Consistency:

1
Consistency Algorithms

• Foundations of Constraint Processing
• CSCE421/821, Fall 2005
• www.cse.unl.edu/choueiry/F05-421-821/
• Berthe Y. Choueiry (Shu-we-ri)
• Avery Hall, Room 123B
• choueiry_at_cse.unl.edu
• Tel [phone]

2
Lecture Sources

• Required reading
• Algorithms for Constraint Satisfaction Problems,
  Mackworth and Freuder AIJ'85
• Sections 3.1, 3.2, 3.3. Chapter 3. Constraint
  Processing. Dechter
• Recommended
• Sections 3.43.10. Chapter 3. Constraint
  Processing. Dechter
• Networks of Constraints Fundamental Properties
  and Application to Picture Processing, Montanari,
  Information Sciences 74
• Consistency in Networks of Relations, Mackworth
  AIJ'77
• Constraint Propagation with Interval Labels,
  Davis, AIJ'87
• Path Consistency on Triangulated Constraint
  Graphs, Bliek Sam-Haroud IJCAI'99

3
Outline

1. Motivation and material
2. Node and arc consistency and their complexity
3. Criteria for performance comparison and CSP
   parameters.
4. Path consistency and its complexity
5. Consistency properties of a CSP
6. Other related results box consistency,
   constraint synthesis, look-ahead, non-binary
   constraints, etc.

4
Consistency checking

• Motivation
• CSP are solved with search (conditioning)
• Search performance is affected by
• - problem size
• - amount of backtracking
• Backtracking may yield
• Thrashing
• Exploring non-promising sub-trees and
  rediscovering the same inconsistencies over and
  over again
• ? Malady of backtrack search

5
(Some) causes of thrashing

• Search order
• V1, V2, V3, V4, V5
• What happens in search if we
• do not check the constraint on V3 (node
  inconsistency)
• do not check constraint between V3 and V5 (arc
  inconsistency)
• do not check constraints between V3V4, V3V5,
  and V4V5 (path inconsistency)

6
Consistency checking

• Goal eliminate inconsistent combinations
• Algorithms (for binary constraints)
• Node consistency
• Arc consistency (AC-1, AC-2, AC-3, ..., AC-7,
  AC-3.1, etc.)
• Path consistency (PC-1, PC-2, DPC, PPC, etc.)
• Constraints of arbitrary arity
• Waltz algorithm ancestor of AC's
• Generalized arc-consistency (Dechter, Section
  3.5.1 )
• Relational m-consistency (Dechter, Section 8.1)

7
Overview of Recommended Reading

• Davis, AIJ'77
• focuses on the Waltz algorithm
• studies its performance and quiescence for given
• constraint types (order relations, bounded diff,
  algebraic, etc.)
• domains types (continuous or finite)
• Mackworth, AIJ'77
• presents NC, AC-1, AC-2, PC-1, PC-2
• Mackworth and Freuder, AIJ'85
• studies their complexity
• ? Mackworth and Freuder concentrate on finite
  domains
• ? More people work on finite domains than on
  continuous ones
• ? Continuous domains can be quite tough

8
Constraint Propagation with Interval Labels
Ernest Davis

• Old' paper (1987) terminology slightly
  different
• Interval labels continuous domains
• Section 8 sign labels (discrete)
• Concerned with Waltz algorithm (e.g., quiescence,
  completeness)
• Constraint types vs. domain types (Table 3)
• Addresses applications from reasoning
• about Physical Systems (circuit, SPAM)
• about time relations (TMM)
• ? Advice read Section 3, more if you wish

9
Waltz algorithm for label inference

• Refine(C(Vi, Vk, Vm, Vn), Vi) - finds a
• new label for Vi consistent with C.
• Revise(C (Vi, Vk, Vm, Vn)) refines the domains of
  Vi, Vk, Vm, Vn by iterating over these variables
  until quiescence (no domains is further refined)
• Waltz Algorithm revises all constraints by
  iterating over each constraint connected to a
  variable whose domain has been revised
• Waltz Algorithm ? ancestor of arc-consistency

10
WARNING

• ? Completeness Running the Waltz Algorithm does
  not solve the problem.
• A2 and B3 is still not a solution!
• ? Quiescence The Waltz algorithm may go into
  infinite loops even if problem is solvable
• x ? 0, 100 x y
• y ? 0, 100 x 2y
• ? Davis provides a classification of problems in
  terms of completeness and quiescence of the Waltz
  algorithm
• f(constraint types, domain types), see Table 3

11
Importance of this paper

• ? Establishes that constraints of bounded
  differences (temporal reasoning) can be
  efficiently solved by the Waltz algorithm (O(n3),
  n number of variables)
• ? Early paper that attracts attention on the
  difficulty of label propagation in interval
  labels (continuous domains).
• This work has been continued by Faltings (AIJ 92)
  who studied the early-quiescence problem of the
  Waltz algorithm.

