Reasoning About Array Structures Using a Dependently Typed Logic

Abstract

This paper explores the use of a dependently-typed design logic for describing and reasoning about systolic arrays. Taking triangular arrays as an example, it is shown how they may usefully be defined in various equivalent ways and how their behaviour may be related to that of the corresponding one-dimensional systolic arrays. This allows complex proofs of the properties of such (dynamic) arrays to be replaced with simpler proofs of the properties of (static) dependency graphs. As an example, the correctness of a systolic sorter is formally established. Overall it is shown that the use of dependent types for describing systolic arrays contributes significantly to the clarity and generality of their specifications.