.

 A  binary CN is  i, j   consistent iff Vi E X  Di / 0 and  any consistent instan tiation of i variables can be extended to a consistent instantiation including any j additional variables
.

A value a E Di is arc consistent iff Vj E X s t  Cij E C there exists b E D j s t  C ij a,b   A domain D i is arc consistent iff  V a E D i,  i, a  is arc consistent  A CN is arc consistent      consistent iff VDj E D  Dj is a non empty arc consistent domain
.

A pair of values i, a , j, b is path consistent iff Vk E X there exists c E D k st C ik a, c and C jk b, c otherwise it is path inconsistent A C N is path consistent consistent iff no path inconsistent pair of values is allowed
- A binary CN is strongly path consistent iffit is node consistent arc consistent and path consistent
- A binary CN is path inverse consistent iff it is consistent i e
∀ i, a ∈D ∀ j, k ∈ X  s t   j   /  i  /  k  /  j   ∃ j, b ∈ D  and  ∃  k, c ∈ D  s t Cij a, b ∧ Cik a, c ∧ Cjk b, c
.

 A  binary  CN is neighborhood inverse consistent iff V i, a ED   i, a   can be extended to a consistent instantiation including the neighborhood of i
.

 A binary CN is restricted path consistent iff Vi E X Di is a non empty arc consistent domain and V i, a E D  for all j E X s t   i, a  has an unique support b in D i for all k E X linked to both i and j  3c E D k s t  C i k a, c  NC jk b, c
.

 A binary CN is k-restricted path consistent iff Vi E X Di is a non empty arc consistent domain and V i, a E D  for all Ci, E C s t   i, a  has at most k supports in Di, 9b E D i s t  C i j  a, b  and
∀k E X linked to both i and j ∃ c E D k s t  C i k a, c  N c j k  b, c
.

A binary CN is max-restricted path consistent iff Vi E X Di is a non empty arc consistent domain and V i, a  E D  for all Ci, E C
Hb E D i s t  C i j  a, b  and
∀k E X linked to both i and j ∃c E Dk s t  C ik a, c  NCjk b, c
.

 A binary CN P is singleton arc consistent iff Vi E X Di / Ø and V i, a E D
P|Dj={a} has an arc consistent sub domain
.

 A binary CN P is singleton restricted path consistent iff Vi E X   Di / I and V i, a E D P | Dj ={a} has a restricted path consistent sub domain
Figure The mentioned local consistencies