Deciding Provability of Linear Logic Formulas

A great deal of insight about a logic can be derived from the study of the diiculty of deciding if a given formula is provable in that logic. Most rst order logics are undecidable and are linear time decidable with no quantiiers and no propositional symbols. However , logics diier greatly in the complexity of deciding propositional formulas. For example, rst-order classical logic is undecidable, propositional classical logic is np-complete, and constant-only classical logic is decidable in linear time. Intuitionistic logic shares the same complexity characterization as classical logic except at the propositional level, where intuitionistic logic is pspace-complete. In this survey we review the available results characterizing various fragments of linear logic. Surprises include the fact that both propositional and constant-only linear logic are undecidable. The results of these studies can be used to guide further proof-theoretic exploration, the study of semantics, and the construction of theorem provers and logic programming languages.