Completeness of Bisimilarity for Contextual Equivalence in Linear Theories

In this paper, we develop new variations of methods from operational semantics, and show how to apply these to a linear type theory which has a lazy operational semantics. In particular, we consider how one can establish contextual equivalences in a linear theory with function types and tensor types by instead establishing bisimulations. Thus bisimilarity is sound for contextual equivalence. Further, we show that bisimilarity is complete for contextual equivalence. We shall show that the notion of a program context in the linear setting is non-trivial. In particular, we give a definition of linear context which is amenable to mechanization in a theorem prover, and explain why a more naive approach to dealing with linear contexts would not be so tractable. The central contributions of the paper are: the formulation, in a linear setting, of a good notion of program context and the associated contextual equivalence; an adaptation of Howe’s method; the notion of a linear precongruence; and a proof that bisimilarity is sound and complete for contextual equivalence.