Space – time programming Jacob Beal 1 and

and call functions Figure 5. Field calculus is a minimal syntax and semantics for construction of aggregate programs with a coherent mapping from aggregate syntax to local syntax. (Online version in colour.) 4. Foundation: field calculus Field calculus [19] provides a theoretical foundation for aggregate programming, grounding on the idea of computing by functional manipulation of computational fields. Its aim is to provide a universal model that is terse enough for mathematical proofs of general properties about aggregate programming and the aggregate/local relationship, just as λ-calculus [36] provides for functional programming, π -calculus for parallel programming [37] or Featherweight Java [38] for object-oriented programming. (a) Syntax and semantics of field calculus Field calculus [19] is a minimal expressive model for computation with fields, developed by refining the set of key operations in the wide range of domain-specific languages for space–time computation surveyed in [1] to find a small covering set. From this survey, it was decided to base field calculus on Proto [34], which is one of the few current general purpose space–time programming languages and which has quite general capabilities. Proto, however, has far too complicated a syntax and semantics (as developed in [39]) and is not practical for mathematical analysis. Field calculus was thus developed by finding a minimal set of operations that, with appropriate syntactic sugar, could cover all of Proto as well as certain capabilities not supported by Proto but supported by other languages. As might be expected from its name, the value of any field calculus expression is a computational field φ, mapping from a potentially time-varying domain of devices to a value for each device at each point in time. From this very consideration, one derives that such an aggregate-level interpretation, which we shall follow in the remainder of this paper, is an improvement and extension over the classical pointwise interpretation of distributed system programming, in which the focus is still on the single device (in a given position of space) computing a single value (at a given time). Fields are computed and manipulated according to the functional (Lisp-like) syntax shown in figure 5 (overbar notation denotes sequences, e.g. ē stands for e1 . . .en with n ≥ 0). Fields may either be literal data values (e.g. 5, ‘username’) representing constant-valued fields over their domain, or else expressions that combine fields using five types of operation. The first two of these are generic, appearing in any functional programming language, while the other three (illustrated in figure 6) are special constructs for support of aggregate programming: — Built-in operators are a collection of primitive operations (symbol b) taking in zero or more input fields and functionally computing an output field. Most specifically, the value of the resulting field at a device is the result of applying the built-in operator to the inputs at the same device. For example, multiplication of two numerical fields φ1 and φ2 yields a field φ mapping each device in space–time to v1 ∗ v2, where v1 and v2 are the values to which φ1 on September 15, 2017 http://rsta.royalsocietypublishing.org/ Downloaded from