogThe Japanese is a PainterThe Italian drinks TeaThe Norwegian lives in the first house on the leftThe owner of the Green house drinks CoffeeThe Green house is on the right of the White houseThe Sculptor breeds SnailsThe Diplomat lives in the Yellow houseThe owner of the middle house drinks MilkThe Norwegian lives next door to the Blue houseThe Violonist drinks Fruit juiceThe Fox is in the house next to the Doctor’sThe Horse is next to the Diplomat’s

Who owns the Zebra?Who drinks Water?


Example: Task SchedulingExample: Task Scheduling

T1 must be done during T3T2 must be achieved before T1 startsT2 must overlap with T3T4 must start after T1 is complete

• Are the constraints compatible?• Find the temporal relation between every two tasks

T1

T2

T3

T4


Finite vs. Infinite CSPFinite vs. Infinite CSP

Finite domains of values finite CSP Infinite domains infinite CSP


Finite vs. Infinite CSPFinite vs. Infinite CSP

Finite domains of values finite CSP Infinite domains infinite CSP We will only consider finite CSP


Constraint GraphConstraint GraphBinary constraints

T

WA

NT

SA

Q

NSW

V

Two variables are adjacent or neighbors if theyare connected by an edge or an arc

T1

T2

T3

T4


CSP as a Search ProblemCSP as a Search Problem

Initial state: empty assignment Successor function: a value is assigned to any

unassigned variable, which does not conflict with the currently assigned variables

Goal test: the assignment is complete Path cost: irrelevant


CSP as a Search ProblemCSP as a Search Problem Initial state: empty assignment Successor function: a value is assigned to

any unassigned variable, which does not conflict with the currently assigned variables

Goal test: the assignment is complete Path cost: irrelevant

n variables of domain size d O(dn) distinct complete assignments


RemarkRemark

Finite CSP include 3SAT as a special case (see class of CS311)

3SAT is known to be NP-complete So, in the worst-case, we cannot expect to solve

a finite CSP in less than exponential time


Commutativity of CSPCommutativity of CSP

1. Generate successors of a node by considering assignments for only one variable

2. Do not store the path to node


Backtracking SearchBacktracking Search

empty assignment

1st variable

2nd variable

3rd variable

Assignment = {}



empty assignment

1st variable

2nd variable

3rd variable

Assignment = {(var1=v11)}



empty assignment

1st variable

2nd variable

3rd variable

Assignment = {(var1=v11),(var2=v21)}



empty assignment

1st variable

2nd variable

3rd variable

Assignment = {(var1=v11),(var2=v21),(var3=v31)}



empty assignment

1st variable

2nd variable

3rd variable




empty assignment

1st variable

2nd variable

3rd variable

Assignment = {(var1=v11),(var2=v22)}



empty assignment

1st variable

2nd variable

3rd variable



Backtracking search

Variable assignments are commutative}, i.e.,[ WA = red then NT = green ] same as [ NT = green then WA =

red ]

Only need to consider assignments to a single variable at each node b = d and there are dn leaves

Depth-first search for CSPs with single-variable assignments is called backtracking search

Backtracking search is the basic uninformed algorithm for CSPs

Can solve n-queens for n ≈ 25


Backtracking search


Backtracking example








Improving backtracking efficiency

General-purpose methods can give huge gains in speed: Which variable should be assigned next? In what order should its values be tried? Can we detect inevitable failure early?


Most constrained variable

Most constrained variable:choose the variable with the fewest legal values

a.k.a. minimum remaining values (MRV) heuristic


Most constraining variable

Tie-breaker among most constrained variables Most constraining variable:

choose the variable with the most constraints on remaining variables


Least constraining value

Given a variable, choose the least constraining value: the one that rules out the fewest values in the

remaining variables

Combining these heuristics makes 1000 queens feasible


Forward checking

Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values


Forward checking



Forward checking



Forward checking



Constraint propagation

Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures:

NT and SA cannot both be blue!

Constraint propagation repeatedly enforces constraints locally


Arc consistency algorithm AC-3

Time complexity: O(n2d3)


Backtracking AlgorithmBacktracking Algorithm

CSP-BACKTRACKING({})

CSP-BACKTRACKING(a) If a is complete then return a X select unassigned variable D select an ordering for the domain of X For each value v in D do

– If v is consistent with a then Add (X= v) to a result CSP-BACKTRACKING(a) If result failure then return result

Return failure

partial assignment of variables


Map ColoringMap Coloring

{}

WA=red WA=green WA=blue

WA=redNT=green

WA=redNT=blue

WA=redNT=greenQ=red

WA=redNT=greenQ=blue

WA

NT

SA

Q

NSWV

T


QuestionsQuestions1. Which variable X should be assigned a value

next?2. In which order should its domain D be sorted?


