Consistency algorithms. Chapter 3

Transcription

1 Consistency algorithms Chapter Fall 00

2 Consistency methods pproximation of inference: rc, path and i-consistecy Methods that transform the original network into tighter and tighter representations Fall 00

3 rc-consistency X, Y, Z, T X Y Y = Z T Z X T X,, Y,,,,,, T = Z Fall 00

4 rc-consistency X, Y, Z, T X Y Y = Z T Z X T X T Y Z = Fall 00 4

5 rc-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 5

6 Revise for arc-consistency i i π i ( Rij j ) Fall 00 6

7 matching diagram describing a network of constraints that is not arc-consistent (b) n arcconsistent equivalent network. Fall 00 7

8 C- Complexity (Mackworth and Freuder, 986): O( enk ) e = number of arcs, n variables, k values (ek^, each loop, nk number of loops), best-case = ek, rc-consistency is: Ω( ek ) Fall 00 8

9 C- Complexity: est case O(ek), O( ek ) since each arc may be processed in O(k) Fall 00 9

10 Example: variables network with constraints: z divides x and z divides y (a) before and (b) after C- is applied. Fall 00 0

11 C-4 Complexity: O( ek ) (Counter is the number of supports to ai in xi from xj. S_(xi,ai) is the set of pairs that (xi,ai) supports) Fall 00

12 Example applying C-4 Fall 00

13 istributed arc-consistency (Constraint propagation) Implement C- distributedly. Node x_j sends the message to node x_i i h j i i π π i ( Rij j i ( Rij j ) ) Node x_i updates its domain: i π ( R ) i i ij j = Messages can be sent asynchronously or scheduled in a topological order i i h j i Fall 00

14 Exercise: make the following network arc-consistent raw the network s primal and dual constraint graph Network = omains {,,,4} Constraints: y < x, z < y, t < z, f<t, x<=t+, Y<f+ Fall 00 4

15 rc-consistency lgorithms C-: brute-force, distributed C-, queue-based C-4, context-based, optimal O( nek O( ek O( ek ) ) ) C-5,6,7,. Good in special cases Important: applied at every node of search (n number of variables, e=#constraints, k=domain size) Mackworth and Freuder (977,98), Mohr and nderson, (985) Fall 00 5

16 Using constraint tightness in analysis t = number of tuples bounding a constraint O( nek ) C-: brute-force, C-, queue-based C-4, context-based, optimal C-5,6,7,. Good in special cases O( ek Important: applied at every node of search (n number of variables, e=#constraints, k=domain size) O(nekt) O(ekt) O(et) Mackworth and Freuder (977,98), Mohr and nderson, (985) ) Fall 00 6

17 Constraint checking rc-consistency < [ ] 4 - : [ ] C: [ ] [... 0 ] < C - : [.. 0 ] C: [ ] < C - < [ ] C - : [ 5.. ] Fall 00 7

18 Is arc-consistency enough? Example: a triangle graph-coloring with values. Is it arc-consistent? Is it consistent? It is not path, or -consistent. Fall 00 8

19 Path-consistency Fall 00 9

20 Path-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 0

21 Revise- R ij R ij π ij( Rik k Rkj ) Complexity: O(k^) est-case: O(t) Worst-case O(tk) Fall 00

22 PC- Complexity: O( n 5 k 5 ) O(n^) triplets, each take O(k^) steps O(n^ k^) Max number of loops: O(n^ k^). Fall 00

23 PC- Complexity: Optimal PC-4: O( n k 5 O( n ) k ) (each pair deleted may add: n- triplets, number of pairs: O(n^ k^) Q is O(n^ k^), processing is O(k^)) size of Fall 00

24 Example: before and after pathconsistency PC- requires processings of each arc while PC- may not Can we do path-consistency distributedly? Fall 00 4

25 Example: before and after pathconsistency PC- requires processings of each arc while PC- may not Can we do path-consistency distributedly? Fall 00 5

26 Path-consistency lgorithms pply Revise- (O(k^)) until no change R ij R ij π ij ( R R ik k kj ) Path-consistency (-consistency) adds binary constraints. PC-: PC-: PC-4 optimal: O( n O( n O( n 5 k 5 k 5 k ) ) ) Fall 00 6

27 I-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 7

28 Higher levels of consistency, global-consistency efinition: Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 8

29 Revise-i Complexity: for binary constraints For arbitrary constraints: O((k) i ) O( k i ) Fall 00 9

30 4-queen example Fall 00 0

31 I-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00

32 I-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00

33 rc-consistency for non-binary constraints: Generalized arc-consistency Click to edit Master text styles Second level Third level Fourth level Fifth level x x π x ( R }) S S {x Complexity: O(t k), t bounds number of tuples. Relational arc-consistency: R ( S { x} π S { x} RS x ) Fall 00

34 Examples of generalized arcconsistency x+y+z <= 5 and z >= implies x<=, y<= Example of relational arc-consistency G, G, Fall 00 4

35 More arc-based consistency Global constraints: e.g., all-different constraints Special semantic constraints that appears often in practice and a specialized constraint propagation. Used in constraint programming. ounds-consistency: pruning the boundaries of domains Fall 00 5

36 Sudoku Constraint Satisfaction Constraint Propagation Inference Variables: empty slots omains = {,,,4,5,6,7,8,9} Constraints: 7 alldifferent 4 6 Each row, column and major block must be alldifferent Well posed if it has unique solution: 7 constraints Fall 00 6

