## Section: Scientific Foundations

### Design of two level-logic systems

The team has spent a number of years in designing a strong new logic that can be used to reason (inductively and co-inductively) on syntactic expressions containing bindings. This work has been published is a series of papers by McDowell and Miller [46] [45] , Tiu and Miller [51] [57] , and Gacek, Miller, and Nadathur [2] [38] . Besides presenting formal properties of these logic, these papers also documented a number of examples where this logic demonstrated superior approaches to reasoning about a number of complex formal systems, ranging from programming languages to the $\lambda $-calculus and $\pi $-calculus.

The team has also been working on three different prototype theorem proving system that are all related to this stronger logic. These systems are the following.

Abella, which is an interactive theorem prover for the full logic.

Bedwyr, which is a model checker for the “finite” part of the logic.

Tac, which is a sophisticate tactic for automatically completing simple proofs involving induction and unfolding.

We are now in the process of attempting to make all of these system
communicate properly. Given that these systems have been authored by
different team members at different times and for different reasons,
they do not formally share the same notions of syntax and proof. We
are now working to revisit all of these systems and revise them so
that they all work on the *same* logic and so that they can share
their proofs with each other.

Currently, Chaudhuri, Miller, and Accattoli are working with our technical staff member, Heath, to redesign and restructure these systems so that they can cooperate in building proofs.