Section: New Results

Synthetic connectives and their proof system

Participant : Dale Miller.

In recent years, focused proof systems have being used to expand our understanding of how introduction rules and structural rule relate to each other. In these proof systems, inference rules and logical connectives are polarized as negative or positive in such a way that the invertible inference rules all belong to the negative polarity. Groups of negative connectives can then be grouped into one negative synthetic connective: similarly, positive connectives can be grouped into a positive synthetic connective. Such synthetic connectives admit cut-elimination. Remarkably, focused proof systems for classical and intuitionistic logics can be organized so that negative formulas are, in fact, treated linearly. That is, if weakening or contraction is applied to a formula, that formula is positive.

Focused proof systems can be used to design richly varying collections of synthetic connectives. These proof systems also provide for new means of describing parallelism within proofs and mixing computation and deduction. The ability to treat negative formulas linearly provides important information for the design of automated theorem provers. Synthetic connectives and their associated inference rules will also allow for the design of broad spectrum proof certificates that theorem provers will be able to print and simple proof checkers will be able to validate. Miller's conference paper [20] develops this approach to proof certificates.