  Proving program termination is typically done by finding a well-founded
ranking function for the program states. Existing termination provers typically
find ranking functions using either linear algebra or templates. As such they
are often restricted to finding linear ranking functions over mathematical
integers. This class of functions is insufficient for proving termination of
many terminating programs, and furthermore a termination argument for a program
operating on mathematical integers does not always lead to a termination
argument for the same program operating on fixed-width machine integers. We
propose a termination analysis able to generate nonlinear, lexicographic
ranking functions and nonlinear recurrence sets that are correct for
fixed-width machine arithmetic and floating-point arithmetic Our technique is
based on a reduction from program \emph{termination} to second-order
\emph{satisfaction}. We provide formulations for termination and
non-termination in a fragment of second-order logic with restricted
quantification which is decidable over finite domains. The resulted technique
is a sound and complete analysis for the termination of finite-state programs
with fixed-width integers and IEEE floating-point arithmetic.
