## Section: New Results

### New insights into -quantification

Participants : David Baelde, Andrew Gacek, Dale Miller.

The team has been actively extending the scope of effectiveness
-quantification. As Tiu and Miller have shown in
[65] , 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 [27] , [28] has developed a minimalist generalization of previous work by Miller and Tiu: he has found what seems to be the simplest extension to 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
[38] and examples of it were presented in
[39] . 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.