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arXiv · 2003.08877

Abstraction, Up-to Techniques and Games for Systems of Fixpoint Equations

Paolo Baldan·Barbara König·Tommaso Padoan
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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.

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Paolo Baldan, Barbara König, Tommaso Padoan. 2020-03-19. Abstraction, Up-to Techniques and Games for Systems of Fixpoint Equations. https://doi.org/10.4230/lipics.concur.2020.25

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