Section: New Results
Focusing proof systems for linear, intuitionistic, and classical logics
When working with proof search and logic programming within linear logic, the completeness of focusing proofs by Andreoli  provides a critical normal form for proofs in all of linear logic. Chaudhuri and his colleagues have applied the focusing proof system of linear logic to forward and backward reasoning systems. Miller and Saurin have provided a new and modular proof of focusing for linear logic in [Oops!] . This proof allows for direct extensions: for example, the focusing result in Baelde and Miller's paper [Oops!] were based on this new modular proof.
In contrast, a flexible and general definition of focusing proofs in intuitionistic and classical proofs have not been provided. Althought there had been a number of focusing-style proof systems that have been defined for these two logics, a general framework to relate all of them was needed. In [Oops!] , Liang and Miller provided just such a framework. In particular, the proof systems LJF and LKF were introduced that provides for a great deal of flexibility in the description of how focusing could be done in these two logics. In particular, polarities could be mixed (a result that was a challenge to get for intuitionistic logic). Many other focusing systems for intuitionistic logic could be mapped compositionally into the LJF via the simple insertion of “delay” operators (simple logical expressions that stop the focusing process).