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Levels of Consistency

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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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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 v[i] = x and v[j] = 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 Arc-consistency (AC) aka 2-consistency

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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) Vi Vj Vk There might be no constraint Cik Therefore 3-consistency may create it! It may create nogood tuples {(i/x,k/z),…} Therefore increases size of model/problem. May result in more constraints to check! M. Singh, TAI-95

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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-consistencyAR33 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) GAC is meaningful only wrt n-ary constraints (in a sense) From Gent, Miguel & Nightingale

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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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Alan Frisch (York)

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Bounds ConsistencyAlan 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 Note: similar to PC but deletes values rather than adds tuples! 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}

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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 basic and least efficient algorithm for SAC (from Bartaks FLAIRS04 paper) The SAC1 algorithm

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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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