Termination of Integer Linear Programs

On the Termination of Integer Loops

“…For straight-line while loops, the most remarkable results are those of Tiwari [2004] andBraverman [2006]. Tiwari proved that the problem is decidable for linear deterministic updates when the domain of the variables is R. Braverman proved that this holds also for Q, and for the homogeneous case it holds for Z (see Section 1).…”

“…For function classes that are sufficiently simple to analyze, one can hope that termination of such loops would be decidable; in fact, the main motivation for this paper has been the remarkable results by Tiwari [2004] and Braverman [2006] on the termination of linear loops, a kind of loops where the update function f is linear. The loop conditions in these works are conjunctions of linear inequalities.…”

“…Over the integers it is also sufficient to have only ≥ or only > in the loop guard. However, replacing > by ≥ (or vice versa) alters the homogeneous loop to a non-homogeneous one, which is why including both inequality types is important in the context of Braverman [2006].…”

“…For example, the termination analysis tools Terminator [Cook et al 2006], COSTA [Albert et al 2007], and Julia [Spoto et al 2010] are based on proving termination of such loops by means of a linear ranking function. It is known that the linear-ranking approach cannot completely resolve the problem [Podelski and Rybalchenko 2004;Braverman 2006], since not every terminating program has such a ranking function -this is the case, for example, for loop (1) above. Moreover, the linear-programming based approaches are not sensitive to the assumption that the data are integers.…”

On the Termination of Integer Loops

“…Over the integers it is also sufficient to have only ≥ or only > in the loop guard. However, replacing > by ≥ (or vice versa) alters the homogeneous loop to a non-homogeneous one, which is why including both inequality types is important in the context of Braverman [2006]. Going back to program analysis, we note that it is typical in this field to assume that some degree of approximation is necessary in order to express the effect of the loop body by linear arithmetics alone.…”