Termination of Linear Loops over the Integers

What’s decidable about linear loops?

“…Linear loops have been extensively studied and play an important role in the foundations of program analysis and software verification. In particular, methods to prove terminationÐor, equivalently, reachabilityÐvia a variety of techniques, such as spectral analysis or the synthesis of ranking functions, have been developed in, e.g., [Ben-Amram et al 2019;Ben-Amram and Genaim 2013Bradley et al 2005;Braverman 2006;Chen et al 2015;Colón and Sipma 2001;Hosseini et al 2019;Podelski and Rybalchenko 2004a,b;Tiwari 2004]. Several of these approaches have been implemented in software verification tools, such as Microsoft's Terminator [Cook et al 2006a,b].…”

Reflections on Termination of Linear Loops

“…Termination analysis of linear loops The universal termination problem for linear loops (or total deterministic affine transition systems, in the terminology of Section 4) was posed by Tiwari [29]. The case of linear loops over the reals was resolved by Tiwari [29], over the rationals by Braverman [4], and finally over the integers by Hosseini et al [14]. In principle, we can combine any of these techniques with our algorithm for computing DATS-reflections of transition formulas to yield a sound (but incomplete) termination analysis.…”

“…Termination analysis Termination analysis, and in particular conditional termination analysis, has been widely studied. Work on the subject can be divided into practical termination analyses that work on real programs (but offer few theoretical guarantees) [6,8,30,31,32,20,11,13,2], and work on simplified model (such as linear, octagonal, and polyhedral loops) with strong guarantees (but cannot be applied directly to real programs) [25,3,21,1,29,14,4]. This paper aims to help bridge the gap between the two, by showing how to apply analyses for linear loops to general programs, while preserving some of their good theoretical properties, in particular monotonicity.…”

A Calculus for Modular Loop Acceleration

“…Loop termination was recently proven to be decidable for the subclass of loops in which the guards and updates use only linear arithmetic and the guards are restricted to conjunctions of atoms [25,39]. Our approach is less restrictive regarding the input loops: we allow for non-linear guards and updates, and we allow for arbitrary Boolean structure in the guards.…”

“…The reason is that some witness of non-termination for the loop must be reachable from an initial program configuration so that the non-termination proof carries over from the loop to the input program. However, the decision procedures in [25,39] are not optimized to this end: They produce a certificate of eventual nontermination, i.e., a formula that describes initial configurations that give rise to witnesses of nontermination by applying the loop body a finite, but unknown number of times. The problem of transforming a single witness of eventual nontermination into a witness of non-termination has been solved partially in [36].…”