On Decidable Growth-Rate Properties of Imperative Programs

TL;DR

This paper extends the decidability of growth-rate properties in imperative programs by including resets, shifting the analysis approach to a top-down method with PSPACE-complete complexity.

Contribution

It introduces a logical analysis method for programs with resets, expanding the class of decidable growth-rate properties in imperative languages.

Findings

Handling resets makes the problem PSPACE-complete

Analysis shifts from proving to disproving bounds

Top-down algorithm improves previous methods

Abstract

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple "core" programming language - an imperative language with bounded loops, and arithmetics limited to addition and multiplication - it was possible to decide precisely whether a program had certain growth-rate properties, namely polynomial (or linear) bounds on computed values, or on the running time. This work emphasized the role of the core language in mitigating the notorious undecidability of program properties, so that one deals with decidable problems. A natural and intriguing problem was whether more elements can be added to the core language, improving its utility, while keeping the growth-rate properties decidable. In particular, the method presented could not handle a command that resets a variable to zero. This paper shows how to handle resets. The analysis is given in a logical style (proof rules), and its complexity is shown to be PSPACE-complete (in contrast, without resets, the problem was PTIME). The analysis algorithm evolved from the previous solution in an interesting way: focus was shifted from proving a bound to disproving it, and the algorithm works top-down rather than bottom-up.