1. CONSTRAINT Networks Chapters 1-2
Compsci-275
Spring 2009
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2. Class Information 11:00 - 12:20 pm Instructor: Rina Dechter
Days: Monday & Wednesday
Time: 11: :20 pm
11: :20 pm
Class page:
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3. Text book (required) Rina Dechter, Constraint Processing,
Morgan Kaufmann
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4. Outline Motivation, applications, history
CSP: Definition, and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
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5. Outline Motivation, applications, history
CSP: Definition, representation and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
Spring 2009

6. Example: student course selection
Context: You are a senior in college
Problem: You need to register in 4 courses for the Spring semester
Possibilities: Many courses offered in Math, CSE, EE, CBA, etc.
Constraints: restrict the choices you can make
Unary: Courses have prerequisites you have/don't have Courses/instructors you like/dislike
Binary: Courses are scheduled at the same time
n-ary: In CE: 4 courses from 5 tracks such as at least 3 tracks are covered
You have choices, but are restricted by constraints
Make the right decisions!!
ICS Graduate program
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7. Student course selection (continued)
Given
A set of variables: 4 courses at your college
For each variable, a set of choices (values)
A set of constraints that restrict the combinations of values the variables can take at the same time
Questions
Does a solution exist? (classical decision problem)
How many solutions exists?
How two or more solutions differ?
Which solution is preferrable?
etc.
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8. The field of Constraint Programming
How did it started:
Artificial Intelligence (vision)
Programming Languages (Logic Programming),
Databases (deductive, relational)
Logic-based languages (propositional logic)
SATisfiability
Related areas:
Hardware and software verification
Operation Research (Integer Programming)
Answer set programming
Graphical Models; deterministic
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9. Practical applications
Radio resource management (RRM)
Databases (computing joins, view updates)
Temporal and spatial reasoning
Planning, scheduling, resource allocation
Design and configuration
Graphics, visualization, interfaces
Hardware verification and software engineering
HC Interaction and decision support
Molecular biology
Robotics, machine vision and computational linguistics
Transportation
Qualitative and diagnostic reasoning
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10. Outline Motivation, applications, history
CSP: Definitions and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
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11. A Constraint Networks Example: map coloring
Variables - countries (A,B,C,etc.)
Values - colors (red, green, blue)
Constraints:
A B
red green
red yellow
green red
green yellow
yellow green
yellow red
Constraint graph C A B D E F G A E
A collection of constraints that are related to each other in a particular domain is called “a constraint network”
They are defined by a set of variables, their possible values and a set of constraints.
I will demonstrate this through a simple example of map-coloring…
You have a map and you want to color the counties in the map by a given set of colors such that adjacent counties have different colors.
We can associate each county with a variable, the colors that can be assigned are the “domain” of the variables.
For each two variables that represent adjacent countries we will have a not-equal constraint,
A solution is an assignment of a value to each variable such that no constraint is violated.
We can associate a graph with a constraint network: the variables are the nodes, and an edge connect any two variables
Appearing in a single constraint (constraint can be defined on more than two variables.). D F B G C
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12. Constraint Satisfaction Tasks
Example: map coloring
Variables - countries (A,B,C,etc.)
Values - colors (e.g., red, green, yellow)
Constraints: A B C D E…
red
green
blue
blu
gre …
Are the constraints consistent?
Find a solution, find all solutions
Count all solutions
Find a good solution
In summary, … as set of constraints can captured by a graph and it represents its set of all solutions
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13. Information as Constraints
I have to finish my talk in 30 minutes
180 degrees in a triangle
Memory in our computer is limited
The four nucleotides that makes up a DNA only combine in a particular sequence
Sentences in English must obey the rules of syntax
Susan cannot be married to both John and Bill
Alexander the Great died in 333 B.C.
What are constraints?
“Constraints” have negative connotation in human discourse. They mean “restrictions,”
such as in the sentence: “I have to finish my talk in 30 minute.”
But any piece of information is “constraint”
If you have a piece of information that does not constrain the set of possible scenarios then it is ill-defined
and the information is worth nothing. In logic we say that it is a tautology.
Any fact, expressed positively, has value only so-far-as it restricts
the set of possible scenarios or possible worlds. So I will use the notion of “constraint: in that broad sense.
As we heard from Geoff and Jim, the English language has syntax rules, which are constraints, and
even historical fact are constraints (on the set of possible histories).
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14. Constraint Network; Definition
A 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
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15. Example: The N-queens problem
The network has four variables, all with domains Di = {1, 2, 3, 4}.
(a) The labeled chess board. (b) The constraints between variables.
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16. A solution and a partial consistent tuple
Not all consistent instantiations are part of a solution:
A consistent instantiation that is not part of a solution.
The placement of the queens corresponding to the
solution (2, 4, 1,3).
c) The placement of the queens corresponding to the solution
(3, 1, 4, 2).
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17. Example: Crossword puzzle
Variables: x1, …, x13
Domains: letters
Constraints: words from
{HOSES, LASER, SHEET, SNAIL, STEER, ALSO, EARN, HIKE, IRON, SAME, EAT, LET, RUN, SUN, TEN, YES, BE, IT, NO, US}
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18. Constraint graphs of the crossword puzzle and the 4-queens problem.
