Long-run cost analysis by approximation of linear operators over dioids

Abstract

We present a static analysis technique for modeling and approximating the long-run resource usage of programs. The approach is based on a quantitative semantic framework where programs are represented as linear operators over dioids. We show how to extract the long-run cost of a program from the matrix representation of its semantics. An essential contribution is to provide abstraction techniques which make it feasible to compute safe overapproximations of this cost. A theorem is proved stating that such abstractions yield correct approximations of the program's long-run cost. The theoretical developments are illustrated on a concrete example taken from the analysis of the cache behaviour of a simple bytecode language.

Injecting Abstract Interpretations into Linear Cost Models

“…If we do not assume any additional lattice order on states, we are able to define a simple notion of partition based abstraction. This abstraction technique has been developed in [18,7], and is suitable for simple analyses that consist in "forgetting" information when going from concrete to abstract states. If we want to use more elaborate abstractions, and in particular reuse the classical abstractions of standard abstract interpretation theory, we have to find an abstraction technique that copes with two distinct notions of order: the dioid order on costs, and the lattice order on states.…”

Long-Run Cost Analysis by Approximation of Linear Operators over Dioids

“…Property (i) immediately follows from the fact that the nth power is ⊕-lsc whereas an intermediate lemma is needed to prove property (ii) [10]. Although the definition of cost dioids may seem rather restrictive, we now show that many classes of dioids found in the literature are indeed cost dioids.…”

“…For dioids of type (3) we also have to prove that if equation X n = q has a solution, this solution is unique [10].…”

“…The proof of theorem relies on the fact that the correctness is preserved when the concrete and abstract matrices are iterated simultaneously [10].…”

“…The key points of the proof are to ensure that this limit exist, and to show that a small part of a trace can be neglected for very long traces. This is proved by bounding q(t) [10].…”