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Injective and pseudo-injective polynomial equations: From permutations to dynamical systems

Antonio E. Porreca, Marius Rolland

Abstract

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively), giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. These algorithms exploit the notion of unroll of a FDDS, an alternative representation based on a forest of infinite trees constructed by computing the transition function of the system backwards. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$ and some generalisations. These results are some of the important final steps for solving more general polynomial equations on FDDS.

Injective and pseudo-injective polynomial equations: From permutations to dynamical systems

Abstract

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively), giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing division and computing -th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. These algorithms exploit the notion of unroll of a FDDS, an alternative representation based on a forest of infinite trees constructed by computing the transition function of the system backwards. This ultimately leads to an efficient solution to equations of the type for connected and some generalisations. These results are some of the important final steps for solving more general polynomial equations on FDDS.

Paper Structure

This paper contains 11 sections, 45 theorems, 43 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomials over the permutations
  4. Roots over the permutations
  5. Sufficient condition for the injectivity of polynomials over the permutations
  6. Necessary condition for the injectivity of polynomials over FDDSs
  7. Pseudo-injective polynomials over the permutations
  8. Equations with cycles encoded compactly
  9. Equations over the unrolls
  10. Beyond permutations
  11. Conclusions

Key Result

Lemma 1

$\Lambda_{p} \colon \mathbb{D} \to \mathbb{D}$ is a semiring endomorphism. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Product of two FDDSs $A$ and $B$, where the states are given a temporary label in order to show how the result is computed (for brevity, the vertices $(u,v)$ of the product are denoted here by $uv$). Remark that $B$ is cancelable (state $6$ is a fixed point) and $AB$ is pseudo-cancelable ($L(AB) = \{2,4\}$ and $\ell(AB) = \min (L(AB)) = 2$). $A$ is also trivially pseudo-cancelable, since it is connected ($L(A) = \{2\}$ and $\ell(A)=2$).
  • Figure 2: Example of a run of \ref{['alg:root-permutation']} on inputs $B = 2C_{2} + 12C_{3} + 26 C_{6}$ and $k =2$.
  • Figure 3: Table of values of the sequences $\alpha, \beta$ and $\delta$ for the subsets of $\{2,3,6\}$. The column with head $I$ gives the subsets, and the columns of head $\alpha_I, \delta_I$ and $\beta_I$ give the value of the respective sequence for $I$.
  • Figure 4: A run of \ref{['algo:poly_cycle']} over equation $C_2X^2 + (C_4+C_6)X = 16C_2+4C_4+18C_6+C_{12}$. Each row of the table is an iteration of the loop This run returns the solution $X = 4C_1 + C_3$. Remark that this equation also admits $2C_2+C_3$ and $2C_1+C_2+C_3$ as solution, but \ref{['algo:poly_cycle']} only returns the solution which maximizes the number of connected component.
  • Figure 5: Table of values of the variables of Algorithm \ref{['algo:aLcm']} during the calculation of $\mathop{\mathrm{alcm}}\nolimits_{a}b$ with $b = 43008$ and $a = 3584$. Each row corresponds to one iteration of the loop.
  • ...and 1 more figures

Theorems & Definitions (88)

  • Lemma 1
  • proof
  • Definition 2: $\le_{\mathrm{ct}}$ for permutations
  • Definition 4: Prefix and super-prefix
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • proof
  • ...and 78 more