Correspondence principle for idempotent calculus and some computer applications

TL;DR

This paper explores the idempotent calculus and its applications, highlighting a correspondence principle that simplifies solving nonlinear problems by translating them into linear ones over semirings, with implications for software and hardware design.

Contribution

It develops an approach to object-oriented design for algorithms based on the idempotent calculus correspondence principle, extending the theory and applications of idempotent semirings.

Findings

Nonlinear problems like the Bellman equation become linear over suitable semirings.

The theory includes new integration, linear algebra, spectral theory, and functional analysis.

Applications span computer algorithms, software, and hardware design.

Abstract

This paper is devoted to heuristic aspects of the so-called idempotent calculus. There is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semirings in the spirit of N. Bohr's correspondence principle in Quantum Mechanics. Some problems nonlinear in the traditional sense (for example, the Bellman equation and its generalizations) turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions. The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis. It has a wide range of applications. Besides a survey of the subject, in this paper the correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus.