DeepAI AI Chat
Log In Sign Up

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

by   Paolo Baldan, et al.

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.


page 1

page 2

page 3

page 4


Fixpoint Games on Continuous Lattices

Many analysis and verifications tasks, such as static program analyses a...

Formal Semantics of a Classical-Quantum Language

We investigate the formal semantics of a simple imperative language that...

Operations on Fixpoint Equation Systems

We study operations on fixpoint equation systems (FES) over arbitrary co...

On-The-Fly Solving for Symbolic Parity Games

Parity games can be used to represent many different kinds of decision p...

A Monoidal View on Fixpoint Checks

Fixpoints are ubiquitous in computer science as they play a central role...