Finding whether a linearconstraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linearconstraint loop with a precondition, specifically with partiallyspecified input, has a linear ranking function is EXPSPACEhard over the integers, and PSPACEhard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear rankingfunction supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACEhard for convex polyhedral invariants and EXPSPACEhard for downwardclosed sets of natural numbers as invariants.