12
Basic consistency algorithms

• Examining finite, binary CSPs and their
  complexity
• Notation
• Given variables i, j, k with values x, y, z, the
  predicate
• Pijk(x, y, z) is true iff the 3-tuple ?x, y, z? ?
  Cijk
• Node consistency checking Pi(x)
• Arc consistency checking Pij(x,y)
• Path consistency bit-matrix manipulation

13
Worst-case asymptotic complexity

• Worst-case complexity as a
  function of the input parameters
• Upper bound f(n) is O(g(n)) means that f(n) ?
  c.g(n)
• f grows as g or slower
• Lower bound f(n) is ? (h(n)) means that f(n) ?
  c.h(n)
• f grows as g or faster
• Input parameters for a CSP
• n number of variables
• a (max) size of a domain
• dk degree of Vk (? n-1)
• e number of edges (or constraints) ? (n-1),
  n(n-1)/2

14
Outline

1. Motivation and material
2. Node arc consistency and their complexity NC,
   AC-1, AC-3, AC-4
3. Criteria for performance comparison and CSP
   parameters.
4. Path consistency and its complexity
5. Consistency properties of a CSP
6. Other related results box consistency,
   constraint synthesis, look-ahead, non-binary
   constraints, etc.

15
Node consistency (NC)

• Procedure NC(i)
• Di ? Di ? x Pi(x)
• Begin
• for i ? 1 until n do NC(i)
• end

16
Complexity of NC

• Procedure of NC(i)
• Di ?Di ? x Pi(x)
• Begin
• for i ? 1 until n do NC(i)
• end
• For each variable, we check a values
• We have n variables, we do n.a checks
• NC is O(a.n)

17
Arc-consistency Adapted from Dechter

• Definition Given a constraint graph G,
• A variable Vi is arc-consistent relative to Vj
  iff for every value a?DVi, there exists a value
  b?DVj (a, b)?CVi,Vj.
• The constraint CVi,Vj is arc-consistent iff
• Vi is arc-consistent relative to Vj and
• Vj is arc-consistent relative to Vi.
• A binary CSP is arc-consistent iff every
  constraint (or sub-graph of size 2) is
  arc-consistent

18
Procedure Revise

• Revises the domains of a variable i
• Procedure Revise(i,j)
• Begin
• DELETE ? false
• for each x ? Di do
• if there is no y ? Dj such that Pij(x,
  y) then
• begin
• delete x from Di
• DELETE ? true
• end
• return DELETE
• end
• Revise is directional
• What is the complexity of Revise?

a2
19
Revise example R. Dechter

• Apply the Revise procedure to the following
  example

20
Effect of Revise Adapted from Dechter

• Question
• Given
• two variables Vi and Vj
• their domains DVi and DVj, and
• the constraint CVi,Vj,
• write the effect of the Revise procedure as a
  sequence of operations in relational algebra
• Hint
• Think about the domain DVi as a unary constraint
  CVi and
• consider the composition of this unary constraint
  and the binary one..

Solution DVi ? DVi ? ??Vi (CVi,Vj
DVi) This is actually equivalent to DVi ? ?Vi
(CVi,Vj DVi)
21
Arc consistency (AC-1)

• Procedure AC-1
• 1 begin
• 2 for i ? 1 until n do NC(i)
• 3 Q ? (i, j) (i,j) ? arcs(G), i ? j
• 4 repeat
• 5 begin
• 6 CHANGE ? false
• 7 for each (i, j) ? Q do CHANGE ? (
  REVISE(i, j) or CHANGE )
• 8 end
• 9 until CHANGE
• 10 end
• AC-1 does not update Q, the queue of arcs
• No algorithm can have time complexity below
  O(ea2)

22
Arc consistency

• AC may discover the solution
• Example borrowed from Dechter

23
Arc consistency

• 2. AC may discover inconsistency
• Example borrowed from Dechter

24
NC AC Example courtesy of M. Fromherz

• In the temporal problem below
• AC propagates bound
• Unary constraint xgt3 is imposed
• AC propagates bounds again

