Precise Cost Analysis via Local Reasoning

Abstract

The classical approach to static cost analysis is based on first transforming a given program into a set of cost relations, and then solving them into closed-form upper-bounds. The quality of the upper-bounds and the scalability of such cost analysis highly depend on the precision and efficiency of the solving phase. Several techniques for solving cost relations exist, some are efficient but not precise enough, and some are very precise but do not scale to large cost relations. In this paper we explore the gap between these techniques, seeking for ones that are both precise and efficient. In particular, we propose a novel technique that first splits the cost relation into several atomic ones, and then uses precise local reasoning for some and less precise but efficient reasoning for others. For the precise local reasoning, we propose several methods that define the cost as a solution of a universally quantified formula. Preliminary experiments demonstrate the effectiveness of our approach. 1 class DynamicArray { 2 int[ ] data; 3 int size; 4 void add(int[ ] elems) { 5 for (int i=0; i<elems.length; i++) { 6 if (data.length == size) { 7 int[ ] tmp = new int[2 * data.length]; 8 copy(tmp,size,data); 9 data = tmp; 10 } 11 data[size] = elems[i]; 12 size++; 13 } 14 } 15 int r; 16 void qsort() { 17 qs(0, size−1); 18 } 19 void qs(int from, int to) { 20 if (to − from < r) 21 insertionSort(from,to);

A practical comparator of cost functions and its applications

“…The motivation for this application is threefold: (1) we might not have techniques to automatically infer an upper bound from the CRS but we can still check that a given CF is an upper bound, (2) we sometimes can check that f is an upper bound for the CRS , and f is strictly smaller than the upper bound that an automatic analyzer can infer, and (3) we can use it in resource usage certification (see Section 5.4 below). The soundness of this application is guaranteed by [10], where it is proven that, given a cost function f (x), we have that it is an upper bound of a cost relation C if, for each equation C(x) = e + k j=1 C(ȳ j ), ϕ for C, we have that ϕ |= ∀x,ȳ j .f (x) ≥ e + k j=1 f (ȳ j ), i.e., by replacing the given cost function in the equation, the above CF comparison holds.…”