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19. Example: configuration and design
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20. Configuration and design
Want to build: recreation area, apartments, houses, cemetery, dump
Recreation area near lake
Steep slopes avoided except for recreation area
Poor soil avoided for developments
Highway far from apartments, houses and recreation
Dump not visible from apartments, houses and lake
Lots 3 and 4 have poor soil
Lots 3, 4, 7, 8 are on steep slopes
Lots 2, 3, 4 are near lake
Lots 1, 2 are near highway
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21. Configuration and design
Variables: Recreation, Apartments, Houses, Cemetery, Dump
Domains: {1, 2, …, 8}
Constraints: derived from conditions
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22. Huffman-Clowes junction labelings (1975)
3-dimensional interpretation of 2-dimentional drawing in block world pictures examples.
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23. Figure 1.5: Solutions: (a) stuck on left wall, (b) stuck on right wall, (c) suspended in mid-air, (d) resting on floor.
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24. Example: Sudoku Constraint propagation 2
Variables: 81 slots
Domains = {1,2,3,4,5,6,7,8,9}
Constraints:
27 not-equal
Constraint propagation 2
As I will disuss later, algorithms for answering constraint satisfaction queries are not likely to be efficient.
Namely, we may have to wait a very long time when we ask if there is a solution.
But one reason the area of constraint network gained its popularity is by highlighting some computational pattern that
is popular with human. We call it constraint propagation. ….
So, while constraint propagation are huge effective helper in solving many problems associated with deterministic constraints.
Each row, column and major block must be alldifferent
“Well posed” if it has unique solution: 27 constraints
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25. Outline Motivation, applications, history
CSP: Definition, representation and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
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26. Mathematical background
Sets, domains, tuples
Relations
Operations on relations
Graphs
Complexity
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27. Two graphical representation and views of a relation: R = {(black, coffee), (black, tea), (green, tea)}.
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28. Operations with relations
Intersection
Union
Difference
Selection
Projection
Join
Composition
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29. Figure 1.8: Example of set operations intersection, union, and difference applied to relations.
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30. selection, projection, and join operations on relations.
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31. Outline Motivation, applications, history
CSP: Definition, representation and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
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32. Constraint’s representations
Relation: allowed tuples
Algebraic expression:
Propositional formula:
Semantics: by a relation
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33. Constraint Graphs: Primal, Dual and Hypergraphs
A (primal) constraint graph: a node per variable
arcs connect constrained variables.
A dual constraint graph: a node per constraint’s scope, an arc connect nodes sharing variables =hypergraph
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34. Graph Concepts Reviews: Hyper Graphs and Dual Graphs
A hypergraph
Dual graphs
A primal graph
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35. Propositional Satisfiability
 = {(¬C), (A v B v C), (¬A v B v E), (¬B v C v D)}.
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36. Constraint graphs of 3 instances of the Radio frequency assignment problem in CELAR’s benchmark
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37. Figure 2.7: Scene labeling constraint network
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38. A combinatorial circuit: M is a multiplier, A is an adder
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39. The constraint graph and constraint relations of the scheduling problem example.
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40. More examples Given P = (V, D, C ), where Example I: Define C ?
{1, 2, 3, 4}
{ 3, 5, 7 }
{ 3, 4, 9 }
{ 3, 6, 7 }
v2 > v4 V4 V2
v1+v3 < 9 V3 V1
v2 < v3
v1 < v2
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41. Example: temporal reasoning
Give one solution: …….
Satisfaction, yes/no: decision problem
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42. Outline Motivation, applications, history
CSP: Definition, representation and simple modeling examples
Mathematical concepts (relations, graphs)
Representing constraints
Constraint graphs
The binary Constraint Networks properties
Spring 2009

43. Properties of binary constraint networks:
A graph  to be colored by two colors,
an equivalent representation ’ having a newly inferred constraint
between x1 and x3.
Equivalence and deduction with constraints (composition)
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44. Relations vs netwroks Can we represent the relations
x1,x2,x3 = (0,0,0)(0,1,1)(1,0,1)(1,1,0)
X1,x2,x3,x4 = (1,0,0,0)(0,1,0,0) (0,0,1,0)(0,0,0,1)
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45. Relations vs netwroksCan we represent
Can we represent the relations
x1,x2,x3 = (0,0,0)(0,1,1)(1,0,1)(1,1,0)
X1,x2,x3,x4 = (1,0,0,0)(0,1,0,0) (0,0,1,0)(0,0,0,1)
Most relations cannot be represented by networks:
Number of relations 2^(k^n)
Number of networks: 2^((k^2)(n^2))
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46. The minimal and projection networks
The projection network of a relation is obtained by projecting it onto each pair of its variables (yielding a binary network).
Relation = {(1,1,2)(1,2,2)(1,2,1)}
What is the projection network?
What is the relationship between a relation and its projection network?
{(1,1,2)(1,2,2)(2,1,3)(2,2,2)}, solve its projection network?
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47. Projection network (continued)
Theorem: Every relation is included in the set of solutions of its projection network.
Theorem: The projection network is the tightest upper bound binary networks representation of the relation.
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48. Projection network
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49. The Minimal Network (partial order between networks)
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50. The N-queens constraint network.
The network has four variables, all with domains Di = {1, 2, 3, 4}.
(a) The labeled chess board. (b) The constraints between variables.
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51. Figure 2.11: The 4-queens constraint network: (a) The constraint graph. (b) The minimal binary constraints. (c) The minimal unary constraints (the domains).
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52. Minimal network
The minimal network is perfectly explicit for binary and unary constraints:
Every pair of values permitted by the minimal constraint is in a solution.
Binary-decomposable networks:
A network whose all projections are binary decomposable
The minimal network repesenst fully binary-decomposable networks.
Ex: (x,y,x,t) = {(a,a,a,a)(a,b,b,b,)(b,b,a,c)} is binary representable but what about its projection on x,y,z?
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