Complexity Bounds for Some Finite Forms of Kruskalʹs Theorem

Well-founded (partial) orders form an important and convenient mathematical basis for proving termination of algorithms. Well-partial orders provide a powerful method for proving the well-foundedness of partial orders (and hence for proving termination), since every partial ordering which extends a given well-partial ordering on the same domain is automatically well-founded. In this article it is shown by purely combinatorial means that the maximal order type of the homeomorphicembeddability relation on a given set of terms over a finite signature yields an appropriate ordinal recursive Hardy bound on the lengths of bad sequences which satisfy an effective growth rate condition. This result yields theoretical upper bounds for the computational complexity of algorithms, for which termination can be proved by Kruskalʹs theorem.