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  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  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  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  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.