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Idempotent mathematics and interval analysis

Grigori Litvinov, Viktor Maslov, Andrei Sobolevskii

TL;DR

This work develops Idempotent Mathematics as a calculus over idempotent semirings, illustrating how optimization, control, and Hamilton–Jacobi problems become linear under idempotent algebra. It formalizes the dequantization (or correspondence) principle that links traditional and idempotent frameworks, and constructs both weak and strong interval extensions to carry imprecise data through idempotent computations. The text establishes foundational algebraic structures, introduces set-valued and interval arithmetic, and demonstrates how idempotent linear algebra, including Bellman equations and spectral theory, can be analyzed and computed with convergence guarantees. These developments yield a robust, parallelizable toolkit for solving order-preserving optimization problems and related linear-algebraic equations in a unified semiring setting.

Abstract

A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.

Idempotent mathematics and interval analysis

TL;DR

This work develops Idempotent Mathematics as a calculus over idempotent semirings, illustrating how optimization, control, and Hamilton–Jacobi problems become linear under idempotent algebra. It formalizes the dequantization (or correspondence) principle that links traditional and idempotent frameworks, and constructs both weak and strong interval extensions to carry imprecise data through idempotent computations. The text establishes foundational algebraic structures, introduces set-valued and interval arithmetic, and demonstrates how idempotent linear algebra, including Bellman equations and spectral theory, can be analyzed and computed with convergence guarantees. These developments yield a robust, parallelizable toolkit for solving order-preserving optimization problems and related linear-algebraic equations in a unified semiring setting.

Abstract

A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.

Paper Structure

This paper contains 12 sections, 15 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Dequantization and idempotent correspondence principle
  3. Idempotent semirings: Basic definitions
  4. Idempotency and linearity
  5. Idempotent superposition principle
  6. Idempotent matrix theory: Some results
  7. Correspondence principle for computations
  8. Set-valued idempotent arithmetic
  9. Weak interval extension of an idempotent semiring
  10. A stronger notion of interval extension
  11. Cancellation and semifields
  12. Application to linear algebra

Key Result

Theorem 1

Let $A$ be a semi-definite matrix. Then for $k \geqslant n - 1$, so the closure matrix $A^* = \bigoplus_{k = 0}^\infty A^k$ exists and is equal to $\bigoplus_{k = 0}^{n - 1} A^k$.

Theorems & Definitions (15)

  • Theorem 1: Carré
  • Theorem 2: Dudnikov, Samborskiı
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Theorem 3
  • ...and 5 more