37 Example for alldiff = {,4,5,6} = {,4} C= {,,4,5} = {,,4} E = {,4} F= {,,,4,5,6} lldiff (,,C,,E} rc-consistency does nothing pply GC to sol(,,c,,e,f)? = {6}, F = {}. lg: bipartite matching kn^.5 (Lopez-Ortiz, et. l, IJCI-0 pp 45 ( fast and simple algorithm for bounds consistency of alldifferent constraint) Fall 00 7

38 Global constraints lldifferent Sum constraint (variable equal the sum of others) Global cardinality constraint (a value can be assigned a bounded number of times to a set of variables) The cummulative constraint (related to scheduling tasks) Fall 00 8

39 ounds consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 9

40 ounds consistency Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 40

41 oolean constraint propagation ( V ~) and () is arc-consistent relative to but not vice-versa rc-consistency by resolution: res(( V ~),) = Given also ( V C), path-consistency: res(( V ~),( V C) = ( V C) Relational arc-consistency rule = unit-resolution G, G, Fall 00 4

42 Constraint propagation for oolean constraints: Unit propagation Click to edit Master text styles Second level Third level Fourth level Fifth level Click to edit Master text styles Second level Third level Fourth level Fall 00 4

43 Example (if there is time) M: The unicorn is mythical I: The unicorn is immortal L: The unicorn is mammal H: The unicorn is horned G: The unicorn is magical (M I ) ( M ( I L)) ((I L) H) (H G) Logic Puzzle IV Is the unicorn mythical? Is it magical? Is it horned? (M I ) ( M ( I L)) ((I L) H) (H G) ( M I ) (M ( I L)) ((I L) H) (H G) ( M I ) (M I ) (M L) ((I L) H) (H G) (I L) ((I L) H) (H G) H G Hence, the unicorn is not necessarily mythical, but it is horned and magical! Fall 00 4

44 Fall Consistency for numeric constraints (Gausian elimination) 0,, 7 0,0], [ 5 0, [5,9] [,5], 0 [5,5], [,0], = + + = + = + z y y x adding by obtained z x consistency path z y z y y x adding by y x consistency arc y x y x

45 Changes in the network graph as a result of arc-consistency, path-consistency and 4- consistency. Fall 00 45

46 istributed arc-consistency (Constraint propagation) Implement C- distributedly. i i π i ( Rij j ) Node x_j sends the message to node x_i h j i π i ( Rij j ) Node x_i updates its domain: i i h j i Relational and generalized arc-consistency can be implemented distributedly: sending messages between constraints over the dual graph R π }( R S { x} S { x S x ) Fall 00 46

47 Relational rc-consistency The message that R sends to R is R R C R C R updates its relation and domains and sends messages to neighbors R 4 G R 6 F G F R 5 C F Fall 00 47

48 istributed Relational rc-consistency RC can be applied to the dual problem of any constraint network: Fall 00 48

49 R R C R C R F G R 6 C F Fall FG F C CF R 5

50 4 h 6 Iteration h 5 4 h 5 h 4 4 h 4 h h 4 h h R 4 R 6 FG R 6 F G R C 4 5 h h h 4 F 6 h 4 CF 6 h 5 F C R C R 5 Node 65 4 sends messages C F h 5 h h 5 h h 4 C h 5 C 5 h 4 5 h 6 F Fall 00 50

51 Iteration R Click to edit Master text styles Second level Third level R Fourth level Fifth level C R C R R 6 6 FG F G F C CF R 5 C F Fall 00 5

52 h 5 Iteration h 4 h R Click to edit Master text styles Second level Third level R Fourth h h level h 4 Fifth level C R C h h 5 C 4 h 6 4 h 4 h R R 6 C F F G F Fall FG F 6 h 4 C CF 6 h 5 R 5 5 h 5 h C 5 h 6 F

53 Iteration R Click to edit Master text styles Second level R Third level Fourth level Fifth level C R C R FG F C CF R 5 C F R 6 F G Fall 00 5

54 h 5 Iteration h 4 h R Click to edit Master text styles Second level Third level R Fourth h h level h 4 Fifth level C R C h h 5 C 4 h 6 4 h 4 h R FG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 F G 6 h 4 6 h 5 F Fall 00 54

55 Iteration R Click to edit Master text styles Second level Third level R Fourth level Fifth level C R C R FG F C CF R 5 C F R 6 F G Fall 00 55

56 h 5 Iteration 4 h 4 h R Click to edit Master text styles Second level Third level h Fourth h hlevel 4 Fifth level R C R C h h 5 C 4 h 6 4 h 4 h R FG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 F G 6 h 4 6 h 5 F Fall 00 56

57 Iteration 4 Click to edit Master text styles Second level Third level R Fourth level Fifth level R R C C R FG F C CF R 5 C F R 6 F G Fall 00 57

58 Iteration 5 Click to edit Master text styles Second level Third level R hfourth h level h 4 Fifth level h 5 h 4 h R C R C h h 5 C 4 h 6 4 h 4 h R FG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 F G 6 h 4 6 h 5 F Fall 00 58

59 Iteration 5 Click to edit Master text styles Second level R Third level Fourth level Fifth level R R C C R FG F C CF R 5 C F R 6 F G Fall 00 59

60 Tractable classes Click to edit Master text styles Second level Third level Fourth level Fifth level Fall 00 60