25
Complexity of AC-1

• Procedure AC-1
• 1 begin
• 2 for i ? 1 until n do NC(i)
• 3 Q ? (i, j) (i,j) ? arcs(G), i ? j
• 4 repeat
• 5 begin
• 6 CHANGE ? false
• 7 for each (i, j) ? Q do CHANGE ?
  (REVISE(i, j) or CHANGE)
• 8 end
• 9 until CHANGE
• 10 end
• Note Q is not modified and Q 2e
• 4 ? 9 repeats at most n?a times
• Each iteration has Q 2e calls to REVISE
• Revise requires at most a2 checks of Pij

? AC-1 is O(a3 ? n ? e)
26
AC versions

• AC-1 does not update Q, the queue of arcs
• AC-2 iterates over arcs connected to at least one
  node whose domain has been modified. Nodes are
  ordered.
• AC-3 same as AC-2, nodes are not ordered

27
Arc consistency (AC-3)

• AC-3 iterates over arcs connected to at least one
  node whose domain has been modified
• Procedure AC-3
• 1 begin
• 2 for i ? 1 until n do NC(i)
• 3 Q ? (i, j) (i, j) ? arcs(G), i ? j
• 4 While Q is not empty do
• 5 begin
• 6 select and delete any arc (k, m) from Q
• 7 If Revise(k, m) then Q ? Q ? (i, k) (i,
  k) ? arcs(G), i ? k, i ? m
• 8 end
• 9 end

28
Complexity of AC-3

• Procedure AC-3
• 1 begin
• 2 for i ? 1 until n do NC(i)
• 3 Q ? (i, j) (i, j) ?arcs(G), i ? j
• 4 While Q is not empty do
• 5 begin
• select and delete any arc (k, m) from Q
• 7 If Revise(k, m) then Q ? Q ? (i, k) (i,
  k) ? arcs(G), i ? k, i ? m
• 8 end
• 9 end

• First Q 2e, then it grows and shrinks 4?8
• Worst case
• 1 element is deleted from DVk per iteration
• none of the arcs added is in Q
• Line 7 a?(dk - 1)
• Lines 4 - 8
• Revise a2 checks of Pij
• ? AC-3 is O(a2(2e a(2e-n)))
• Connected graph (e ? n 1), AC-3 is O(a3e)
• If AC-p, AC-3 is O(a2e) ? AC-3 is ?(a2e)
• Complete graph O(a3n2)

29
Example Apply AC-3

• Example Apply AC-3 Thanks to Xu Lin
• DV1 1, 2, 3, 4, 5
• DV2 1, 2, 3, 4, 5
• DV3 1, 2, 3, 4, 5
• DV4 1, 2, 3, 4, 5
• CV2,V3 (2, 2), (4, 5), (2, 5), (3, 5), (2, 3),
  (5, 1), (1, 2), (5, 3), (2, 1), (1, 1)
• CV1,V3 (5, 5), (2, 4), (3, 5), (3, 3), (5, 3),
  (4, 4), (5, 4), (3, 4), (1, 1), (3, 1)
• CV2,V4 (1, 2), (3, 2), (3, 1), (4, 5), (2, 3),
  (4, 1), (1, 1), (4, 3), (2, 2), (1, 5)

30
Applying AC-3 Thanks to Xu Lin

• Queue CV2, V4, CV4,V2, CV1,V3, CV2,V3, CV3,
  V1, CV3,V2
• Revise(V2,V4) DV2 ?DV2 \ 5 1, 2, 3, 4
• Queue CV4,V2, CV1,V3, CV2,V3, CV3, V1, CV3,
  V2
• Revise(V4, V2) DV4 ? DV4 \ 4 1, 2, 3, 5
• Queue CV1,V3, CV2,V3, CV3, V1, CV3, V2
• Revise(V1, V3) DV1 ? 1, 2, 3, 4, 5
• Queue CV2,V3, CV3, V1, CV3, V2
• Revise(V2, V3) DV2 ? 1, 2, 3, 4

31
Applying AC-3 (cont.) Thanks to Xu Lin

• Queue CV3, V1, CV3, V2
• Revise(V3, V1) DV3 ? DV3 \ 2 1, 3, 4, 5
• Queue CV2, V3, CV3, V2
• Revise(V2, V3) DV2 ? DV2
• Queue CV3, V2
• Revise(V3, V2) DV3 ? 1, 3, 5
• Queue CV1, V3
• Revise(V1, V3) DV1 ? 1, 3, 5 END

32
Main Improvements

• Mohr Henderson (AIJ 86)
• AC-3 O(a3e) ? AC-4 O(a2e)
• AC-4 is optimal
• Trade repetition of consistency-check operations
  with heavy bookkeeping on which and how many
  values support a given value for a given
  variable
• Data structures it uses
• m values that are active, not filtered
• s lists all vvp's that support a given vvp
• counter given a vvp, provides the number of
  support provided by a given variable
• How it proceeds
• Generates data structures
• Prepares data structures
• Iterates over constraints while updating support
  in data structures

