A novel way to build expert systems with infinite-valued attributes

Antonio Hernando; José Luis Galán-García; Gabriel Aguilera-Venegas; Antonio Hernando; José Luis Galán-García; Gabriel Aguilera-Venegas

doi:10.3934/math.2024145

AIMS Mathematics

2024, Volume 9, Issue 2: 2938-2963. doi: 10.3934/math.2024145

Previous Article Next Article

Research article Special Issues

A novel way to build expert systems with infinite-valued attributes

1.
Depto. de Sistemas Informáticos, E.T.S.I. de Sistemas Informáticos, Universidad Politécnica de Madrid, Madrid, Spain
2.
Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Málaga, Málaga, Spain

Received: 20 October 2023 Revised: 19 December 2023 Accepted: 27 December 2023 Published: 02 January 2024

An expert system is a computer program that uses the knowledge of an expert to solve problems in a specific domain. Expert systems are used in a wide variety of fields, such as medicine, financial diagnosis and engineering. The attributes of an expert system are the characteristics of the problems that the system can solve. In traditional expert systems, attributes typically have a finite number of possible values. However, in scenarios where an attribute can assume a value from an infinite (or significantly large finite) set, the expert system cannot be represented using propositional logic. Until now, no method had been identified to implement such a system on a Computer Algebra System. Here, we break new ground by presenting a model that not only addresses this gap but also provides a fresh perspective on previous results. In fact, these prior results can be viewed as specific instances within the broader framework of our proposed solution. In this paper, we put forth an algebraic approach for the development of expert systems capable of handling attributes with infinite values, thereby expanding the problem-solving capacity of these systems.

Keywords:

Citation: Antonio Hernando, José Luis Galán-García, Gabriel Aguilera-Venegas. A novel way to build expert systems with infinite-valued attributes[J]. AIMS Mathematics, 2024, 9(2): 2938-2963. doi: 10.3934/math.2024145

Related Papers:

[1]	Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad . On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172
[2]	Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad . New results on a coupled system for second-order pantograph equations with $\mathcal{ABC}$ fractional derivatives. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071
[3]	Abdelkader Moumen, Hamid Boulares, Tariq Alraqad, Hicham Saber, Ekram E. Ali . Newly existence of solutions for pantograph a semipositone in $\Psi$ -Caputo sense. AIMS Mathematics, 2023, 8(6): 12830-12840. doi: 10.3934/math.2023646
[4]	Ayub Samadi, Chaiyod Kamthorncharoen, Sotiris K. Ntouyas, Jessada Tariboon . Mixed Erdélyi-Kober and Caputo fractional differential equations with nonlocal non-separated boundary conditions. AIMS Mathematics, 2024, 9(11): 32904-32920. doi: 10.3934/math.20241574
[5]	Hui Huang, Kaihong Zhao, Xiuduo Liu . On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses. AIMS Mathematics, 2022, 7(10): 19221-19236. doi: 10.3934/math.20221055
[6]	Cuiying Li, Rui Wu, Ranzhuo Ma . Existence of solutions for Caputo fractional iterative equations under several boundary value conditions. AIMS Mathematics, 2023, 8(1): 317-339. doi: 10.3934/math.2023015
[7]	Mohamed Houas, Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad . Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 2023, 8(3): 5216-5232. doi: 10.3934/math.2023262
[8]	Ahmed M. A. El-Sayed, Wagdy G. El-Sayed, Kheria M. O. Msaik, Hanaa R. Ebead . Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral equations. AIMS Mathematics, 2025, 10(3): 4970-4991. doi: 10.3934/math.2025228
[9]	Isra Al-Shbeil, Abdelkader Benali, Houari Bouzid, Najla Aloraini . Existence of solutions for multi-point nonlinear differential system equations of fractional orders with integral boundary conditions. AIMS Mathematics, 2022, 7(10): 18142-18157. doi: 10.3934/math.2022998
[10]	Yujun Cui, Chunyu Liang, Yumei Zou . Existence and uniqueness of solutions for a class of fractional differential equation with lower-order derivative dependence. AIMS Mathematics, 2025, 10(2): 3797-3818. doi: 10.3934/math.2025176

Abstract

1. Introduction

The key to solving the general quadratic congruence equation is to solve the equation of the form $x^2\equiv a\mod p$ , where $a$ and $p$ are integers, $p > 0$ and $p$ is not divisible by $a$ . For relatively large $p$ , it is impractical to use the Euler criterion to distinguish whether the integer $a$ with $(a, p) = 1$ is quadratic residue of modulo $p$ . In order to study this issue, Legendre has proposed a new tool-Legendre's symbol.

