## Section: New Results

### Foundational aspects of focusing proof systems

Participants : David Baelde, Kaustuv Chaudhuri, 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. In particular,
they present a compact sequent calculus LKU for classical logic
organized around the concept of *polarization.* Focused sequent
calculi for classical, intuitionistic, and multiplicative-additive
linear logics are derived as fragments of the host system by varying
the sensitivity of specialized structural rules to polarity
information. They identify a general set of criteria under which cut
elimination holds in such fragments. From cut elimination leads to a
unified proof of the completeness of focusing. Furthermore, each
sublogic can interact with other fragments through cut. Under certain
circumstances, for example, it is possible for a classical lemma to be
used in an intuitionistic proof while preserving intuitionistic
provability. They also examine the possibility of defining some
classical-linear *hybrid logics.*

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 [27] 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 [61] 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, Miller, and Saurin have
shown in [32] 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.