33
Step 1 Generate 3 data structures

• m and s have as many rows as there are vvps in
  the problem
• counter has as many rows as there are tuples in
  the constraints

34
Step 2 Prepare data structures

• Data structures s and counter.
• Checks for every constraint CVi,Vj all tuples
  Viai, Vjbj_)
• When the tuple is allowed, then update
• s(Vj,bj) ? s(Vj,bj) ? (Vi, ai) and
• counter(Vj,bj) ? (Vj,bj) 1
• Update counter ((V2, V3), 2) to value 1
• Update counter ((V3, V2), 2) to value 1
• Update s-htable (V2, 2) to value ((V3, 2))
• Update s-htable (V3, 2) to value ((V2, 2))
• Update counter ((V2, V3), 4) to value 1
• Update counter ((V3, V2), 5) to value 1
• Update s-htable (V2, 4) to value ((V3, 5))
• Update s-htable (V3, 5) to value ((V2, 4))

35
Constraints CV2,V3 and CV3,V2
S S
(V4, 5) Nil
(V3, 2) ((V2, 1), (V2, 2))
(V3, 3) ((V2, 5), (V2, 2))
(V2, 2) ((V3,1), (V3,3), (V3, 5), (V3,2))
(V3, 1) ((V2, 1), (V2, 2), (V2, 5))
(V2, 3) ((V3, 5))
(V1, 2) Nil
(V2, 1) ((V3, 1), (V3, 2))
(V1, 3) Nil
(V3, 4) Nil
(V4, 2) Nil
(V3, 5) ((V2, 3), (V2, 2), (V2, 4))
(v4, 3) Nil
(V1, 1) Nil
(V2, 4) ((V3, 5))
(V2, 5) ((V3, 3), (V3, 1))
(V4, 1) Nil
(V1, 4) Nil
(V1, 5) Nil
(V4, 4) Nil
Counter Counter
(V4, V2), 3 0
(V4, V2), 2 0
(V4, V2), 1 0
(V2, V3), 1 2
(V2, V3), 3 4
(V4, V2), 5 1
(V2, V4), 1 0
(V2, V3), 4 0
(V4, V2), 4 1
(V2, V4), 2 0
(V2, V3), 5 0
(V2, V4), 3 2
(V2, V4), 4 0
(V2, V4), 5 0
(V3, V1), 1 0
(V3, V1), 2 0
(V3, V1), 3 0

(V3, V2), 4 0
Etc Etc.
Updating m Note that (V3, V2),4 ? 0 thus we
remove 4 from the domain of V3 and update (V3,
4) ? nil in m Updating counter Since 4 is
removed from DV3 then for every (Vk, l) (V3,
4) ? s(Vk, l), we decrement counter(Vk, V3),
l by 1
36
Summary of arc-consistency algorithms

• AC-4 is optimal (worst-case) Mohr Henderson,
  AIJ 86
• Warning worst-case complexity is pessimistic.
• Better worst-case complexity AC-4
• Better average behavior AC-3 Wallace,
  IJCAI 93
• AC-5 special constraints Van Hentenryck,
  Deville, Teng 92
• functional, anti-functional, and monotonic
  constraints
• AC-6, AC-7 general but rely heavily on data
  structures for bookkeeping Bessière Régin
• Now, back to AC-3 AC-2000, AC-2001, AC-3.1,
  AC3.3, etc.
• Non-binary constraints
• GAC (general) Mohr Masini 1988
• all-different (dedicated) Régin, 94

37
Outline

• Motivation and material
• Node and arc consistency and their complexity
• Criteria for performance comparison CSP
  parameters.
• Which AC algorithm to choose? Project results may
  tell
• Path consistency and its complexity
• Consistency properties of a CSP
• Other related results box consistency,
  constraint synthesis, look-ahead, non-binary
  CSPs, etc.

38
CSP parameters n, a, t, d

• n is number of variables
• a is maximum domain size
• t is constraint tightness
• d is constraint density
• where e is the constraints, emin(n-1), and
  emax n(n-1)/2
• Lately, we use constraint ratio p e/emax
• ? Constraints in random problems often generated
  uniform
• ? Use only connected graphs (throw the
  unconnected ones away)

39
Criteria for performance comparison

• Bounding time and space complexity (theoretical)
• worst-case
• average-case
• best- case
• Counting the number of constraint checks
  (theoretical, empirical)
• Measuring CPU time (empirical)

40
Performance comparison Courtesy of Lin XU
AC-4
AC-7
AC-3
41
Wisdom (?) Free adaptation from Bessière

• When a single constraint check is very expensive
  to make, use AC-7.
• When there is a lot of propagation, use AC-4
• When little propagation and constraint checks are
  cheap, use AC-3x
• Advice? Instructor's personal
  opinion
• Use AC-3..