Let $p$ be an odd prime, the quadratic character modulo $p$ is called the Legendre's symbol, which is defined as follows:

$\left(\frac{a}{p}\right)= \begin{cases}1, & \text { if } a \text { is a quadratic residue modulo } p; \\ -1, & \text { if } a \text { is a quadratic non-residue modulo } p ;\\ 0, & \text { if } p \mid a.\end{cases}$

The Legendre's symbol makes it easy for us to calculate the level of quadratic residues. The basic properties of Legendre's symbol can be found in any book on elementary number theory, such as ^[1,2,3].

The properties of Legendre's symbol and quadratic residues play an important role in number theory. Many scholars have studied them and achieved some important results. For examples, see the ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]}.

One of the most representative properties of the Legendre's symbol is the quadratic reciprocal law:

Let $p$ and $q$ be two distinct odd primes. Then, (see Theorem 9.8 in ^[1] or Theorems 4–6 in ^[3])

$\left(\frac{p}{q}\right)\cdot\left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}.$

For any odd prime $p$ with $p\equiv1\; \rm mod\; 4$ there exist two non-zero integers $\alpha\left(p\right)$ and $\beta\left(p\right)$ such that

$\begin{align*} p = \alpha^2\left(p\right)+\beta^2\left(p\right). \end{align*}$

(1)

In fact, the integers $\alpha\left(p\right)$ and $\beta\left(p\right)$ in the (1) can be expressed in terms of Legendre's symbol modulo $p$ (see Theorems 4–11 in ^[3])

$\alpha\left(p\right) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a^3+a}{p}\right)\; \; \; \rm and\; \; \; \it \beta\left(\it p\right) = \frac{\rm1}{\rm 2}\sum\limits_{a = \rm1}^{p-\rm1}\left(\frac {\it a^{\rm3}+\it ra}{\it p}\right),$

where $r$ is any integer, and $\left(r, p\right) = 1$ , $\left(\frac{r}{p}\right) = -1$ , $\left(\frac{\ast}{p}\right) = \chi_2$ denote the Legendre's symbol modulo $p$ .

Noting that Legendre's symbol is a special kind of character. For research on character, Han ^[7] studied the sum of a special character $\chi\left(ma+\bar{a}\right)$ , for any integer $m$ with $\left(m, p\right) = 1$ , then

$\left|\sum\limits_{a = 1}^{p-1}\chi\left(ma+\bar{a}\right)\right|^2 = 2p+\left(\frac{m}{p}\right)\sum\limits_{a = 1}^{p-1}\chi(a)\sum\limits_{b = 1}^{p-1}\left(\frac{b(b-1)(a^2b-1)}{p}\right),$

which is a special case of a general polynomial character sums $\sum_{a = N+1}^{N+M}\chi\left(f\left(a\right)\right)$ , where $M$ and $N$ are any positive integers, and $f\left(x\right)$ is a polynomial.

In ^[8], Du and Li introduced a special character sums $C\left(\chi, m, n, c;p\right)$ in the following form:

