## Section: New Results

### Foundational aspects of focusing proof systems

Participants : David Baelde, Dale Miller, Alexis Saurin.

Since focusing proof systems seem to be behind much of our computational logic framework, the team has spent some energies developing further some foundational aspects of this approach to proof systems.

Chuck Liang and Miller have recently finished the paper [6] in which a comprehensive approach to focusing in intuitionistic and classical logic was developed.

Given the team's ambitious to automate logics that require induction and co-induction, we have also looked in detail at the proof theory of fixed points. In particular, David Baelde's recent PhD thesis [11] contains a number of important, foundational theorems regarding focusing and fixed points. In particular, he has examined the logic MALL (multiplicative and additive linear logic). To strengthen this decidable logic into a more general logic, Girard added the exponentials, which allowed for modeling unbounded (“infinite”) behavior. Baelde considers, however, the addition of fixed points instead and he has developed the proof theory of the resulting logic. We see this logic as being behind much of the work that the team will be doing in the coming few years.

Alexis Saurin's recent PhD [12] also contains a wealth of new material concerning focused proof system. In particular, he provides a new and modular approach to proving the completeness of focused proof systems as well as develops the theme of multifocusing.

A particular outcome of our work on focused proof search is the use of
*maximally multifocused proofs* to help provide sequent calculus proofs a canonicity. In particular, Chaudhuri (a former Parsifal post doc), Miller, and Saurin have shown in
[15] that it is possible to show that maximally multifocused sequent proofs can be placed in one-to-one correspondence with more traditional proof net structures for
subsets of MALL.