42
Outline

1. Motivation and material
2. Node and arc consistency and their complexity
3. Criteria for performance comparison and CSP
   parameters.
4. Path consistency and its complexity
5. Consistency properties of CSPs
6. Other related results box consistency,
   constraint synthesis, look-ahead, non-binary
   CSPs, etc.

43
AC is not enough Example borrowed from Dechter

• Arc-consistent?
• Satisfiable?
• ? seek higher levels of consistency

V
V
1
1
b a
a b
V
V
V
V
2
3
2
a b
3
b a
a b
a b
44
Consistency of a path

• A path (V0, V1, V2, , Vm) of length m is
  consistent iff
• for any value x?DV0 and for any value y?DVm that
  are consistent (i.e., PV0 Vm(x, y))
• ? a sequence of values z1, z2, , zm-1 in the
  domains of variables V1, V2, , Vm-1, such that
  all constraints between them (along the path, not
  across it) are satisfied
• (i.e., PV0 V1(x, z1) ? PV1 V2(z1, z2) ?
  ? PVm-1 Vm(zm-1, zm) )

45
Note

• ? The same variable can appear more than once in
  the path
• ? Every time, it may have a different value
• ? Constraints considered PV0,Vm and those along
  the path
• ? All other constraints are neglected

46
Example consistency of a path

• ? Check path length 2, 3, 4, 5, 6, ....

?
?
?
?
?
?
?
?
?
?
?
?
47
Path consistency definition

• ? A path of length m is path consistent
• ? A CSP is path consistent Property of a
  CSP
• Definition A CSP is path consistent (PC) iff
  every path is
• consistent (i.e., any length of path)
• Question should we enumerate every path of any
  length?
• Answer No, only length 2, thanks to Mackworth
  AIJ'77

48
Making a CSP Path Consistent (PC)

• Special case Complete graph
• Theorem In a complete graph, if every path of
  length 2 is consistent, the network is path
  consistent Mackworth AIJ'77
• ? PC-1 two operations, composition and
  intersection
• ? Proof by induction.
• Special case Triangulated graph
• Theorem In a triangulated graph, if every path
  of length 2 is consistent, the network is path
  consistent Bliek Sam-Haroud 99
• ? PPC (partially path consistent) ? PC

49
Tools for PC-1

• Two operators
• Constraint composition ( )
• R13 R12 R23
• Constraint intersection ( ? )
• R13? R13, old ? R13, induced

50
Path consistency (PC-1)

• Achieved by composition and intersection (of
  binary relations expressed as matrices) over all
  paths of length two.
• Procedure PC-1
• 1 Begin
• 2 Yn ? R
• 3 repeat
• 4 begin
• 5 Y0 ? Yn
• 6 For k ? 1 until n do
• 7 For i ? 1 until n do
• 8 For j ? 1 until n do
• 9 Ylij ? Yl-1ij ? Yl-1ik Yl-1kk Yl-1kj
• 10 end
• 11 until Yn Y0
• 12 Y ? Yn
• 10 end

51
Properties of PC-1

• Discrete CSPs Montanari'74
• PC-1 terminates
• PC-1 results in a path consistent CSP
• PC-1 terminates. It is complete, sound (for
  finding PC network)
• PC-2 Improves to PC-1similarly to (AC-1 ? AC-3)
• Complexity of PC-1..

52
Complexity of PC-1

• Procedure PC-1
• 1 Begin
• 2 Yn ? R
• 3 repeat
• 4 begin
• 5 Y0 ? Yn
• 6 For k ? 1 until n do
• 7 For i ? 1 until n do
• 8 For j ? 1 until n do
• 9 Ylij ? Yl-1ij ? Yl-1ik Yl-1kk
  Yl-1kj
• 10 end
• 11 until Yn Y0
• 12 Y ? Yn
• 10 end

• Line 9 a3
• Lines 610 n3. a3
• Line 3 at most n2 relations x a2 elements
• PC-1 is O(a5n5)
• PC-2 is O(a5n3) and ?(a3n3)

