The Data Field Model

Indexed data structures are prevalent in many programming applications. Collection-oriented languages provide means to operate directly on these structures , rather than having to loop or recurse through them. This style of programming will often yield clear and succinct programs. However, these programming languages will often provide only a limited choice of indexed data types and primitives, and the exact semantics of these primitives will sometimes vary with the data type and language. In this paper we develop a unifying semantical model for indexed data structures. The purpose is to support the construction of abstract data types and language features for such structures from rst principles, such that they are largely generic over many kinds of data structures. The use of these abstract data types can make programs and their semantics less dependent of the actual data structure. This makes programs more portable across diierent architectures and facilitates the early design phase. The model is a general-isation of arrays, which we call data elds: these are functions with explicit information about their domains. This information can be conventional array bounds but it could also deene other shapes, for instance sparse. Data elds can be interpreted as partial functions, and we deene a metalanguage for partial functions. In this language we deene abstract versions of collection-oriented operations, and we show a number of identities for them. This theory is used to guide the design of data elds and their operations so they correspond closely to the more abstract notion of partial functions. We deene '-abstraction, a lambda-like syntax for deening data elds in a shape-independent manner, and prove a theorem which relates '-abstraction and-abstraction semantically. We also deene a small data eld language whose semantics is given by formal data elds, and give examples of data eld programming for parallel algorithms with arrays and sparse structures, database quering and computing, and speciication of symbolic drawings.

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