Abstraction, Up-to Techniques and Games for Systems of Fixpoint Equations
Abstract
Systems of fixpoint equations over complete lattices, consisting of (mixed) least and greatest fixpoint equations, allow one to express a number of verification tasks such as model-checking of various kinds of specification logics or the check of coinductive behavioural equivalences. In this paper we develop a theory of approximation for systems of fixpoint equations in the style of abstract interpretation: a system over some concrete domain is abstracted to a system in a suitable abstract domain, with conditions ensuring that the abstract solution represents a sound/complete overapproximation of the concrete solution. Interestingly, up-to techniques, a classical approach used in coinductive settings to obtain easier or feasible proofs, can be interpreted as abstractions in a way that they naturally fit in our framework and extend to systems of equations. Additionally, relying on the approximation theory, we can provide a characterisation of the solution of systems of fixpoint equations over complete lattices in terms of a suitable parity game, generalising some recent work that was restricted to continuous lattices. The game view opens the way to the development of on-the-fly algorithms for characterising the solution of such equation systems.
Keywords
Cite
@article{arxiv.2003.08877,
title = {Abstraction, Up-to Techniques and Games for Systems of Fixpoint Equations},
author = {Paolo Baldan and Barbara König and Tommaso Padoan},
journal= {arXiv preprint arXiv:2003.08877},
year = {2021}
}