On Typed Calculi with a Merge Operator

Extensions of the simply typed lambda calculus with a merge operator are de ned and investigated. A merge of functions denotes a function which can be applied to more than one type and computes the result by merging the results of the component functions. We propose rst two calculi in which the inconsistency between the return values of the component functions is solved using a non-symmetric rule, and show that they do not have good proof-theoretic properties: one does not have the Strong Normalization Property, and the other does not have the Transitivity of Coercions Property. Then, using the type system, we restrict the two calculi in a way that terms which may cause inconsistency are excluded, and show that the two calculi become equivalent and have all the desired properties. A merge of functions in object oriented programming corresponds to writing a method in a subclass by modifying those in superclasses, and the above restriction corresponds to only allowing adding some behaviors in the modi cation. This result conforms with the programming methodology in which one should write a subclass method so that it does not override the behavior of superclass methods.