Course: Engineering Artificial Intelligence

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

Course:Engineering Artificial Intelligence

Dr. Radu Marinescu

Lecture 4


Today’s class

o Constraint satisfaction problems (CSP)

R. Marinescu October-2009 page 2


Each row, column and major block must be alldifferent

“Well posed” if it has unique solution: 27 constraints

2 34 62

Constraint propagation

•Variables: 81 slots

•Domains = {1,2,3,4,5,6,7,8,9}

•Constraints: •27 not-equal

Sudoku


Sudoku

Each row, column and major block must be alldifferent

“Well posed” if it has unique solution: 27 constraints


Constraint satisfaction problems

o CSP• state – is defined by variable Xi with values from domain Di

• goal test – is a set of constraints specifying allowable combinations of values for subsets of variables

o Applications for CSPs• Scheduling tasks (transportation, maintenance)• Robot planning tasks• Puzzles• Tail-Gate assignment (airports)• Radio-link frequency assignment (telecom)• …



Example: map coloring

o variables: WA, NT, Q, SA, NSW, V, To domains: Di = {red, green, blue}o constraints: adjacent regions must have different colors

• e.g., WA ≠ NT or (WA, NT) in {(red, green), (red, blue), (green, red), (green, blue), (blue, red), (blue, green)}



Example: map coloring

o Solutions are complete and consistent assignments• E.g., WA = red, NT = green, SA = blue, Q = red, NSW = green, V = red, T = green



Constraint graph


Binary CSP

variable Vi withvalues in domain Di

General class of problems:

Unary constraint arc

Binaryconstraintarc

(Unary constraintjust cut down domains)

This diagram is called a constraint graph


Constraint graph


Binary CSP

variable Vi withvalues in domain Di

General class of problems:

Unary constraint arc

Binaryconstraintarc

(Unary constraintjust cut down domains)

This diagram is called a constraint graph

Basic problem:Find dj in Di for each Vi s.t. all constraints satisfied(finding consistent labeling for variables)


N-Queens as CSP


1 Q

2 Q

3 Q

4 Q

1 2 3 4

Place N queens on an NxNchessboard so that none canattack the other

Classic “benchmark” problem

Variables

Domains

Constraints

are board positions in NxN chessboard

Queen or blank

Two positions on a line (vertical, horizontal,diagonal) cannot both be Q


Scheduling as CSP


Choose time for activities e.g.,observations on Hubble telescope, or terms to takerequired classes

5

4

3

2

1

time

activity

Variables

Domains

Constraints

are activities

sets of start times (or “chunks” of time)

1. Activities that use the sameresource cannot overlap in time

2. Preconditions satisfied


3-SAT as CSP


Variables

Domains

Constraints

clauses

Boolean variable assignments that makeclauses trueClauses with shared Boolean variables mustagree on value of variables

Find values for Booleanvariables A, B, C, … thatsatisfy the formula

The original NP-complete problem

...CBACBA


Varieties of CSPs

o Discrete variables• Finite domains

n variables, domain size d → O(dn) complete assignments E.g., map coloring, n-queens, …

• Infinite domains Integers, strings E.g., job scheduling (variables are start/end days of each job)

o Continuous variables• Usually, the domain of the variables are real numbers



Varieties of constraints

o Unary constraints involve one variable• E.g., SA ≠ green

o Binary constraints involve pairs of variables• E.g., SA ≠ WA

o Higher-order constraints involve 3 or more variables

• E.g., crypt-arithmetic column constraints (see next)



Example: crypt-arithmetic puzzle


Variables

Domains

Constraints

F T U O R W X1 X2 X3

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

AllDiff(F, T, U, O, R, W)O + O = R + 10∙X1

X1 + W + W = U + 10∙X2

X2 + T + T = O + 10∙X3

X3 = FF ≠ 0T ≠ 0


Good news / Bad news

o Good news – very general & interesting class problems

o Bad news – includes NP-hard (intractable) problems

So, good behavior is a function of domain not the formulation as a CSP



CSP example

Given 40 courses (8.01, 8.02, …, 6.840) and 10 terms (Fall 1, Spring 1, ,,, Spring 5). Find a legal schedule

Constraints

Pre-requisites

Courses offered on limited terms

Limited number of courses per term

Avoid time conflicts

Note: CSPs are not for expressing (soft) preferences, e.g. minimize difficulty, balance subject areas, etc



