Latticed $k$-Induction with an Application to Probabilistic Programs

TL;DR

This paper introduces latticed $k$-induction, a generalized verification technique based on fixed point theory, enabling the automatic verification of complex infinite-state probabilistic programs that previous methods could not handle.

Contribution

It develops latticed $k$-induction, extending classical $k$-induction and Park induction to a more general lattice-theoretic framework, including transfinite ordinals.

Findings

Successfully verified non-trivial probabilistic programs automatically.

Generalized $k$-induction to transfinite ordinals for broader applicability.

Enabled verification of programs previously unverifiable without stronger invariants.

Abstract

We revisit two well-established verification techniques, $k$-induction and bounded model checking (BMC), in the more general setting of fixed point theory over complete lattices. Our main theoretical contribution is latticed $k$-induction, which (i) generalizes classical $k$-induction for verifying transition systems, (ii) generalizes Park induction for bounding fixed points of monotonic maps on complete lattices, and (iii) extends from naturals $k$ to transfinite ordinals $\kappa$, thus yielding $\kappa$-induction. The lattice-theoretic understanding of $k$-induction and BMC enables us to apply both techniques to the fully automatic verification of infinite-state probabilistic programs. Our prototypical implementation manages to automatically verify non-trivial specifications for probabilistic programs taken from the literature that - using existing techniques - cannot be verified without synthesizing a stronger inductive invariant first.