Normalisation for Dynamic Pattern Calculi

The Pure Pattern Calculus (PPC) [10, 11] extends the -calculus, as well as the family of algebraic pattern calculi [20, 6, 12], with first-class patterns i.e. patterns can be passed as arguments, evaluated and returned as results. The notion of matching failure of PPC in [11] not only provides a mechanism to define functions by pattern matching on cases but also supplies PPC with parallelor-like, non-sequential behaviour. Therefore, devising normalising strategies for PPC to obtain well-behaved implementations turns out to be challenging. This paper focuses on normalising reduction strategies for PPC. We define a (multistep) strategy and show that it is normalising. The strategy generalises the leftmost-outermost strategy for -calculus and is strictly finer than parallel-outermost. The normalisation proof is based on the notion of necessary set of redexes, a generalisation of the notion of needed redex encompassing non-sequential reduction systems.