Choice of variables & values

VARIABLES DOMAINS


A. Terms? Legal combinations of for example 4courses (but this is huge set of values)

B. Terms Slots? Courses offered during that term

Subdivide terms into slots e.g. 4 of them(Fall 1,1) (Fall 1,2)(Fall 1,3) (Fall 1,3)

C. Courses? Terms or term slots (Term slots allowexpressing constraints on limited numberof courses/term)


Constraints


Use courses as variables and term slots as values

Prerequisite > 6.001 6.034

Term before

Term after

For pairs of courses thatmust be ordered

Courses offered only in some terms > Filter domain

Avoid time conflicts> 6.001 6.034

Term not equal

For pairs offered at sameor overlapping times


Solving CSPs

o Solving CSPs involve some combination of:

• Constraint propagation, to eliminate values that could not be part of any solution

• Search, to explore valid assignments



Constraint propagation (aka Arc Consistency)

o Arc consistency eliminates values from domain of variable that can never be part of a consistent solution

o Directed arc (Vi, Vj) is arc consistent if


ji VV

arc on the constraint by the allowed is y)(x,st , ji DyDx





o We can achieve consistency on arc by deleting values from Di

(domain of variable at tail of constraint arc) that fail condition


ji VV






o We can achieve consistency on arc by deleting values from Di

(domain of variable at tail of constraint arc) that fail condition

o Assume domain size at most d and there are e binary constraints

o A simple algorithm for arc consistency is O(ed3) – note that just verifying arc consistency takes O(ed2) for each arc


ji VV



Constraint propagation example


Graph coloringinitial domains are indicated R, G, B

R, G G

V1

V2 V3

different-color constraint





R, G G

V1

V2 V3


Arc examined Value deleted

R, G, B

R, G G

V1

V2 V3

Each undirected arc is really two directed constraint arcs, theeffects shown above are from examining BOTH arcs.





R, G G

V1

V2 V3



V1 – V2 none R, G, B

R, G G

V1

V2 V3






R, G G

V1

V2 V3



V1 – V2 none

V1 – V3 V1(G)R, B

R, G G

V1

V2 V3






R, G G

V1

V2 V3



V1 – V2 none

V1 – V3 V1(G)

V2 – V3 V2(G)

R, B

R G

V1

V2 V3






R, G G

V1

V2 V3



V1 – V2 none

V1 – V3 V1(G)

V2 – V3 V2(G)

V1 – V2 V1(R)

B

R G

V1

V2 V3






R, G G

V1

V2 V3



V1 – V2 none

V1 – V3 V1(G)

V2 – V3 V2(G)

V1 – V2 V1(R)

V1 – V3 none

B

R G

V1

V2 V3






R, G G

V1

V2 V3



V1 – V2 none

V1 – V3 V1(G)

V2 – V3 V2(G)

V1 – V2 V1(R)

V1 – V3 none

V2 – V3 none

B

R G

V1

V2 V3

CSP is arc-consistent


But, arc consistency is not enough in general


R, G

R, G R, G

arc consistent but nosolutions

Graph coloring




R, G

R, G R, G


Graph coloring

B, G

R, G R, G

arc consistent but 2 solutions: B,R,G; B,G,R




R, G

R, G R, G


Graph coloring

B, G

R, G R, G


B, G

R, G R, G

arc consistent but 1 solutions: B,R,G;

Assume B, R notallowed




R, G

R, G R, G


Graph coloring

B, G

R, G R, G


B, G

R, G R, G

arc consistent but 1 solutions: B,R,G;

Assume B, R notallowed

Need to search to find solutions (if any)


Searching for solutions – backtracking (BT)

o Standard search formulation• Initial state: the empty assignment {}• Successor function: assign a value to an unassigned variable that

does not conflict with the current assignment fail is no legal assignments

• Goal test: the current assignment is complete

1.This is the same for all CSPs2.Every solution appears at depth n (for n variables)3.Path is irrelevant4.Depth-first search for CSPs is called backtracking5.Variable assignments are commutative (e.g., [WA=red then

NT=blue] is the same as [NT=blue then WA=red]





When we have too many values in domain (and/or constraints are weak) arc consistency doesn’tdo much, so we need to search. Simplest approach is pure backtracking (depth-first search)