$C\left(\chi, m, n, c;p\right) = \sum\limits_{a = 0}^{p-1}\sum\limits_{b = 0}^{p-1}\chi\left(a^2+na-b^2-nb+c\right)\cdot e\left(\frac{mb^2-ma^2}{p}\right),$

and studied the asymptotic properties of it. They obtained

$\begin{equation*} \label{test1} \begin{split} \sum\limits_{c = 1}^{p-1}\left|C\left(\chi, m, n, c;p\right)\right|^{2k} = \left\{ \begin{array}{l} &p^{2k+1}+\frac{k^2-3k-2}{2}\cdot p^{2k}+O\left(p^{2k-1}\right), \; \; \; \; \rm if\; \chi\ \rm is\ \rm the\ \rm Legendre\ \rm symbol\ \rm {modulo} \ \it p \rm; \\ &p^{2k+1}+\frac{k^2-3k-2}{2}\cdot p^{2k}+O\left(p^{2k-1/2}\right), \; \; \rm if\; \chi\ \rm is\ \rm a\ \rm complex\ \rm character\ \rm {modulo} \ \it p.\\ \end{array} \right. \end{split} \end{equation*}$

Recently, Yuan and Zhang ^[12] researched the question about the estimation of the mean value of high-powers for a special character sum modulo a prime, let $p$ be an odd prime with $p\equiv1\; \rm mod\; 6$ , then for any integer $k\geq0$ , they have the identity

$S_k\left(p\right) = \frac{1}{3}\cdot\left[d^k+\left(\frac{-d+9b}{2}\right)^k+\left(\frac{-d-9b}{2}\right)^k\right],$

where

$S_k\left(p\right) = \frac{1}{p-1}\sum\limits_{r = 1}^{p-1}A^k\left(r\right),$

$A\left(r\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+r\bar{a}}{p}\right),$

and for any integer $r$ with $(r, p) = 1$ .

More relevant research on special character sums will not be repeated. Inspired by these papers, we have the question: If we replace the special character sums with Legendre's symbol, can we get good results on $p\equiv1\; \rm mod\; 4$ ?

We will convert $\beta\left(p\right)$ to another form based on the properties of complete residues

$\beta\left(p\right) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a+n\bar{a}}{p}\right),$

where $\bar{a}$ is the inverse of $a$ modulo $p$ . That is, $\bar{a}$ satisfy the equation $x\cdot a\; \equiv1\mod p$ for any integer $a$ with $(a, p) = 1$ .

For any integer $k\geq0$ , $G\left(n\right)$ and $K_k(p)$ are defined as follows:

$G\left(n\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\; \; \; \rm {and} \; \; \; \it K_{\it k}\left(\it p\right) = \frac{\rm 1}{\it p-\rm 1}\sum\limits_{\it n\rm = 1}^{\it p-\rm 1}\it G^{\it k}\left(\it n\right).$

In this paper, we will use the analytic methods and properties of the classical Gauss sums and Dirichlet character sums to study the computational problem of $K_k\left(p\right)$ for any positive integer $k$ , and give a linear recurrence formulas for $K_k\left(p\right)$ . That is, we will prove the following result.

Theorem 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ , then we have

$K_k\left(p\right) = \left(4p+2\right)\cdot K_{k-2}\left(p\right)-8\left(2\alpha^2-p\right)\cdot K_{k-3}\left(p\right)+\left(16\alpha^4-16p\alpha^2+4p-1\right)\cdot K_{k-4}\left(p\right),$

for all integer $k\geq4$ with

$K_0\left(p\right) = 1, \; K_1\left(p\right) = 0, \; K_2\left(p\right) = 2p+1, \; K_3\left(p\right) = -3(4\alpha^2-2p),$

where

$\alpha = \alpha\left(p\right) = \sum\limits_{a = 1}^{\frac{p-1}{2}}\left(\frac{a+\bar{a}}{p}\right).$

Applying the properties of the linear recurrence sequence, we may immediately deduce the following corollaries.

