Finding All Minimal Safe Inductive Sets.

Computing minimal (or even just small) certificates is a central problem in automated reasoning and, in particular, in automated formal verification. For unsatisfiable formulas in CNF such certificates take the form of Minimal Unsatisfiable Subsets (MUSes) and have a wide range of applications. As a formula can have multiple MUSes that each provide different insights on unsatisfiability, commonly studied problems include computing a smallest MUS (SMUS) or computing all MUSes (AllMUS) of a given unsatisfiable formula. In this paper, we consider certificates to safety properties in the form of Minimal Safe Inductive Sets (MSISes), and we develop algorithms for exploring such certificates by computing a smallest MSIS (SMSIS) or computing all MSISes (AllMSIS) of a given safe inductive invariant. More precisely, we show how the well-known MUS enumeration algorithms CAMUS and MARCO can be adapted to MSIS enumeration.