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Binary C ONSTRAINT P ROCESSING Chapter 2

Binary C ONSTRAINT P ROCESSING Chapter 2. ICS-275A Fall 2007. Class 1 Constraint Networks. The constraint network model Inference. Class description. Instructor: Rina Dechter Days: Monday & Wednesday Time: 11:00 - 12:20 pm 11:00 - 13:00 pm (weeks 1,2)

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Binary C ONSTRAINT P ROCESSING Chapter 2

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  1. Binary CONSTRAINTPROCESSINGChapter 2 ICS-275A Fall 2007

  2. Class 1Constraint Networks The constraint network model Inference.

  3. Class description • Instructor: Rina Dechter • Days: Monday & Wednesday • Time: 11:00 - 12:20 pm 11:00 - 13:00 pm (weeks 1,2) • Class page: http://www.ics.uci.edu/~dechter/ics-275a/spring-2007/ Spring 2007

  4. Text book (required) Rina Dechter, Constraint Processing, Morgan Kaufmann Spring 2007

  5. E A B red green red yellow green red green yellow yellow green yellow red A D B F G C Constraint Networks A Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: Constraint graph A E D F B G C Spring 2007

  6. Constraint Satisfaction Tasks Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: Are the constraints consistent? Find a solution, find all solutions Count all solutions Find a good solution Spring 2007

  7. 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. Spring 2007

  8. Constraint Network • A constraint network is: R=(X,D,C) • Xvariables • Ddomain • Cconstraints • Rexpresses 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 Spring 2007

  9. 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} Spring 2007

  10. 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. Spring 2007

  11. Not all consistent instantiations are part of a solution: (a) A consistent instantiation that is not part of a solution. (b) 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). Spring 2007

  12. Configuration and design Spring 2007

  13. 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 Spring 2007

  14. Huffman-Clowes junction labelings (1975) Spring 2007

  15. Figure 1.5: Solutions: (a) stuck on left wall, (b) stuck on right wall, (c) suspended in mid-air, (d) resting on floor. Spring 2007

  16. Constraint graphs of the crossword puzzle and the 4-queens problem. Spring 2007

  17. Mathematical background • Sets, domains, tuples • Relations • Operations on relations • Graphs • Complexity Spring 2007

  18. Two graphical views of relationR = {(black, coffee), (black, tea), (green, tea)}. Spring 2007

  19. Figure 2.4: The constraint graph and constraint relations of the scheduling problem example. Spring 2007

  20. Operations with relations • Intersection • Union • Difference • Selection • Projection • Join • Composition Spring 2007

  21. Figure 1.8: Example of set operations intersection, union, and difference applied to relations. Spring 2007

  22. selection, projection, and join operations on relations. Spring 2007

  23. Constraint’s representations • Relation: allowed tuples • Algebraic expression: • Propositional formula: • Semantics: by a relation Spring 2007

  24. 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 Spring 2007

  25. Graph Concepts Reviews:Hyper Graphs and Dual Graphs • A hypergraph • Dual graphs • A primal graph Spring 2007

  26. Propositional Satisfiability  = {(¬C),(A v B v C),(¬A v B v E), (¬B v C v D)}. Spring 2007

  27. Constraint graphs of 3 instances of the Radio frequency assignment problem in CELAR’s benchmark Spring 2007

  28. Figure 2.7: Scene labeling constraint network Spring 2007

  29. Figure 2.9: A combinatorial circuit: M is a multiplier, A is an adder. Spring 2007

  30. 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) Spring 2007

  31. 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) Spring 2007

  32. 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)) Spring 2007

  33. 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? Spring 2007

  34. 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. Spring 2007

  35. Projection network Spring 2007

  36. The Minimal Network(partial order between networks) Spring 2007

  37. Figure 2.11: The 4-queens constraint network: (a) The constraint graph. (b) The minimal binary constraints. (c) The minimal unary constraints (the domains). Spring 2007

  38. 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? Spring 2007

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