Subtyping With Antimirov’s Algorithm

I start with an explanation of the notion of ‘subtyping’ to make clear what it means to subtype a type r against a type s. It will be pointed out, how types are represented in the subtyping algorithm. Then I will explain the main idea of the subtyping algorithm and make clear what subconcepts are needed. In detail, those are the concepts of checking for -inclusion, leading names and partial derivatives of regular expressions and regular inequalities. These are treated in four separate sections. In the course of these sections, I will introduce small running examples. In section 8, I will construct a big example for a complete recursive application of the algorithm. The description will be closed by a short conclusion. 1 What Does ‘Subtyping’ Mean? A typed language like XQuery defines built-in types and allows the definition of user defined types. The compilation of a piece of program code written in a typed language therefore requires some kind of check, whether the use of the declared instances and variables in this code obeys the type rules. The two tasks connected with this requirement are type inference and type checking. Type inference means deriving the type of a simple or composite expression e, which requires the application of type rules on e. Type checking means to check whether the use of the declared instances is coherent with the type rules in any explicit or implicit assignment. In the following, I will concentrate on a single but central task concerning type checking, the so called subtyping. Subtyping means to check for two types r and s whether the set of instances matched by r is a subset of the set of instances matched by s. A subtyping facility is an important feature because the translation process from XQuery to the XQuery core language makes excessive use of typeswitch commands. Without any subtyping mechanism, we would not be able to implement efficient typechecking. Furthermore, in any language, where explicit typechecking is implemented (e.g. represented by the typeswitch command), an implementation of subtyping must be available, too. 2 Type and Type Instance Representation Since types are represented by regular expressions, the quite hand-waving definition above becomes more clear: The types r and s are represented by regular expressions, so both r and s define languages as sets of sentences. We denote the language defined by a regular expression p as L(p), where L(p) is a set of sentences. In the context of subtyping, these statements are instances of types. For XQuery this means the sentences defined by L(p) are XML tree structures. We can now formulate our first