Consistency algorithms

Size: px

Start display at page:

Download "Consistency algorithms"

Clarissa Fields
5 years ago
Views:

1 Consistency algorithms Chapter SQ 00

2 rc-consistency X, Y, Z, T X Y Y = Z T Z X T X,, Y,,,,,, T = Z SQ 00

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

4 rc-consistency Sound < < Incomplete lways converges (polynomial) < D D < D < D < C C = = C C

5 rc-consistency Definition: Given a constraint graph G, variable V i is arc-consistent relative to V j iff for every value ad Vi, there exists a value bd Vj (a, b)r Vi,Vj. Vi Vj Vi Vj The constraint R Vi,Vj is arc-consistent iff V i is arc-consistent relative to V j and V j is arc-consistent relative to V i. binary CSP is arc-consistent iff every constraint (or sub-graph of size ) is arc-consistent 5

6 Revise for arc-consistency D i D i i ( Rij D j ) SQ 00 6

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

8 C- O( enk ) Complexity (Mackworth and Freuder, 986): e = number of arcs, n variables, k values (ek^, each loop, nk number of loops), best-case = ek, rc-consistency is: ( ek Complexity of C-: O(enk^ ) ) SQ 009 8

9 rc consistency. C may discover the solution V V V V V V V V V V V V V V V V V V V V V 9

10 rc consistency. C may discover inconsistency X {,, } X<Y Z<X Y {,, } {,, } Y<Z Z 0

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

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

13 rc-consistency Constraint checking < 4 [ ] - : [ ] C: [ ] [... 0 ] < C - : [.. 0 ] C: [ ] < C - < [ ] C - : [ 5.. ]

14 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) SQ 00 4

15 Exercise: make the following network arc-consistent Draw the network s primal and dual constraint graph Network = Domains {,,,4} Constraints: y < x, z < y, t < z, f<t, x<=t+, Y<f+ SQ 00 5

16 rc-consistency lgorithms C-: brute-force, distributed C-, queue-based C-4, context-based, optimal C-5,6,7,. Good in special cases O( nek O( ek 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) ) O( ek ) ) SQ 00 6

17 Using constraint tightness in analysis t = number of tuples bounding a constraint O(nekt) C-: brute-force, O( nek ) C-, queue-based O( ek ) O(ekt) C-4, context-based, optimal O(et) 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) SQ 00 7

18 DRC on the dual join-graph R R C R C R 4 D 4 5 D R 6 D F G D C F SQ DFG F C CF R 5

19 Distributed Relational rc-consistency DRC can be applied to the dual problem of any constraint network: SQ 00 9

20 4 h 6 D R h Iteration h h 5 4 h 5 h 4 4 h 4 h h 4 h h R 4 R D D 6 D R 6 D F G 6 h 4 D h 4 C 4 5 DFG F 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 SQ 009 0

21 Iteration R R C R C R 4 D D 4 5 D R 6 6 DFG D F G F C CF R 5 C F SQ 009

22 Iteration R h h h 4 h 5 h 4 h R C R C h h 5 C 4 h 6 D 4 h 4 h R 4 D 4 5 D D R 6 D F G C F SQ DFG F 6 h 4 D C CF 6 h 5 F R 5 5 h 5 h C 5 h 6 F

23 Iteration R R C R C R 4 D D 4 5 D 6 DFG F C CF R 5 C F R 6 D F G SQ 009

24 Iteration R h h h 4 h 5 h 4 h R C R C h h 5 C 4 h 6 D 4 h 4 h R 4 D D 4 5 D 6 DFG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 D F G 6 h 4 D 6 h 5 F SQ 009 4

25 Iteration R R R C C R 4 D D 4 5 D 6 DFG F C CF R 5 C F R 6 D F G SQ 009 5

26 Iteration 4 h 5 h 4 h R R h h h 4 C R C h h 5 C 4 h 6 D 4 h 4 h R 4 D D 4 5 D 6 DFG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 D F G 6 h 4 D 6 h 5 F SQ 009 6

27 Iteration 4 R R R C C R 4 D D 4 5 D 6 DFG F C CF R 5 C F R 6 D F G SQ 009 7

28 Iteration 5 R h h h 4 h 5 h 4 h R C R C h h 5 C 4 h 6 D 4 h 4 h R 4 D D 4 5 D 6 DFG F C CF R 5 C F 5 h 5 h C 5 h 6 F R 6 D F G 6 h 4 D 6 h 5 F SQ 009 8

