## Section: New Results

### Combining inference systems: a generalization of Nelson-Oppen and MCSAT

Participant : Stéphane Graham-Lengrand.

Nelson-Oppen [79] and Model-Constructing Satisfiability (MCSAT) [89], [65]
are two methodologies that allow the reasoning mechanisms of different theories to collaborate, in order to tackle hybrid problems.
While these methodologies are often used and implemented for the practical applications of Automated Reasoning,
their rather sophisticated foundations are traditionally explained in terms of model theory.
SRI International pioneered some work
providing such methodologies with new and more general foundations in terms of *inference systems* [57],
closer to proof theory and to Parsifal's research.
The more recent MCSAT methodology was not captured,
more generally lacked any kind of theorem about the generic combination of arbitrary theories,
and was also thought to be incompatible with the Nelson-Oppen approach,
so that SMT-solvers are either working with one methodology or the other, unable to get the best of both worlds.

In 2016 we designed a combination methodology, based on *inference systems*, that supersedes both Nelson-Oppen and MCSAT [34].
We showed its soundness and completeness, and identified for this the properties that the theories to combine are required to satisfy.
This generalized MCSAT with the generic combination mechanism that it lacked,
and showed that it is perfectly compatible with the Nelson-Oppen methodology, which can now cohabit within the same solver.