On the syntactic and functional correspondence between hybrid (or layered) normalisers and abstract machines

We show how to connect the syntactic and the functional correspondence for normalisers and abstract machines implementing hybrid (or layered) reduction strategies, that is, strategies that depend on subsidiary sub-strategies. Many fundamental strategies in the literature are hybrid, in particular, many full-reducing strategies, and many full-reducing and complete strategies that deliver a fully reduced result when it exists. If we follow the standard program-transformation steps the abstract machines obtained for hybrids after the syntactic correspondence cannot be refunctionalised, and the junction with the functional correspondence is severed. However, a solution is possible based on establishing the shape invariant of well-formed continuation stacks. We illustrate the problem and the solution with the derivation of substitution-based normalisers for normal order, a hybrid, full-reducing, and complete strategy of the pure lambda calculus. The machine we obtain is a substitution-based, eval/apply, open-terms version of Pierre Crégut's full-reducing Krivine machine KN.