Consistency algorithms Chapter 3 Fall 2010 1
Consistency methods Approximation of inference: Arc, path and i-consistecy Methods that transform the original network into tighter and tighter representations Fall 2010 2
Arc-consistency X Y 1, 2, 3 1, 2, 3 1 X, Y, Z, T 3 X Y = Y = Z T Z X T 1, 2, 3 1, 2, 3 T Z Fall 2010 3
Arc-consistency X Y 1 3 1 X, Y, Z, T 3 X Y = Y = Z T Z X T 2 3 T Z Fall 2010 4
Arc-consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 5
Revise for arc-consistency ← ∩ π ⊗ D D ( R D ) i i i ij j Fall 2010 6
A matching diagram describing a network of constraints that is not arc-consistent (b) An arc- consistent equivalent network. Fall 2010 7
AC-1 Complexity ( Mackworth and Freuder, 1986 ): 3 O ( enk ) e = number of arcs, n variables, k values ( ek^2 , each loop, nk number of loops), best-case = ek , Ω 2 ( ek ) Arc-consistency is: Fall 2010 8
AC-3 3 Complexity: since each arc may be processed in O(2k) O ( ek ) Best case O(ek), Fall 2010 9
Example: A 3 variables network with 2 constraints: z divides x and z divides y (a) before and (b) after AC-3 is applied. Fall 2010 10
AC-4 Complexity: 2 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 2010 11
Example applying AC-4 Fall 2010 12
Distributed arc-consistency (Constraint propagation) ← ∩ π ⊗ D D ( R D ) Implement AC-1 distributedly. i i i ij j ← π ⊗ j h ( R D ) Node x_j sends the message to node x_i i i ij j Node x_i updates its domain: ← ∩ π ⊗ = D D ( R D ) i i i ij j ← ∩ j Messages can be sent asynchronously or scheduled D D h in a topological order i i i Fall 2010 13
Exercise: make the following network arc-consistent Draw the network’s primal and dual constraint graph Network = Domains {1,2,3,4} Constraints: y < x, z < y, t < z, f<t, x<=t+1, Y<f+2 Fall 2010 14
Arc-consistency Algorithms 3 O ( nek ) AC-1 : brute-force, distributed 3 O ( ek ) AC-3 , queue-based 2 O ( ek ) AC-4 , context-based, optimal AC-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 (1977,1983), Mohr and Anderson, (1985)… Fall 2010 15
Using constraint tightness in analysis t = number of tuples bounding a constraint O ( nekt ) 3 O ( nek ) AC-1 : brute-force, O ( ekt ) 3 O ( ek ) AC-3 , queue-based O ( et ) AC-4 , context-based, optimal AC-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 (1977,1983), Mohr and Anderson, (1985)… Fall 2010 16
Constraint checking Arc-consistency → B 1- B: [ 5 .. 14 ] 13 A < B 14 A C: [ 6 .. 15 ] [ 5.... 18] 2- A: [ 2 .. 10 ] 2 B < C [ 1.... 10 ] C: [ 6 .. 14 ] 2 < C - A < 5 3- B: [ 5 .. 13 ] 14 6 [ 4.... 15] C Fall 2010 17
Is arc-consistency enough? Example: a triangle graph-coloring with 2 values. Is it arc-consistent? Is it consistent? It is not path, or 3-consistent. Fall 2010 18
Path-consistency Fall 2010 19
Path-consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 20
Revise-3 ← ∩ π ⊗ ⊗ R R ( R D R ) ij ij ij ik k kj Complexity: O(k^3) Best-case: O(t) Worst-case O(tk) Fall 2010 21
PC-1 5 k 5 O ( n ) Complexity: O(n^3) triplets, each take O(k^3) steps O(n^3 k^3) Max number of loops: O(n^2 k^2) . Fall 2010 22
PC-2 Complexity: 3 k 5 O ( n ) Optimal PC-4: 3 k 3 O ( n ) (each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2) size of Q is O(n^3 k^2), processing is O(k^3)) Fall 2010 23
Example: before and after path- consistency PC-1 requires 2 processings of each arc while PC-2 may not Can we do path-consistency distributedly? Fall 2010 24
Example: before and after path- consistency PC-1 requires 2 processings of each arc while PC-2 may not Can we do path-consistency distributedly? Fall 2010 25
Path-consistency Algorithms Apply Revise-3 (O(k^3)) until no change ← ∩ π ⊗ ⊗ R R ( R D R ) ij ij ij ik k kj Path-consistency (3-consistency) adds binary constraints. 5 k 5 O ( n ) PC-1: 3 k 5 O ( n ) PC-2: 3 k 3 O ( n ) PC-4 optimal: Fall 2010 26
I-consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 27
Higher levels of consistency, global-consistency Click to edit Master text styles Definition: Second level ● Third level ● Fourth level ● Fifth level Fall 2010 28
Revise-i i Complexity: for binary constraints O ( k ) i O (( 2 k ) ) For arbitrary constraints: Fall 2010 29
4-queen example Fall 2010 30
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 2010 31
I-consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 32
Arc-consistency for non-binary constraints: Generalized arc-consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level ← ∩ π ⊗ D D ( R D ) − x x x S S { x } Complexity: O(t k), t bounds number of tuples. Relational arc-consistency: ← π ⊗ R ( R D ) − − S { x } S { x } S x Fall 2010 33
Examples of generalized arc- consistency x+y+z <= 15 and z >= 13 implies x<=2, y<=2 Example of relational arc-consistency ∧ → ¬ ⇒ ¬ ∨ ¬ A B G , G , A B Fall 2010 34
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. Bounds-consistency: pruning the boundaries of domains Fall 2010 35
Sudoku – Constraint Satisfaction • Variables: empty slots • Constraint • Domains = • Propagation {1,2,3,4,5,6,7,8,9} • Constraints: • Inference 27 all- • different 2 3 2 4 6 Each row, column and major block must be alldifferent “Well posed” if it has unique solution: 27 constraints Fall 2010 36
Example for alldiff A = {3,4,5,6} B = {3,4} C= {2,3,4,5} D= {2,3,4} E = {3,4} F= {1,2,3,4,5,6} Alldiff (A,B,C,D,E} Arc-consistency does nothing Apply GAC to sol(A,B,C,D,E,F)? A = {6}, F = {1}…. Alg: bipartite matching kn^1.5 (Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and simple algorithm for bounds consistency of alldifferent constraint) Fall 2010 37
Global constraints Alldifferent 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 2010 38
Bounds consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 39
Bounds consistency Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Fall 2010 40
Boolean constraint propagation (A V ~B) and (B) B is arc-consistent relative to A but not vice-versa Arc-consistency by resolution: res((A V ~B),B) = A Given also (B V C), path-consistency: res((A V ~B),(B V C) = (A V C) Relational arc-consistency rule = unit-resolution ∧ → ¬ ⇒ ¬ ∨ ¬ A B G , G , A B Fall 2010 41
Constraint propagation for Boolean 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 Fall 2010 42 ● Fourth level
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 L)) ((I L) → H) (H → G) (M → I ) ( ∧ ∧ ∧ ∨ ∧ A 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 2010 43
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