An Operational Account of Call-By-Value Minimal and Classical $\lambda$-calculus in ''Natural Deduction'' Form

Abstract : We give a decomposition of the equational theory of call-by-value lambda-calculus into a confluent rewrite system made of three independent subsystems that refines Moggi's computational calculus: - the purely operational system essentially contains Plotkin's beta-v rule and is necessary and sufficient for the evaluation of closed terms; - the structural system contains commutation rules that are necessary and sufficient for the reduction of all ''computational'' redexes of a term, in a sense that we define; - the observational system contains rules that have no proper computational content but are necessary to characterize the valid observational equations on finite normal forms. We extend this analysis to the case of lambda-calculus with control and provide with the first presentation as a confluent rewrite system of Sabry-Felleisen and Hofmann's equational theory of lambda-calculus with control. Incidentally, we give an alternative definition of standardization in call-by-value lambda-calculus that, unlike Plotkin's original definition, prolongs weak head reduction in an unambiguous way.