∂ for Data: Differentiating Data Structures

Authors:
Michael Abbott;Thorsten Altenkirch;Conor Mcbride;Neil Ghani
Affiliations:
Diamond Light Source, Rutherford Appleton Laboratory, UK. michael@araneidae.co.uk;(Correspd.) School of Computer Science & IT, The University of Nottingham, UK. txa@cs.nott.ac.uk/ ctm@cs.nott.ac.uk;School of Computer Science & IT, The University of Nottingham, UK. txa@cs.nott.ac.uk/ ctm@cs.nott.ac.uk;Department of Mathematics and Computer Science, University of Leicester, UK. ng13@mcs.le.ac.uk
Venue:
Fundamenta Informaticae - Typed Lambda Calculi and Applications 2003, Selected Papers
Year:
2005

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Abstract

This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its one-hole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of a data type, which leads to the notion of a container, introduced in (Abbott, Altenkirch, and Ghani, 2003a) and investigated extensively in (Abbott, 2003). Using containers we can provide a notion of linear map which is the concept missing from McBride's first analysis. We verify the usual laws of differential calculus including the chain rule and establish laws for initial algebras and terminal coalgebras.