Simulating Linear Logic in 1-Only Linear Logic

Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. It turned out that full propositional Linear Logic is undecidable (Lincoln, Mitchell, Scedrov, and Shankar) and, hence, it is more expressive than (modalized) classical or intuitionistic logic. In this paper we focus on the study of the simplest fragments of Linear Logic, such as the one-literal and constant-only fragments (the latter contains no literals at all). Here we demonstrate that all these extremely simple fragments of Linear Logic (one-literal, ⊥-only, and even unit-only) are exactly of the same expressive power as the corresponding full versions: (a) On the level of the multiplicatives {⊗, . .. . . . . . . . . . . . . . . . . . . . . . . .. ,−◦} we get NP -completeness. (b) Enriching this basic set of connectives by additives {&,⊕} yields PSPACE-completeness. (c) Using in addition the storage operator !, we can prove the undecidability of all these three fragments. We present also a complete computational interpretation (in terms of acyclic programs with stack) for ⊥-free Intuitionistic Linear Logic. Based on this interpretation, we prove the fairness of our encodings and establish the foregoing complexity results.