29 Iteration 5 R R R C C R 4 D D 4 5 D 6 DFG F C CF R 5 C F R 6 D F G SQ 009 9

30 Distributed rc-consistency rc-consistency can be formulated as a distributed algorithm: C D F G a Constraint network SQ 00 0

31 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 D D G R 6 D F G F R 5 C F SQ 00

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

33 Path-consistency SQ 00

34 Path-consistency SQ 00 4

35 Revise- R ij R ij ij( Rik Dk Rkj ) Complexity: O(k^) est-case: O(t) Worst-case O(tk) SQ 009 5

36 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^). ) SQ 009 6

37 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^) size of Q is O(n^ k^), processing is O(k^)) SQ 009 7

38 Example: before and after pathconsistency PC- requires processings of each arc while PC- may not Can we do path-consistency distributedly? SQ 009 8

39 Example: before and after pathconsistency PC- requires processings of each arc while PC- may not Can we do path-consistency distributedly? SQ 009 9

40 Path-consistency lgorithms pply Revise- (O(k^)) until no change O( n O( n O( n 5 k 5 k 5 k ) ) ) R ij R ij ij ( R D R ik k kj Path-consistency (-consistency) adds binary constraints. PC-: PC-: PC-4 optimal: ) SQ 00 40

41 I-consistency SQ 00 4

42 Higher levels of consistency, globalconsistency SQ 00 4

43 Revise-i O( k i ) Complexity: for binary constraints For arbitrary constraints: O((k) i ) SQ 00 4

44 4-queen example SQ 00 44

45 i-consistency SQ 00 45

46 rc-consistency for non-binary constraints: Generalized arc-consistency D x D x ( R D }) x S S{x Complexity: O(t k), t bounds number of tuples. Relational arc-consistency: R }( R D S { x} S { x S x ) SQ 00 46

47 Examples of generalized arc-consistency x+y+z <= 5 and z >= implies x<=, y<= Example of relational arc-consistency G, G, x+y <= SQ 00 47

48 What is ST? Given a sentence: Sentence: conjunction of clauses c c4 c5 c6 c c c4 c Clause: disjunction of literals Literal: a term or its negation c c, c 6 Term: oolean variable c c 0 Question: Find an assignment of truth values to the oolean variables such the sentence is satisfied. SQ 00 48

49 oolean constraint propagation If lex goes, then ecky goes: If Chris goes, then lex goes: Query: Example: party problem C Is it possible that Chris goes to the party but ecky does not? Is propositional theory, C,, C (or, ) (or, C ) satisfiable? SQ 00 49

50 CSP is NP-Complete Verifying that an assignment for all variables is a solution Provided constraints can be checked in polynomial time Reduction from ST to CSP Many such reductions exist in the literature (perhaps 7 of them) SQ 00 50

51 Problem reduction Example: CSP into ST (proves nothing, just an exercise) Notation: variable-value pair = vvp vvp term V = {a, b, c, d} yields x = (V, a), x = (V, b), x = (V, c), x 4 = (V, d), V = {a, b, c} yields x 5 = (V, a), x 6 = (V, b), x 7 = (V,c). The vvp s of a variable disjunction of terms V = {a, b, c, d} yields (Optional) t most one VVP per variable x x x x4 x x x x x x x x x x x x x x x x SQ 00 5

52 Constraint: CSP into ST (cont.) C V V {( a, a),( a, b),( b, c),( c, b),( d, a)}. Way : Each inconsistent tuple one disjunctive clause x For example: how many? x 7. Way : a) Consistent tuple conjunction of terms x b) Each constraint disjunction of these conjunctions x 5 x x x x x x x x x x 5 6 transform into conjunctive normal form (CNF) Question: find a truth assignment of the oolean variables such that the sentence is satisfied SQ 00 5

53 Constraint propagation for oolean constraints: Unit propagation SQ 00 5

54 Consistency for numeric constraints SQ ,, 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

55 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 SQ 00 55

56 ounds consistency SQ 00 56

57 rc-consistency Constraint checking < 4 [ ] - : [ ] C: [ ] [... 0 ] < C - : [.. 0 ] C: [ ] < C - < [ ] C - : [ 5.. ] Overview 57

58 ounds consistency for lldifferent constraints For alldiff bounds consistency can be enforced in O(nlog n) SQ 00 58

59 Tractable classes SQ 00 59

60 Changes in the network graph as a result of arc-consistency, path-consistency and 4-consistency. SQ 00 60

Consistency algorithms. Chapter 3

Consistency algorithms. Chapter 3 Consistency algorithms Chapter Fall 00 Consistency methods pproximation of inference: rc, path and i-consistecy Methods that transform the original network into tighter and tighter representations Fall