Original Article

Formal Aspects of Computing

, Volume 25, Issue 2, pp 189-218

First online:

The data type of spatial objects

  • Kenneth JohnsonAffiliated withINRIA Rennes-Bretagne Atlantique, Campus Universitaire de BeaulieuAston University Email author 
  • , John V. TuckerAffiliated withDepartment of Computer Science, Swansea University

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A spatial object consists of data assigned to points in a space. Spatial objects, such as memory states and three dimensional graphical scenes, are diverse and ubiquitous in computing. We develop a general theory of spatial objects by modelling abstract data types of spatial objects as topological algebras of functions. One useful algebra is that of continuous functions, with operations derived from operations on space and data, and equipped with the compact-open topology. Terms are used as abstract syntax for defining spatial objects and conditional equational specifications are used for reasoning. We pose a completeness problem: Given a selection of operations on spatial objects, do the terms approximate all the spatial objects to arbitrary accuracy? We give some general methods for solving the problem and consider their application to spatial objects with real number attributes.


Abstract data types Space Spatial objects Continuous functions Compact-open topology Algebras Terms Equations Conditional equations Topological algebras Dense subalgebras Approximation by terms