Lower-Bound Synthesis Using Loop Specialization and Max-SMT

Abstract This paper presents a new framework to synthesize lower-bounds on the worst-case cost for non-deterministic integer loops. As in previous approaches, the analysis searches for a metering function that under-approximates the number of loop iterations. The key novelty of our framework is the specialization of loops, which is achieved by restricting their enabled transitions to a subset of the inputs combined with the narrowing of their transition scopes. Specialization allows us to find metering functions for complex loops that could not be handled before or be more precise than previous approaches. Technically, it is performed (1) by using quasi-invariants while searching for the metering function, (2) by strengthening the loop guards, and (3) by narrowing the space of non-deterministic choices. We also propose a Max-SMT encoding that takes advantage of the use of soft constraints to force the solver look for more accurate solutions. We show our accuracy gains on benchmarks extracted from the 2020 Termination and Complexity Competition by comparing our results to those obtained by the "Image missing" system.