Partial Type Assignment in Left Linear Apllicative Term Rewriting Systems

This paper introduces a notion of partial type assignment on left linear applicative term rewriting systems that is based on the extension defined by Mycroft of Curry’s type assignment system. The left linear applicative TRS we consider are extensions to those suggested by most functional programming languages in that they do not discriminate against the varieties of function symbols that can be used in patterns. As such there is no distinction between function symbols (such as append and plus) and constructor symbols (such as cons and succ). Terms and rewrite rules will be written as trees, and type assignment will consist of assigning types to function symbols, nodes and edges between nodes. The only constraints on this system are imposed by the relation between the type assigned to a node and those assigned to its incoming and out-going edges. We will show that every typeable term has a principal type, and formulate a needed and sufficient condition typeable rewrite rules should satisfy in order to gain preservance of types under rewriting. As an example we will show that the optimisation function performed after bracket abstraction is typeable. Finally we will present a type check algorithm that checks if rewrite rules are correctly typed, and finds the principal pair for typeable terms.