DeepAI AI Chat
Log In Sign Up

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

03/19/2020
by   Paolo Baldan, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

10/26/2018

Fixpoint Games on Continuous Lattices

Many analysis and verifications tasks, such as static program analyses a...
07/02/2021

Formal Semantics of a Classical-Quantum Language

We investigate the formal semantics of a simple imperative language that...
04/14/2023

Operations on Fixpoint Equation Systems

We study operations on fixpoint equation systems (FES) over arbitrary co...
01/24/2022

On-The-Fly Solving for Symbolic Parity Games

Parity games can be used to represent many different kinds of decision p...
05/04/2023

A Monoidal View on Fixpoint Checks

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