RG

B

R G R G R G

R G R G R G R G

V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3





RG

B

R G R G R G

R G R G R G R G

V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3

Inconsistentwith V1 = R

Backup atinconsistentassignment





RG

B

R G R G R G

R G R G R G R G

V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3







RG

B

R G R G R G

R G R G R G R G

V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3



Inconsistentwith V2 = G





RG

B

R G R G R G

R G R G R G R G

V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3



Inconsistentwith V2 = G

consistent


Combine backtracking and constraint propagation

o A node in BT tree is partial assignment in which the domain of each variable has been set (tentatively) to singleton set

o Use constraint propagation (arc-consistency) to propagate the effect of this tentative assignment i.e., eliminate values inconsistent with current values






o Question: how much propagation to do?






o Question: how much propagation to do?o Answer: not much, just local propagations from domains

with unique assignments, which is called forward checking (FC). This conclusion is not necessarily obvious, but it generally holds in practice



Backtracking with forward checking (BT-FC)


When examining assignment Vi = dk, remove any values inconsistent with that assignmentFrom neighboring domains in the constraint graph

RV1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3





RV1 assignments

V2 assignments

V3 assignments

R

G G

V1

V2 V3

G





RV1 assignments

V2 assignments

V3 assignments

R

G

V1

V2 V3

G

We have a conflictwhenever a domainbecomes empty





V1 assignments

V2 assignments

V3 assignments

When backing up, need torestore domain values,since deletions were done to reach consistency withtentative assignmentsconsidered during search

G

R, G, B

R, G R, G

V1

V2 V3





V1 assignments

V2 assignments

V3 assignments

G

G

R R

V1

V2 V3





V1 assignments

V2 assignments

V3 assignments

G

G

R R

V1

V2 V3

R





V1 assignments

V2 assignments

V3 assignments

G

G

R R

V1

V2 V3

R





V1 assignments

V2 assignments

V3 assignments

R, G, B

R, G R, G

V1

V2 V3

B





V1 assignments

V2 assignments

V3 assignments

B

R, G R, G

V1

V2 V3

B

R





V1 assignments

V2 assignments

V3 assignments

B

R G

V1

V2 V3

B

R





V1 assignments

V2 assignments

V3 assignments

B

R G

V1

V2 V3

B

R

G





V1 assignments

V2 assignments

V3 assignments

B

G R

V1

V2 V3

B

G

R





V1 assignments

V2 assignments

V3 assignments

B

G R

V1

V2 V3

B

G

R

No need to check previous assignments

Generally preferable to pure BT


BT-FC with dynamic ordering

o Traditional backtracking uses fixed ordering of variables & values, e.g., random order or place variables with many constraints first

o We can usually do better by choosing an order dynamically as the search proceeds






1. Most constrained variable• When doing forward-checking, pick variable with fewest legal

values to assign next (minimizes branching factor)






1. Most constrained variable• When doing forward-checking, pick variable with fewest legal

values to assign next (minimizes branching factor)

2. Least constraining value• Choose value that rules out the fewest values from neighboring

domains



Constraint optimization problems

o COP formulation• Variables: X1, X2, …, Xn with domains D1, D2, …, Dn

• Constraints: C1, C2, …, Cm

• Objective function: f(X1, …, Xn)

o Goal: find a solution that satisfies all constraints and minimizes the objective function

o Example• Maintenance scheduling in smart buildings: schedule a set of

maintenance tasks (eg, clean rooms, repair outlet) such that there is no overlap between tasks and the sum of costs is minimized

o Solution techniques• Depth-first Branch-and-Bound search



Local search

o In many problems, the path to the goal is irrelevant; the goal state itself is the solution

o State space = set of "complete" configurationso Find configuration satisfying constraints, e.g., n-

queens

o In such cases, we can use local search algorithms which keep a single "current" state, try to improve it

o Constant space. Good for offline and online search



Local search for CSPs

o Hill-climbing, simulated annealing typically work with “complete” states, i.e., all variables assigned

o Apply to CSPs• Allow states with unsatisfied constraints• Operators reassign variable values• Variable selection

Randomly select any conflicted variable• Value selection by min-conflict heuristic

Choose value that violates the fewest constraints i.e., hill climbing: h(n) = total number of violated constraints



Example: 4-Queens problem

o States: 4 queens in 4 columns (44 = 256 states)o Actions: move queen to columno Goal test: no attackso Evaluation: h(n) = number of attacks

o Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)



Summary

o 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

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

o Variable ordering and value selection heuristics help significantly

o Forward checking prevents assignments that guarantee later failure

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


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Course: Engineering Artificial Intelligence