On focusing and polarities in linear logic and intuitionistic logic

There are a number of cut-free sequent calculus proof systems known that are complete for first-order intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important aspects of many of these proof systems. First, we suggest that one way to unify these proof systems is to first translate intuitionistic logic formulas into linear logic formulas, then assign a bias (positive or negative) to atomic formulas, and then examine the nature of focused proofs in the resulting linear logic setting. Second, we provide a single focusing proof system for intuitionistic logic and show that these other intuitionistic proof systems can be accounted for by assigning bias to atomic formulas and by inserting certain markers that halt focusing on formulas. Using either approach, we are able to account for proof search mechanisms that allow for forward-chaining (program-directed search), backward-chaining (goaldirected search), and combinations of these two. The keys to developing this kind of proof system for intuitionistic logic involves using Andreoli’s completeness result for focusing proofs and Girard’s notion of polarity used in his LC and LU proof systems.