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

Abstract

In this paper we present a semantics-based framework for analysing the quantitative behaviour of programs with respect to resource usage. We start from an operational semantics in which costs are modelled using a dioid structure. The dioid structure of costs allows the definition of the quantitative semantics as a linear operator. We then develop a theory of approximation of such a semantics, which is akin to what is offered by the theory of abstract interpretation for analysing qualitative properties, in order to compute effectively global cost information from the program. We focus on the notion of long-run cost, which models the asymptotic average cost of a program. The abstraction of the semantics has to take two distinct notions of order into account: the order on costs and the order on states. We prove that our abstraction technique provides a correct approximation of the concrete long-run cost of a program.