## The Data Field Model (2001)

Venue: | Coyne R D, Rosenman M A, Radford A D, Balachandran M and Gero J S Knowledge-based |

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Lisper01thedata,

author = {Björn Lisper and Per Hammarlund},

title = {The Data Field Model},

institution = {Coyne R D, Rosenman M A, Radford A D, Balachandran M and Gero J S Knowledge-based},

year = {2001}

}

### OpenURL

### Abstract

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 first 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 different architectures and facilitates the early design phase. The model is a generalisation of arrays, which we call data fields: these are functions with explicit information about their domains. This information can be conventional array bounds but it could also define other shapes, for instance sparse. Data fields can be interpreted as partial functions, and we define a metalanguage for partial functions. In this language we define 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 fields and their operations so they correspond closely to the more abstract notion of partial functions. We define phi-abstraction, a lambda-like syntax for defining data fields in a shape-independent manner, and prove a theorem which relates phi-abstraction and lambda-abstraction semantically. We also define a small data field language whose semantics is given by formal data fields, and give examples of data field programming for parallel algorithms with arrays and sparse structures, database quering and computing, and specification of symbolic drawings.

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Citation Context ...tly) indexed data structures. Similar to FP in spirit is the Bird-Meertens formalism (BMF). Here an algebra with unary and binary functions forms a base for a set of theories for different data types =-=[4, 5]-=-. In particular, there is a theory for functions over lists. This formalism was originally developed to support the formal calculation of programs from specifications, but it can also serve as an abst... |

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Citation Context ...mensional arrays, there are natural orderings. The situation is less clear for multidimensional arrays and sparse structures. 3 Data Fields, Informally The term "data fields" is borrowed fro=-=m Crystal [10]. Our data-=- fields are pairs (f; b), where f is a function and the "bound" b is a set representation. We denote the set defined by b by ffbgg, and the corresponding predicate by [[b]] bool . ffbgg cont... |

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Citation Context ...e all based on a single type of indexed data structure, and a purpose of our work is to develop a more generic model which is less dependent on the choice of structure. The work on Shape Polymorphism =-=[34] is a step-=- in that direction, where a category-theoretical model is used to specify shapely types and identify operations which are polymorphic over these types. The "usual" algebraic types of lists, ... |

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Citation Context ...andling of singularities, like empty arrays. This model is highly APL-specific and it is not easily generalized to other indexed structures. Another approach to arrays is taken by Fitzpatrick et. al. =-=[18]: they def-=-ine arrays as pairs of "shapes" and functions (essentially a simple instance of data fields) and consider algebraic transformations of high-level functional array algorithm specifications in... |

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Citation Context ...serve as an abstract model for explicit data parallel programming, with a cost model [60]. The systematic transformation in BMF of specifications into parallel algorithms has been studied by Gorlatch =-=[21]-=-. Formal parallelisation in BMF is achieved by transforming functions into list homomorphisms: such functions can be computed in parallel in logarithmic time. A potential weakness of BMF in this conte... |

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Citation Context ... related to the partial function view of data fields. A Mathematics of Arrays (MOA) [51] is a model where APL-like arrays are defined in this manner, and operations on arrays are formally defined. In =-=[9]-=- space-time recursion equations are studied, a kind of recursively defined partial functions which give semantics to systolic arrays. This work later developed into the language Crystal [10, 71], wher... |

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Citation Context ... resulting structure. The parallel for construct in Sisal is an example: adding X and Y elementwise for indices 1 to n can be expressed as for i in 1,n returns array of X[i]+Y[i] The FORALL statement =-=[1]-=- in HPF is very related. The exact semantics of elementwise applied operations varies, in particular when they take several arguments with different extents. For operations on lists as above, the sema... |

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Citation Context ...ed "Data Field Haskell", a dialect of Haskell where the arrays are replaced with an instance of data fields (essentially the sparse/dense arrays of Sect. 7.5), according to the framework def=-=ined here [27, 30, 29, 47]-=-. The rest of this paper is organized as follows. In Sect. 2 we give a taxonomy of collection-oriented operations and identify the major syntactical styles for these. Sect. 3 provides a first, informa... |

2 | Development of parallel algorithms in data field Haskell
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Citation Context ...ed "Data Field Haskell", a dialect of Haskell where the arrays are replaced with an instance of data fields (essentially the sparse/dense arrays of Sect. 7.5), according to the framework def=-=ined here [27, 30, 29, 47]-=-. The rest of this paper is organized as follows. In Sect. 2 we give a taxonomy of collection-oriented operations and identify the major syntactical styles for these. Sect. 3 provides a first, informa... |

1 |
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Citation Context ...appings are really set-theoretical functions over a finite domain and they correspond to sparse data fields. A formalization of the Connection Machine Lisp model was done by Boug'e and Paulin-Mohring =-=[6]-=-. These formalisms are all based on a single type of indexed data structure, and a purpose of our work is to develop a more generic model which is less dependent on the choice of structure. The work o... |

1 |
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Citation Context ...han the absolute output levels: most differences will be zero and need therefore not be sent, which greatly reduces the communication and also the arithmetics needed when updating the activities. See =-=[22]-=-. The following is an iterative data field algorithm: recall beta w taue piin einit deltat f == 39 let relax pi e s = let pinew = f e beta deltapi = sparse (pinew-pi) deltab = forall(i,j)-?deltapi!j d... |