By by Patrik Haslum.
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Additional resources for Admissible heuristics for automated planning
That for some s, h (s) = v > v = hm (s). Since there are only finitely many states, it can be assumed that v is the smallest value for which this holds. If |s| > m, hm (s) = v = hm (s ) for some subset s of s such that |s | = m. ) It must be the case that hm (s ) < v for every size m subset s containing s , since otherwise hm (s) v (contrary to assumption). Therefore, it is enough to consider the simpler case when |s| = m , and hm (s) = hm (s ) for some size m subset of s. , hm (s) = hm ((s − add(a)) ∪ pre(a)) + cost(a) (replacing s by the result of 43 regressing s through a).
SemSyn, which uses a kind of bidirectional search (Parker 2004), falls somewhere in between. 2 above, page 21). Thus, the planning systems are actually solving somewhat different problems. That the hm and planning graph heuristics, which were both originally developed for regression planning, are applicable also in the context of other plan search methods is again due to the close correspondence between solution paths in the regression search space and plans. 1, page 10) solving a regression state is equivalent to solving a planning problem with the set of atoms in the state as the goal and therefore an admissible regression heuristic yields a lower bound on the cost of the solution to a certain planning problem, which is valid regardless of how the search for this solution is made.
Thus hm (s ) hm ((s − add(a)) ∪ pre(a)) + cost(a) ( rather than = since there may be actions with smaller cost applicable to s but not to s) and thus hm ((s −add(a))∪pre(a)) v−cost(a) > v −cost(a) (due to the assumption that v > v and the fact that cost(a) > 0). However, since (s −add(a))∪pre(a) ⊆ (s−add(a))∪pre(a), hm ((s−add(a))∪pre(a)) hm ((s −add(a))∪pre(a)), and thus hm ((s − add(a)) ∪ pre(a)) v − cost(a) > v − cost(a) = hm ((s − add(a)) ∪ pre(a)), contradicting the assumption that v is the smallest value such that hm (s) = v > v = hm (s).