Editing Monotonic function (section)

== In the context of search algorithms ==
In the context of [[search algorithm]]s monotonicity (also called consistency) is a condition applied to [[heuristic function]]s.  A heuristic <math>h(n)</math> is monotonic if, for every node {{mvar|n}} and every successor {{mvar|n'}} of {{mvar|n}} generated by any action {{mvar|a}}, the estimated cost of reaching the goal from {{mvar|n}} is no greater than the step cost of getting to {{mvar|n'}} plus the estimated cost of reaching the goal from {{mvar|n'}},

<math display="block">h(n) \leq c\left(n, a, n'\right) + h\left(n'\right) .</math>

This is a form of [[triangle inequality]], with {{mvar|n}}, {{mvar|n'}}, and the goal {{mvar|G<sub>n</sub>}} closest to {{mvar|n}}.  Because every monotonic heuristic is also [[admissible heuristic|admissible]], monotonicity is a stricter requirement than admissibility.  Some [[heuristic algorithm]]s such as [[A* search algorithm|A*]] can be proven [[asymptotically optimal algorithm|optimal]] provided that the heuristic they use is monotonic.<ref>Conditions for optimality:  Admissibility and consistency pg. 94–95 {{Harv|Russell|Norvig|2010}}.</ref>