Static Analysis

Abstract

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Synthesis of ranking functions using extremal counterexamples

“…In the framework of Alias et al [2010], this would be equivalent to maximizing the number of transitions where ρ decreases strictly.…”

“…quasi lexicographic ordering) over vectors of integers. Our approach is similar to the one of Alias et al [2010]: for each dimension (d), generate a quasi ranking function of maximal termination power ρ d (x) = λ d · x + λ 0,d , after restricting the invariant on successive states to transitions that leave the previous components constant. The algorithm is described in Algorithm 2.…”

“…, ρ m (x) , which is shown to be strictly decreasing with respect to lexicographic ordering. Again, the function ρ can be allowed to depend on the program point, and complete automated synthesis methods exist for this class Alias et al [2010], Ben-Amram and Genaim [2014].…”

“…First, if the program has K basic blocks, n integer variables and one searches for a m-dimensional ranking function, then the number of unknowns in the constraint system is K.m.n, which might be too big Alias et al [2010]. Thus, it is desirable to limit the set of points to consider to heads of loops, or, more generally, a cut-set of K program points -a set such that removing these points cuts all cycles in the program Shamir [1979].…”

“…Lexicographic linear ranking functions Alias et al [2010], Cook et al [2013], Bradley et al [2005b,a], Ben-Amram and Genaim [2014] are both much more powerful than linear ranking functions (e.g. one can easily prove the termination of the Ackermann-Péter function) and not much harder to obtain.…”