Corollary 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\frac{1}{1+\sum _{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)} = \frac{16\alpha^2p-28\alpha^2-8p^2+14p} {16\alpha^4-16\alpha^2p+4p-1}.$

Corollary 2. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\begin{align*} \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\sum\limits_{m = 0}^{p-1}\left(1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right)\cdot e\left(\frac{nm^2}{p}\right) = -\sqrt{p}. \end{align*}$

Corollary 3. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then we have

$\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\sum\limits_{m = 0}^{p-1}\left[1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right]^2\cdot e\left(\frac{nm^2}{p}\right) = \left(4\alpha^2-2p\right)\cdot\sqrt{p}.$

Corollary 4. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 8$ . Then we have

$\begin{align*} \sum\limits_{n = 1}^{p-1}\left(1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right)\right)\cdot \sum\limits_{m = 0}^{p-1}e\left(\frac{nm^4}{p}\right) = \sqrt{p}\left(-1+B(1)\right)-p, \end{align*}$

where

$B(1) = \sum\limits_{m = 0}^{p-1}e\left(\frac{m^4}{p}\right).$

If we consider such a sequence $F_k\left(p\right)$ as follows: Let $p$ be a prime with $p\equiv1\; \rm mod\; 8$ , $\chi_4$ be any fourth-order character modulo $p$ . For any integer $k\geq0$ , we define the $F_k\left(p\right)$ as

$F_k\left(p\right) = \sum\limits_{n = 1}^{p-1}\frac{1}{G^{k}(n)},$

we have

$\begin{align*} F_{k}\left(p\right) = &\frac{1}{16\alpha^4-16\alpha^2p+4p-1}F_{k-4}\left(p\right)-\frac{(4p+2)}{16\alpha^4-16\alpha^2p+4p-1}F_{k-2}\left(p\right)\\ &+\frac{4(4\alpha^2-2p)}{16\alpha^4-16\alpha^2p+4p-1}F_{k-1}\left(p\right). \end{align*}$

2. Some lemmas

Lemma 1. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then for any fourth-order character $\chi_4 \; \rm{mod} \; \it p$ , we have the identity

$\tau^2(\chi_4)+\tau^2(\bar{\chi_4}) = 2\sqrt{p}\cdot\alpha,$

where

$\tau(\chi_4) = \sum\limits_{a = 1}^{p-1}\chi_4(a)e\left(\frac{a}{p}\right)$

denotes the classical Gauss sums, $e(y) = e^{2\pi iy}, i^2 = -1$ , and $\alpha$ is the same as in the Theorem 1.

Proof. See Lemma 2.2 in ^[9].

Lemma 2. Let $p$ be an odd prime. Then for any non-principal character $\psi$ modulo $p$ , we have the identity

$\tau(\psi^2) = \frac{\psi^2(2)}{\tau(\chi_2)}\cdot\tau(\psi)\cdot\tau(\psi\chi_2),$

where $\chi_2 = \left(\frac{\ast}{p}\right)$ denotes the Legendre's symbol modulo $p$ .

Proof. See Lemma 2 in ^[12].

Lemma 3. Let $p$ be a prime with $p\equiv 1\; \rm mod\; 4$ , then for any integer $n$ with $\left(n, p\right) = 1$ and fourth-order character $\chi_{4}\; \rm mod \; \it p$ , we have the identity

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})).$

Proof. For any integer $a$ with $(a, p) = 1$ , we have the identity

$1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a) = 4,$

if $a$ satisfies $a\equiv b^4\; \rm mod \; \it p$ for some integer $b$ with $(b, p) = 1$ and

$1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a) = 0,$

otherwise. So from these and the properties of Gauss sums we have

$\begin{align*} \mathop\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = &\sum\limits_{a = 1}^{p-1}\left(\frac{a^2}{p}\right)\left(\frac{a^4+n}{p}\right)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2\left(a^4\right)\chi_2\left(a^4+n\right)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}(1+\chi_{4}(a)+\chi_2(a)+\bar{\chi_{4}}(a))\cdot\chi_2(a)\cdot\chi_2(a+n)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}(1+\chi_{4}(na)+\chi_2(na)+\bar{\chi_{4}}(na))\cdot\chi_2(na)\cdot\chi_2(na+n)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2(a)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\chi_{4}(na)\chi_2(a)\chi_2(a+1) \end{align*}$

(2)

$\begin{align*} &+\sum\limits_{a = 1}^{p-1}\chi_2(na)\chi_2(a)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(na)\chi_2(a)\chi_2(a+1)\nonumber\\ = &\sum\limits_{a = 1}^{p-1}\chi_2(1+\bar{a})+\sum\limits_{a = 1}^{p-1}\chi_{4}(na)\chi_2(a)\chi_2(a+1)\nonumber\\ &+\sum\limits_{a = 1}^{p-1}\chi_2(n)\chi_2(a+1)+\sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(na)\chi_2(a)\chi_2(a+1)\nonumber. \end{align*}$