QuestionsQuestions

1. Which variable X should be assigned a value next?

2. In which order should its domain D be sorted?3. What are the implications of a partial

assignment for yet unassigned variables? ( Constraint Propagation)


Choice of VariableChoice of Variable

Map coloring

WA

NT

SA

Q

NSWV

T

WA

NT

SA



8-queen



#1: Minimum Remaining Values (aka Most-constrained-variable heuristic):

Select a variable with the fewest remaining

values



#2: Degree Heuristic (aka Most-constraining-variable heuristic):

Select the variable that is involved in the largest number of constraints on other unassigned variables

WA

NT

SA

Q

NSWV

T

SA


{}

Choice of ValueChoice of Value

WA

NT

SA

Q

NSWV

T

WA

NT


Choice of ValueChoice of Value

#3: Least-constraining-value heuristic: Prefer the value that leaves the largest subset

of legal values for other unassigned variables

{blue}

WA

NT

SA

Q

NSWV

T

WA

NT


Constraint Propagation …Constraint Propagation …

… is the process of determining how the possible values of one variable affect the possible values of other variables


Forward CheckingForward Checking

After a variable X is assigned a value v, look at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain any value that is inconsistent with v



WA NT Q NSW V SA T

RGB RGB RGB RGB RGB RGB RGB

TWA

NT

SA

Q

NSW

V



WA NT Q NSW V SA T


R GB RGB RGB RGB GB RGB

TWA

NT

SA

Q

NSW

V


WA NT Q NSW V SA T



R B G RB RGB B RGB


TWA

NT

SA

Q

NSW

V



WA NT Q NSW V SA T



R B G RB RGB B RGB

R B G R B RGB

Impossible assignments that forward checking do not detect

TWA

NT

SA

Q

NSW

V


4-Queens Problem4-Queens Problem

1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}



1

3

2

4

32 41

X1{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}


Arc consistency

Simplest form of propagation makes each arc consistent X Y is consistent iff

for every value x of X there is some allowed y


Arc consistency




Arc consistency


If X loses a value, neighbors of X need to be rechecked



Arc consistency


If X loses a value, neighbors of X need to be rechecked Arc consistency detects failure earlier than forward checking

Can be run as a preprocessor or after each assignment



Local Search for CSPLocal Search for CSP

12

33223

22

22

202

Pick initial complete assignment (at random)Repeat

• Pick a conflicted variable var (at random)• Set the new value of var to minimize the number of conflicts• If the new assignment is not conflicting then return it

(min-conflicts heuristics)


RemarkRemark

Local search with min-conflict heuristic works extremely well for million-queen problems

The reason: Solutions are densely distributed in the O(nn) space, which means that on the average a solution is a few steps away from a randomly picked assignment


Infinite-Domain CSPInfinite-Domain CSP

Variable domain is the set of the integers (discrete CSP) or of the real numbers (continuous CSP)

Constraints are expressed as equalities and inequalities

Particular case: Linear-programming problems


ApplicationsApplications

CSP techniques allow solving very complex problems

Numerous applications, e.g.: Crew assignments to flights Management of transportation fleet Flight/rail schedules Task scheduling in port operations Design Brain surgery

Timetabling problems Factory scheduling See www.ilog.com Transportation scheduling


Stereotaxic Brain Surgery


Stereotaxic Brain Surgery

• 2000 < Tumor < 22002000 < B2 + B4 < 22002000 < B4 < 22002000 < B3 + B4 < 22002000 < B3 < 22002000 < B1 + B3 + B4 < 22002000 < B1 + B4 < 22002000 < B1 + B2 + B4 < 22002000 < B1 < 22002000 < B1 + B2 < 2200

• 0 < Critical < 5000 < B2 < 500

T

C

B1

B2

B3B4

T


Constraint ProgrammingConstraint Programming

“Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.”Eugene C. Freuder, Constraints, April 1997


When to Use CSP When to Use CSP Techniques?Techniques?

When the problem can be expressed by a set of variables with constraints on their values

When constraints are relatively simple (e.g., binary)

When constraints propagate well


Summary CSPs are a special kind of problem:

states defined by values of a fixed set of variables goal test defined by constraints on variable values

Backtracking = depth-first search with one variable assigned per node

Variable ordering and value selection heuristics help significantly

Forward checking prevents assignments that guarantee later failure

Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies

Iterative min-conflicts is usually effective in practice


Additional ReferencesAdditional References

Surveys: Kumar, AAAI Mag., 1992; Dechter and Frost, 1999

Text: Marriott and Stuckey, 1998; Russell and Norvig, 2nd ed.

Applications: Freuder and Mackworth, 1994

Conference series: Principles and Practice of Constraint Programming (CP)

Journal: Constraints (Kluwer Academic Publishers)

Internet Constraints Archive http://www.cs.unh.edu/ccc/archive

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PSU CS 370 – Introduction to Artificial Intelligence 1 Constraint Satisfaction Problems Chapter 5.

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