53
Some improvements

• Mohr Henderson (AIJ 86)
• PC-2 O(a5n3) ? PC-3 O(a3n3)
• Open question PC-3 optimal?
• Han Lee (AIJ 88)
• PC-3 is incorrect
• PC-4 O(a3n3) space and time
• Singh (ICTAI 95)
• PC-5 uses ideas of AC-6 (support bookkeeping)
• Note PC is seldom used in practical applications
  unless in presence of special type of constraints
  (e.g., bounded difference)

54
Path consistency as inference of binary
constraints

• Path consistency corresponds to inferring a new
  constraint
• (alternatively, tightening an existing
  constraint) between every two
• variables given the constraints that link them to
  a third variable
• ? Considers all subgraphs of 3 variables
• ? 3-consistency

B lt C
55
Path consistency as inference of binary
constraints

• Another example

56
Question Adapted from Dechter

• Given three variables Vi, Vk, and Vj and the
  constraints CVi,Vk, CVi,Vj, and CVk,Vj, write the
  effect of PC as a sequence of operations in
  relational algebra.

B
B
A lt B
A lt B
A
A
B lt C
B lt C
C
C
-3 lt A C lt 0
A 3 gt C
Solution CVi,Vj ? CVi,Vj ? ??ij(CVi,Vk
CVk,Vj)
57
Constraint propagation courtesy of Dechter

• After Arc-consistency
• After Path-consistency
• Are these CSPs the same?
• Which one is more explicit?
• Are they equivalent?
• The more propagation,
• the more explicit the constraints
• the more search is directed towards a solution

58
PC can detect unsatisfiability

• Arc-consistent?
• Path-consistent?

V1
a b
?
?
?
V2
V3
?
a b
a b
a b
?
?
a b
V4
59
Outline

• Motivation and material
• Node and arc consistency and their complexity
• Criteria for performance comparison and CSP
  parameters.
• Which AC algorithm to choose? Project results may
  tell!
• Path consistency and its complexity
• Consistency properties of a CSP
• Other related results box consistency,
  constraint synthesis, look-ahead, non-binary
  constraints, etc.

60
Consistency properties of a CSP

• (Arc Consistency, AC)
• (Path Consistency, PC)
• Minimality
• Decomposability (ref. lossless join
  decomposition)
• When PC approximates minimality and
  decomposability.
• Example temporal reasoning
• (strong) k-consistency
• (i,j)-consistency

61
Minimality

• PC tightens the binary constraints
• The tightest possible binary constraints yield
  the minimal network
• Minimal network a.k.a. central problem
• Given two values for two variables, if they are
  consistent, then they appear in at least one
  solution.
• Note
• Minimal ? path consistent
• The definition of minimal CSP is concerned with
  binary CSPs

62
Minimal CSP

• Minimal network a.k.a. central problem
• Given two values for two variables, if they are
  consistent, then they appear in at least one
  solution.
• Informally
• In a minimal CSP the remainder of the CSP does
  not add any further constraint to the direct
  constraint CVi, Vj between the two variables Vi
  and Vj
  Mackworth AIJ'77
• A minimal CSP is perfectly explicit as far as
  the pair Vi and Vj is concerned, the rest of the
  network does not add any further constraint to
  the direct constraint CVi, Vj
  Montanari'74
• The binary constraints are explicit as possible.
  Montanari'74

63
Decomposability

• Any combination of values for k variables that
  satisfy the constraints between them can be
  extended to a solution.
• Decomposability generalizes minimality
• Minimality any consistent combination of
  values for
• any 2 variables is extendable to a
  solution
• Decomposability any consistent combination of
  values for
• any k variables is extendable to a
  solution

Minimal ?
Decomposable ?
Path Consistent
?
?
n-consistent ?
Strong n-consistent ?
Solvable
64
Relations to (theory of) DB
CSP Database
Minimal Pair-wise consistent
Decomposable Complete join
65
PC approximates..

• In general
• Decomposability ? minimality ? path consistent
• PC is used to approximate minimality (which is
  the central problem)
• When is the approximation the real thing?
• Special cases
• When composition distributes over intersection,
  Montanari'74
• PC-1 on the completed graph guarantees
  minimality and decomposability
• When constraints are convex
  Bliek Sam-Haroud 99
• PPC on the triangulated graph guarantees
  minimality and decomposability (and the existing
  edges are as tight as possible)

66
PC algorithms and property

• Result
• CSP0 is not path consistent or
• PC-1(CSP0) is path consistent
• It is tight, but could be tighter
• The graph is complete

• Result
• CSP0 is not path consistent or
• PPC(CSP0) is path consistent
• It is generally less tight than PC-1(CSP0)
• The graph is triangulated

