  We present a new method for inferring complexity properties for a class of
programs in the form of flowcharts annotated with loop information.
Specifically, our method can (soundly and completely) decide if computed values
are polynomially bounded as a function of the input; and similarly for the
running time. Such complexity properties are undecidable for a Turing-complete
programming language, and a common work-around in program analysis is to settle
for sound but incomplete solutions. In contrast, we consider a class of
programs that is Turing-incomplete, but strong enough to include several
challenges for this kind of analysis. For a related language that has
well-structured syntax, similar to Meyer and Ritchie's LOOP programs, the
problem has been previously proved to be decidable. The analysis relied on the
compositionality of programs, hence the challenge in obtaining similar results
for flowchart programs with arbitrary control-flow graphs. Our answer to the
challenge is twofold: first, we propose a class of loop-annotated flowcharts,
which is more general than the class of flowcharts that directly represent
structured programs; secondly, we present a technique to reuse the ideas from
the work on tructured programs and apply them to such flowcharts. The technique
is inspired by the classic translation of non-deterministic automata to regular
expressions, but we obviate the exponential cost of constructing such an
expression, obtaining a polynomial-time analysis. These ideas may well be
applicable to other analysis problems.
