## Section: New Results

### New insights into -quantification

Participants : David Baelde, Dale Miller.

The team has been actively extending the scope of effectiveness
-quantifications. As Tiu and Miller have shown in
[55] , the
quantifier (developed in previous years within the team) provides a completely satisfactory treatment of binding structures in the
*finite* -calculus. Moving this quantifier to treat infinite behaviors via induction and co-induction, required new advances in the underlying proof theory of
-quantification.

The team has explored two different approaches to this problem. David Baelde [11] , [14] has developed a minimalist generalization of previous work by Miller and Tiu: he has found what seems to be the simplest extension that earlier work that allows to interact properly with fixed points and their inference rules (namely, induction and co-induction). His logical approach allows for a rather careful and rigid understanding of scope in the treatment of the meta-theory of logics and computational specifications.

Another angle has been developed as a result of our close international collaborations. Alwen Tiu, now at the Australian National University, has developed a logic, called
which extends the earlier, “minimal” approach by introducing the structural rules of strengthening and exchange into the context of generic variables. As a result, the behavior of bindings becomes much more like the behavior of names more generally, while still maintaining
much of the status as being binders. In combination with our close colleagues at the University of Minnesota, we have extended this work to include a new definitional principle, called
*nabla-in-the-head* , that strengthens our ability to declaratively describe the structure of contexts and proof invariants. This new definitional principle was first presented in
[17] and examples of it were presented in
[18] . Our colleague, Andrew Gacek (a PhD student at the University of Minnesota and former intern with Parsifal) has also built the Abella proof editor that allows for the
direct implementation of this new definitional principle. His system is in distribution and has been used by a number of people to develop examples in this logic.