Normal Form Bisimulations By Value

Sangiorgi's normal form bisimilarity is call-by-name, identifies all the call-by-name meaningless terms, and rests on open terms in its definition. The literature contains a normal form bisimilarity for the call-by-value $\lambda$-calculus, Lassen's enf bisimilarity, which validates all of Moggi's monadic laws. The starting point of this work is the observation that enf bisimilarity is not the call-by-value equivalent of Sangiorgi's, because it does not identify the call-by-value meaningless terms. The issue has to do with open terms. We then develop a new call-by-value normal form bisimilarity, deemed net bisimilarity, by exploiting an existing formalism for dealing with open terms in call-by-value. It turns out that enf and net bisimilarities are incomparable, as net bisimilarity identifies meaningless terms but it does not validate Moggi's laws. Moreover, there is no easy way to merge them. To better understand the situation, we provide a detailed analysis of the rich range of possible call-by-value normal form bisimilarities, relating them to Ehrhard's call-by-value relational semantics.