Watch for ?PPC
67
PC algorithms special cases

• Result
  Montanari 74
• CSPa is not path consistent or
• PC-1(CSPa) is
• Path consistent, minimal decomposable
• Graph is complete

With a?(b?c)(a?b)?(b?c)
CSPb

• Result
  BS-H 99
• CSPb is not path consistent or
• PPC(CSPb) is
• Path consistent, minimal, decomposable
• Existing edges are as tight as PC-1(CSPb)
• Graph is triangulated

Triangulatethe graph
PPC
Constraint are convex
68
Special conditions examples

• Distributivity property
• 1 loop in PC-1 (PC-3) is sufficient
• Exploiting special conditions in temporal
  reasoning
• Temporal constraints in the Simple Temporal
  Problem (STP) composition intersection
• Composition distributes over intersection
• PC-1 is a generalization of the Floyd-Warshall
  algorithm (all pairs shortest path)
• Convex constraints
• PPC

69
Distributivity property
Intersection, ? Composition,

• In PC-1, two operations
• RAB (RBC ? R'BC) (RAB RBC) ? (RAB
  RBC)
• When ( ) distributes over ( ? ), then
  Montanari'74
• PC-1 guarantees that CSP is minimal and
  decomposable
• 1 loop in PC-1 is sufficient

B
RBC
RAB
RBC
A
C
70
Temporal reasoning constraints of bounded
difference

• Variables X, Y, Z, etc.
• Constraints a ? Y-X ? b, i.e. Y-X a, b I
• Composition I 1 I2 a1, b1 a2, b2
  a1 a2, b1b2
• Interpretation
• intervals indicate distances
• composition is triangle inequality.
• Intersection I1 ? I2 max(a1, a2), min(b1,
  b2)
• Distributivity I1 (I2 ? I3) (I1 I2) ? (I1
  I3)
• Proof left as an exercise

71
Example constraints of bounded difference

• Composition of intervals
• R13 R12 R23 4, 12
• R01 R13 2,5 3, 5 5, 10
• R01 R'13 2,5 4, 12 6, 17
• Intersection of intervals R13 ? R'13 4, 12
  ? 3, 5 4, 5
• R01 (R13 ? R'13) (R01 R13) ? (R01 R'13)
• R01 (R13 ? R'13) 2, 5 4, 5 6, 10
• (R01 R13) ? (R01 R'13) 5, 10 ? 6,17
  6, 10
• Path consistency guarantees minimality and
  decomposability

72
PC-1 on the STP

• PC-1 generalizes Floyd-Warshall algorithm
  (all-pairs shortest path), where
• composition is scalar addition and
• intersection is scalar minimal

73
Convex constraints temporal reasoning (again!)

• Thanks to Xu Lin (2002)
• Constraints of bounded difference are convex
• We triangulate the graph (good heuristics exist)
• Apply PPC restrict propagations in PC to
  triangles of the graph (and not in the complete
  graph)
• According to Bliek Sam-Haroud 99 PPC becomes
  equivalent to PC, thus it guarantees minimality
  and decomposability

74
PC is not enough

• Arc-consistent?
• Path-consistent?
• Satisfiable?
• ? we should seek (even) higher levels of
  consistency
• ? k-consistency, k 1, 2, 3, .

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
75
k-consistency

• A CSP is k-consistent iff
• given any consistent instantiation of any (k-1)
  distinct variables
• there exists an instantiation of any kth variable
  such that the k values taken together satisfy all
  the constraints among the k variables.
• Example (courtesy of Dechter)

3-consistent?
4-consistent?
Q
Q
Q
Q
Q
76
Warning Does 3-consistency guarantee
2-consistency?
B
red, blue
red, blue
?
?
A
C
red
red

• Question
• Is this CSP 3-consistent?
• is it 2-consistent?
• Lesson
• 3-consistency does not guarantee 2-consistency

77
(Strong) k-consistency definition

• Definitions
• k-consistency vs. strong k-consistency
• k-consistency
• any consistent assignment for (k-1)-variable can
  be extended to any kth variable
• strong k-consistency
• has to be j-consistent for all j ? k

78
(i,j)-consistency definition

• (i,j) consistency
• every consistent instantiation to i variables
• can be extended to every j variables
• Consequently
• 2-consistency i1, j1
• 3-consistency i2, j1
• k-consistency ik-1, j1
• Minimality i2, jn-2
• Decomposability ik, jn-k

79
Consistency properties algorithms summary

• New terms AC, PC, minimal CSP, decomposable
  CSP, complete graph, triangulated graph, Revise,
  AC-1, AC-3, PC-1, PC-2, PPC
• Complexity of AC, PC algorithms
• AC tightens domains, PC tightens binary
  constraints
• Consistency operations in terms of composition,
  intersection in relational algebra
• Path consistency approximates minimality, which
  approximates decomposability
• Situations when path consistency guarantees
  minimality and decomposability
• Strong k-consistency and (i,j)-consistency

80
Outline

• Motivation and material
• Node and arc consistency and their complexity
• Criteria for performance comparison and CSP
  parameters.
• Which AC algorithm to choose? Project results may
  tell!
• Path consistency and its complexity
• Consistency properties of a CSP
• Other related results
• constraint synthesis,
• box consistency,
• look-ahead,
• non-binary CSPs, etc.