Noting that for any non-principal character $\chi$ ,

$\sum\limits_{a = 1}^{p-1}\chi(a) = 0$

and

$\sum\limits_{a = 1}^{p-1}\chi(a)\chi(a+1) = \frac{1}{\tau(\bar{\chi})}\sum\limits_{b = 1}^{p-1}\sum\limits_{a = 1}^{p-1}\bar{\chi}(b)\chi(a)e\left(\frac{b(a+1)}{p}\right).$

Then we have

$\sum\limits_{a = 1}^{p-1}\chi_2(1+\bar{a}) = -1, \; \; \; \sum\limits_{a = 1}^{p-1}\chi_2(a+1) = -1,$

$\begin{align*} \sum\limits_{a = 1}^{p-1}\chi_{4}(a)\chi_2(a)\chi_2(a+1)& = \frac{1}{\tau(\chi_2)}\sum\limits_{b = 1}^{p-1}\sum\limits_{a = 1}^{p-1}\chi_2(b)\chi_{4}(a)\chi_2(a)e\left(\frac{b(a+1)}{p}\right)\nonumber\\ & = \frac{1}{\tau(\chi_2)}\sum\limits_{b = 1}^{p-1}\bar{\chi_{4}}(b)e\left(\frac{b}{p}\right)\sum\limits_{a = 1}^{p-1}\chi_{4}(ab)\chi_2(ab)e\left(\frac{ab}{p}\right) \end{align*}$

(3)

$\begin{align*} & = \frac{1}{\tau(\chi_2)}\cdot\tau(\bar{\chi_{4}})\cdot\tau(\chi_{4}\chi_2)\nonumber. \end{align*}$

For any non-principal character $\psi$ , from Lemma 2 we have

$\begin{align*} \tau(\psi^2) = \frac{\psi^2(2)}{\tau(\chi_2)}\cdot\tau(\psi)\cdot\tau(\psi\chi_2). \end{align*}$

(4)

Taking $\psi = \chi_{4}$ , note that

$\tau(\chi_2) = \sqrt{p}, \ \ \tau(\chi_{4})\cdot\tau(\bar{\chi_{4}}) = \chi_{4}(-1)\cdot p,$

from (3) and (4), we have

$\begin{align*} \sum\limits_{a = 1}^{p-1}\chi_{4}(a)\chi_2(a)\chi_2(a+1)& = \frac{\bar{\chi_{4}}^2(2)\cdot\tau(\chi_{4}^2)\cdot\tau(\chi_2)\cdot\tau(\bar{\chi_{4}})}{\tau(\chi_2)\cdot\tau(\chi_{4})}\nonumber\\ & = \frac{\chi_2(2)\cdot\tau(\chi_2)\cdot\tau^2(\bar{\chi_{4}})}{\tau(\chi_{4})\cdot\tau(\bar{\chi_{4}})}\nonumber\\ & = \frac{\chi_2(2)\cdot\sqrt{p}\cdot\tau^2(\bar{\chi_{4}})}{\chi_4(-1)\cdot p} \end{align*}$

(5)

$\begin{align*} & = \frac{\chi_2(2)\cdot\tau^2(\bar{\chi_{4}})}{\chi_4(-1)\cdot\sqrt{p}}.\nonumber \end{align*}$

Similarly, we also have

$\begin{align*} \sum\limits_{a = 1}^{p-1}\bar{\chi_{4}}(a)\chi_2(a)\chi_2(a+1) = \frac{\chi_2(2)\cdot\tau^2(\chi_{4})}{\chi_4(-1)\cdot\sqrt{p}}. \end{align*}$

(6)

Consider the quadratic character modulo $p$ , we have

$\begin{align*} \left(\frac{2}{p}\right) = \chi_2(2) = \begin{cases} \; \; 1, &\rm{ if\; \ p\equiv\pm1\; mod\; 8};\\ -1, &\rm{ if\; \ p\equiv\pm3\; mod\; 8}.\\ \end{cases} \end{align*}$

(7)

And when $p\equiv1\; \rm mod\; 8$ , we have $\chi_{4}(-1) = 1$ ; when $p\equiv5\; \rm mod\; 8$ , we have $\chi_{4}(-1) = -1.$ Combining (2) and (5)–(7) we can deduce that

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{ p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right).$

This prove Lemma 3.

