A Symbolic Computing Perspective on Software Systems

2026
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Authors: Arthur C. Norman; Stephen M. Watt
Published in: Languages, Compilers, Analysis - From Beautiful Theory to Useful Practice
Publisher: Springer Nature Switzerland

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Abstract

This chapter explores the evolution of software systems through the lens of symbolic computing, celebrating the work of Alan Mycroft. It delves into the intricate world of low-level code optimization and high-level abstraction, focusing on the task of long division of multi-precision integers as a case study. The chapter highlights the interplay between detailed code constructions and global analysis, emphasizing the importance of both low-level hacking and high-level mathematical abstraction. It also discusses the impact of compiler optimizations over the past 30 years, making it possible to express code in a cleaner way. The chapter concludes with a detailed analysis of the performance of different division algorithms, providing insights into the complexities and challenges of achieving high performance in symbolic computation systems.

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Abstract

Symbolic mathematical computing systems have served as a canary in the coal mine of software systems for more than sixty years. They have introduced or have been early adopters of programming language ideas such as dynamic memory management, arbitrary precision arithmetic and dependent types. These systems have the property of being highly complex while at the same time operating in a domain where results are well-defined and clearly verifiable. These software systems span multiple layers of abstraction with concerns ranging from instruction scheduling and cache pressure up to algorithmic complexity of constructions in algebraic geometry. All of the major symbolic mathematical computing systems include low-level code for arbitrary precision arithmetic, memory management and other primitives, a compiler or interpreter for a bespoke programming language, a library of high-level mathematical algorithms, and some form of user interface. Each of these parts invokes multiple deep issues. We present some lessons learned from this environment and free flowing opinions on topics including:

Portability of software across architectures and decades;
Infrastructure to embrace and infrastructure to avoid;
Choosing base abstractions upon which to build;
How to get the most out of a small code base;
How developments in compilers both to optimise and to validate code have always been and remain of critical importance, with plenty of remaining challenges;
The way in which individuals including in particular Alan Mycroft who has been able to span from hand-crafting Z80 machine code up to the most abstruse high-level code analysis techniques are needed, and
Why it is important to teach full-stack thinking to the next generation.

The key message is that the real world is often messier than presentation in papers and we need to be able to cross between very low and very high levels of abstraction to deal with this, as Alan Mycroft has done in his work.

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Title: A Symbolic Computing Perspective on Software Systems
Authors: Arthur C. Norman
Stephen M. Watt
Publisher: Springer Nature Switzerland
Book: Languages, Compilers, Analysis - From Beautiful Theory to Useful Practice
Print ISBN: 978-3-032-08186-5

Electronic ISBN: 978-3-032-08187-2

Copyright Year: 2026
DOI: https://doi.org/10.1007/978-3-032-08187-2_3

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