# On the Monniaux Problem in Abstract Interpretation

## TL;DR

This paper investigates the decidability of the Monniaux Problem in abstract interpretation, proving undecidability for certain program classes and invariants, but identifying decidability in simple linear loops.

## Contribution

It establishes the undecidability of the Monniaux Problem for unguarded affine programs with semilinear invariants and shows decidability for simple linear loops.

## Key findings

- Undecidability for unguarded affine programs with semilinear invariants.
- Decidability for simple linear loops.
- Clarifies boundaries of decidability in abstract interpretation.

## Abstract

The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program $P$, a safety (\emph{e.g.}, non-reachability) specification $\varphi$, and an abstract domain of invariants $\mathcal{D}$, does there exist an inductive invariant $I$ in $\mathcal{D}$ guaranteeing that program $P$ meets its specification $\varphi$. The Monniaux Problem is of course parameterised by the classes of programs and invariant domains that one considers. In this paper, we show that the Monniaux Problem is undecidable for unguarded affine programs and semilinear invariants (unions of polyhedra). Moreover, we show that decidability is recovered in the important special case of simple linear loops.