  Multiphase ranking functions (M$\Phi$RFs) are tuples $\langle f_1,\ldots,f_d
\rangle$ of linear functions that are often used to prove termination of loops
in which the computation progresses through a number of "phases". Our work
provides new insights regarding such functions for loops described by a
conjunction of linear constraints (Single-Path Constraint loops). The decision
problem existence of a M$\Phi$RF asks to determine whether a given SLC loop
admits a M$\Phi$RF, and the corresponding bounded decision problem restricts
the search to M$\Phi$RFs of depth $d$, where the parameter $d$ is part of the
input. The algorithmic and complexity aspects of the bounded problem have been
completely settled in a recent work. In this paper we make progress regarding
the existence problem, without a given depth bound. In particular, we present
an approach that reveals some important insights into the structure of these
functions. Interestingly, it relates the problem of seeking M$\Phi$RFs to that
of seeking recurrent sets (used to prove non-termination). It also helps in
identifying classes of loops for which M$\Phi$RFs are sufficient. Our research
has led to a new representation for single-path loops, the difference
polyhedron replacing the customary transition polyhedron. This representation
yields new insights on M$\Phi$RFs and SLC loops in general. For example, a
result on bounded SLC loops becomes straightforward.
