Shapely types separate data, represented by lists, from shape, or structure. This separation supports shape polymorphism, where operations are defined for arbitrary shapes, and shapely operations, for which the shape of the result is determined by that of the input, permitting static shape checking. The shapely types are closed under the formation of fixpoints, and hence include the usual algebraic types of lists, trees, etc. They also include other standard data structures such as arrays, graphs and records. 1 Introduction The values of a shapely type are uniquely determined by their shape and their data. The shape can be thought of as a structure with holes or positions, into which data elements (stored in a list) can be inserted. The use of shape in computing is widespread, but till now it has not, apparently, been the subject of independent study. The body of the paper presents a semantics for shape, based on elementary ideas from category theory. First, let us consider some examp...

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This chapter describes the design of a prototype knowledge-based system for crystal and molecular structure determination from diffraction data. This system enhances current methods for the determination and interpretation of protein structures by incorporating direct methods probabilistic strategies,

Abstract. High-level array processing is characterized by the composition of generic operations, which treat all array elements in a uniform way. This paper proposes a mechanism that allows programmers to direct effects of such array operations to non-scalar subarrays of argument arrays without sacrificing the high-level programming approach. A versatile notation for axis control is presented, and it is shown how the additional language constructs can be transformed into regular SaC code. Furthermore, an optimization technique is introduced which achieves the same runtime performance regardless of whether code is written using the new notation or in a substantially less elegant style employing conventional language features. 1

Model-based reasoning involves proving the truth of a proposition by computation in the semantic domain. In contrast, rule-based reasoning is proving truth by means of formal manipulation of formulas. A growing body of research in cognitive science suggests that human spatial reasoning is model-based, rather than rule-based. The paper begins with a cognitive perspective of model-based reasoning. A semantic domain for spatial reasoning, based on a theory of symbolic arrays, is defined. A modal logic of spatial assertions for reasoning in indeterminate worlds is then presented, along with possible extensions that address structural hierarchy, temporal modalities, multiple views and analogy. N CG1 CG2 CB CA C O Figure 1: Spatial model of a molecular structure 1 Introduction Psychologists have acknowledged that mental models are fundamental to human problem solving, particularly for their predictive and explanatory power in understanding human interactions with the environment and with...

In this paper we propose a new means to model and operate on nested arrays that allows for a high level of abstraction without introducing a performance penalty. We achieve this by using a nesting structure on array types which allows us to shift the nesting information of arrays from the runtime representation level to the type system level. This information can then be exploited for generic function definitions on the nesting structure of arrays which, as we show, neatly integrates with subtyping based function overloading. Finally, we demonstrate for an example how nested arrays and generic function definitions can be fully stripped out using existing optimisation techniques. 1