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On solving basic equations over the semiring of functional digraphs

Alberto Dennunzio, Enrico Formenti, Luciano Margara, Sara Riva

TL;DR

This work studies solving basic equations over the semiring of functional digraphs, focusing on the permutation case where equations have the form $C_p\cdot X = nC_q$. It introduces the $C_p$ notation and a product rule for unions of cycles, and proves a polynomial-time decision procedure for the basic equation, requiring that $p\mid q$ and that a computed parameter $e$ divides $n$; the computation of $e$ uses arithmetic subroutines $\Pi_F$ and $\Pi_E$ with overall time $O(s^3)$ in the input size. The paper further presents special-case quadratic-time criteria and discusses the implications for factorization and cancellation problems, as well as practical applications to software pipelines for solving polynomial equations on functional digraphs. It also outlines potential extensions to larger classes of functional digraphs and discusses how these results can impact related areas in biology and cryptography through efficient equation solving on graph-based dynamical systems.

Abstract

Endowing the set of functional graphs (FGs) with the sum (disjoint union of graphs) and product (standard direct product on graphs) operations induces on FGs a structure of a commutative semiring R. The operations on R can be naturally extended to the set of univariate polynomials R[X] over R. This paper provides a polynomial time algorithm for deciding if equations of the type AX=B have solutions when A is just a single cycle and B a set of cycles of identical size. We also prove a similar complexity result for some variants of the previous equation.

On solving basic equations over the semiring of functional digraphs

TL;DR

This work studies solving basic equations over the semiring of functional digraphs, focusing on the permutation case where equations have the form . It introduces the notation and a product rule for unions of cycles, and proves a polynomial-time decision procedure for the basic equation, requiring that and that a computed parameter divides ; the computation of uses arithmetic subroutines and with overall time in the input size. The paper further presents special-case quadratic-time criteria and discusses the implications for factorization and cancellation problems, as well as practical applications to software pipelines for solving polynomial equations on functional digraphs. It also outlines potential extensions to larger classes of functional digraphs and discusses how these results can impact related areas in biology and cryptography through efficient equation solving on graph-based dynamical systems.

Abstract

Endowing the set of functional graphs (FGs) with the sum (disjoint union of graphs) and product (standard direct product on graphs) operations induces on FGs a structure of a commutative semiring R. The operations on R can be naturally extended to the set of univariate polynomials R[X] over R. This paper provides a polynomial time algorithm for deciding if equations of the type AX=B have solutions when A is just a single cycle and B a set of cycles of identical size. We also prove a similar complexity result for some variants of the previous equation.
Paper Structure (5 sections, 10 theorems, 35 equations, 3 figures, 3 algorithms)

This paper contains 5 sections, 10 theorems, 35 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and basic facts
  3. Solving basic equations on permutations
  4. Some special cases
  5. Conclusions

Key Result

Theorem 1

The class of FDs (modulo isomorphisms) equipped with the operations of $+$ and $\cdot$ is a commutative semiring.

Figures (3)

  • Figure 1: An example of polynomial equation over FDs (above) and a solution (below).
  • Figure 2: Example of operations on FDs of permutations. Using the C-notation, we have $\color{green}C_2\color{black}\cdot(\color{red}C_3\color{black}+\color{blue}C_2)\color{black}=\color{green}C_2\color{black}\cdot\color{red}C_3\color{black}+\color{green}C_2\color{black}\cdot\color{blue}C_2\color{black}=C_6+2C_2$.
  • Figure 3: Example of functional digraph having multiple factorizations. Using the C-notation, we have $\color{orange}C_2\color{black}\cdot\color{orange}C_2\color{black}=\color{black}(\color{blue}C_1\color{black}+\color{blue}C_1\color{black})\cdot\color{orange}C_2\color{black}=2C_2$.

Theorems & Definitions (21)

  • Definition 1: Sum of FDs
  • Definition 2: Product of FDs
  • Theorem 1: dorigatti2018
  • Theorem 2: dorigatti2018
  • Proposition 1
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • ...and 11 more