Past exam of the computer science course of the University of Oxford / 2021 / Artificial intelligence / 1 / a

All three variants are complete search algorithms under the usual finite-branching factor, nonnegative-heuristic function assumptions. Here $0\le h\le h^{\ast }$, with zero at goals; the supplied positive lower bound on step costs is also essential. The root is processed once before a parent-dependent priority is needed.^[ref] Completeness means finding a solution if one exists, not necessarily terminating when none exists.

Fix a finite solution path. Every search node on that path has a finite priority, so let $K$ bound all of its relevant priorities. For ordinary A-star search and the second variant, the priority is at least $g(n)$. Thus any search node with priority at most $K$ has depth at most $\lfloor K/ϵ\rfloor$. In the third variant the priority is at least $g(n^{\prime })$, so its depth is at most $\lfloor K/ϵ\rfloor +1$. A finitely branching search tree has finitely many search nodes at these depths. Therefore finitely many earlier search nodes can delay each successive search node on the chosen solution path; eventually its goal is generated or selected, as appropriate.

Moving the goal test to generation cannot destroy that existence guarantee, although it can destroy optimality. Likewise the altered parent-dependent priorities remain bounded below by the accumulated cost to the search node or its parent, which is enough for this completeness argument. If “admissible” were interpreted to allow arbitrarily negative estimates, the lower-bound argument would need an additional hypothesis; the conventional nonnegative interpretation is being used explicitly. For the literal upper-bound-only interpretation, all three can fail completeness: attach an infinite unit-cost dead-end chain with $g(n_d)=d$ and $h(n_d)=-2d$ beside a finite goal branch beginning with a cost-$3$ edge and estimate zero. Its successive low priorities can starve the finite branch. Every negative chain estimate is still at most its infinite true remaining cost.