81
Constraint synthesis

• k-consistency an alternative view
• Constraint of arity (k-1) are used to synthesize
  a constraint of arity k, which is then used to
  filter constraints of arity (k-1)
• Examples
• 2-consistency DVi ? DVi ? ?Vi (CVi,Vj DVj)
• Projection of binary constraint is used to
  tighten unary constraint
• 3-consistency CVi,Vj ? CVi,Vj ? ?Vi,Vj(CVi,Vk
  CVk,Vj)
• Binary constraints are used to synthesize a
  ternary constraint
• Projection of the ternary constraint is used to
  tighten binary constraint
• k-consistency Freuder 78, 82
• Constraints of arity (k-1) are used to
  synthesize constraints of arity k
• Projection of k-ary constraint is used to
  tighten (k-1)-ary constraints

82
Solving CSPs by Constraint synthesis Freuder 78

• From k2 to kn,
• achieve k-consistency by using (k-1)-arity
  constraints to synthesize k-arity constraints,
• then use the k-ary constraints to filter
  constraints of arity k-1, k-2, etc.
• Process ends with a unique n-ary constraint whose
  tuples are all the solutions to the CSP

83
Interval constraints

• Domains are (continuous) intervals
• Historically also called continuous CSPs,
  continuous domains
• Domains are infinite
• We cannot enumerate consistent values/tuples
• Davis, AIJ 87 (see recommended reading) showed
  that even AC may be incomplete or not terminate
• We apply consistency (usually, arc-consistency)
  on the boundaries of the interval

84
Look-ahead (already discussed)

• Integration of consistency algorithm with search
• AC
• Forward Checking (FC) Haralick Elliott
• Directional Arc-Consistency (DAC) Dechter
• Maintaining arc consistency (MAC) Sabin
  Freuder
• PC
• Directed path consistency (DPC)
  Dechter

85
Non-binary CSPs

• (Almost) all algorithms and properties discussed
  so far were restricted to binary CSPs
• Consistency properties for non-binary CSPs are
  the topic of current research. Mainly,
  properties and algorithms for
• GAC generalized arc-consistency Mohr Masini
• Relational m-consistency Dechter, Chap 8

86
Generalized Arc-Consistency

• Conceptually project the constraint on each of
  the variables in its scope to tighten the domain
  of the variable.

• When constraint is not defined in extension, GAC
  may be problematic (e.g., NP-hard in TCSP)

87
Relational consistency

• Dechter (see Section 8.1.1) generalizes
  consistency properties to non-binary constraints
  as relational m-consistency
• Relational 1-consistency relational
  arc-consistency GAC
• Relational 2-consistency relational
  path-consistency

88
Look-ahead in non-binary CSPs

• In binary CSPs, when a variable is instantiated,
  Revise (Vf, Vc) updates the domain of a future
  variable Vf that is a neighbor to the current
  variable Vc
• In non-binary CSPs, a constraint that applies to
  a current variable may involve one or more past
  variables and one or more future variables. The
  question is when to trigger Revise?
• There are 5 strategies nFC0, nFC1, , nFC5 in
  increasing aggressivity Bessiere et al. AIJ
  02
• For example, nFC0 triggers Revise only when all
  but one variable has been instantiated

89
Related results summary

• Interval constraints box consistency
• Solving CSPs by Constraint synthesis Freuder
• From k2 to kn-1, achieve k-consistency by using
  (k-1)-arity constraints to synthesize k-arity
  constraints, then filter constraints of arity
  k-1, k-2, etc.
• Integration with search
• AC
• Forward Checking (FC) Haralick Elliott
• Directional Arc-Consistency (DAC) Dechter
• Maintaining arc consistency (MAC) Sabin
  Freuder
• PC Directed path consistency (DPC)
  Dechter
• Non-binary constraints
• Generalized arc consistency (GAC) Mohr
  Masini
• Relational consistency Dechter
• Non-binary constraints and integration with
  search
• nFC0, nFC1, , nFC5
• etc.