Levels of Consistency

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Levels of Consistency
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Levels of Consistency

• Node Consistency (NC)
• Arc-consistency (AC)
• Path Consistency (PC)
• Generalised arc-consistency (GAC)
• Bounds consistency
• Inverse Path Consistency (IPC aka PIC)
• Singleton Arc-consistency (SAC)
• and others

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Node Consistency
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Node Consistency (NC)
aka 1-consistency
Example
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Arc-consistency
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Arc-consistency (AC)
aka 2-consistency

• A constraint Cij is arc consistent if
• for every value x in Di there exists a value y
  in Dj that supports x
• i.e. if vi x and vj y then Cij holds
• note we are assuming Cij is a binary constraint

• A csp (V,D.C) is arc consistent if
• every constraint is arc consistent

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Arc-consistency
AC appears to be the best level of consistency to
maintain (MAC) There is no proof of
this However, there is a body of evidence
For non-trivial problems MAC beats FC beats BT
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Path Consistency
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Path-consistency (aka 3-consistency)
Vj
M. Singh, TAI-95
Vi
Vk
It may create nogood tuples (i/x,k/z), Therefor
e increases size of model/problem. May result in
more constraints to check!
There might be no constraint Cik Therefore
3-consistency may create it!
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Path-consistency (aka 3-consistency)
for x in Di do for z in Dk do supported
false for y in Dj while supported
do supported Cij(x,y) Cik(x,z)
Cjk(y,z) if supported then
post(Cik(x,z))
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See AR33 notes section 7.3
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k-consistency
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Also k-consistency
AR33 section 7.4
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Generalised arc-consistency
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Generalised arc-consistency (GAC)
AR33 section 7.5
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Generalised arc-consistency (GAC)
AR33 section 7.5
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Generalised arc-consistency (GAC)
From Gent, Miguel Nightingale
GAC is meaningful only wrt n-ary constraints (in
a sense)
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Generalised arc-consistency (GAC)
Alan Frisch (York)
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Generalised arc-consistency (GAC)
Alan Frisch (York)
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Bounds Consistency
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Bounds Consistency
Alan Frisch (York)
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Bounds Consistency
Alan Frisch (York)
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Inverse Path Consistency (aka Path Inverse
Consistency)
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Path Inverse Consistency (PIC aka IPC)
Debruyne Bessiere
Note similar to PC but deletes values rather
than adds tuples!
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Path Inverse Consistency (PIC aka IPC)
Debruyne Bessiere
for x in Di do supported false for y in
Dj while supported do for z in Dk while
supported do supported Cij(x,y)
Cik(x,z) Cjk(y,z) if supported then
Di Di \ x
Note similar to PC but deletes values rather
than adds tuples!
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Singleton arc-consistency
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Singleton arc-consistency (SAC)
i.e. we can take any variable, assign it a value
from its domain and then make the problem
arc-consistent and all variables have non-empty
domains
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The SAC1 algorithm
The basic and least efficient algorithm for SAC
(from Bartaks FLAIRS04 paper)
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complexities
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General notes

• we can have inverse consistency for any k
• we can have neighbourhood inverse consistency
• we can have singleton k-consistency

• In our model and CP toolkit we may have mixed
  consistency
• some variables/constraints only forward checked
• some variables in binary constraints AC
• some variable in n-ary constraints GAC
• variables NC

This is not a problem, so long as we are sure
that when we instantiate a variable it is
consistent With respect to the past variables
We can also maintain these levels of
consistency during search
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Conclusion
There are MANY different levels of
consistency This is an ACTIVE area of research
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