Lemma 4. Let $p$ be an odd prime with $p\equiv1\; \rm mod\; 4$ . Then for any integer $k\geq4$ and $n$ with $(n, p) = 1$ , we have the fourth-order linear recurrence formula

$G^k(n) = (4p+2)\cdot G^{k-2}(n)+8(p-2\alpha^2)\cdot G^{k-3}(n)+[(4\alpha^2-2p)^2-(2p-1)^2]\cdot G^{k-4}(n),$

where

$\alpha = \alpha(p) = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a^3+a}{p}\right) = \sum\limits_{a = 1}^{\frac{p-1}{2}}\left(\frac{a+\bar{a}}{p}\right),$

$\left(\frac{\ast} {p}\right) = \chi_2$ denotes the Legendre's symbol.

Proof. For $p\equiv1\; \rm mod\; 4$ , any integer $n$ with $\left(n, p\right) = 1$ , and fourth-order character $\chi_{4}$ modulo $p$ , we have the identity

$\chi_{4}^4(n) = \bar{\chi_{4}}^4(n) = \chi_0(n), \ \ \chi_{4}^2(n) = \chi_2(n),$

where $\chi_0$ denotes the principal character modulo $p$ .

According to Lemma 3,

$\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = -1-\chi_2(n)+\frac{1}{\sqrt{ p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right),$

$G(n) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right).\quad\quad\quad\quad\quad\quad\quad\quad\ \$

We have

$\begin{align*} G(n) = -\chi_2(n)+\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right), \end{align*}$

(8)

$\begin{align*} G^2(n) = &[-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4}))]^2\\ = &1-2\chi_2(n)\cdot\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right)\\ &+\frac{1}{ p}\cdot\left(\chi_2(n)\cdot\tau^4(\bar{\chi_{4}})+\chi_2(n)\cdot\tau^4(\chi_{4})+2p^2\right)\\ = &1-2\chi_2(n)\cdot\frac{1}{\sqrt{p}}\cdot\left(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4})\right)\\ &+\frac{1}{p}\cdot\left(\chi_2(n)\cdot(\tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4}))+2p^2\right). \end{align*}$

According to Lemma 1, we have

$\left(\tau^2(\chi_{4})+\tau^2(\bar{\chi_{4}})\right)^2 = \tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4})+2p^2 = 4p\alpha^2.$

Therefore, we may immediately deduce

$\begin{align*} G^2(n) = &1-2(\chi_2(n)\cdot\left( G(n)+\chi_2(n)\right)\nonumber\\ &+\frac{1}{p}\left(\chi_2(n)\cdot(\tau^4(\bar{\chi_{4}})+\tau^4(\chi_{4}))+2p^2\right)\nonumber\\ = &1-2\chi_2(n)\cdot\left( G(n)+\chi_2(n)\right) \end{align*}$

(9)

$\begin{align*} &+\frac{1}{p}\cdot\left[\chi_2(n)((\tau^2(\bar{\chi_{4}})+\tau^2(\chi_{4}))^2-2p^2)+2p^2\right]\nonumber\\ = &2p-1-2\chi_2(n)\cdot G(n)+\left(4\alpha^2-2p\right)\cdot\chi_2(n)\nonumber, \end{align*}$

$\begin{align*} \; \; \; \; \; \; \; \; \; \; \; \; \; \; G^3(n) = &[-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\cdot\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\cdot\tau^2(\chi_{4}))]^3\nonumber\\ = &\left(2p-1-2\chi_2(n)\cdot G(n)+\left(4\alpha^2-2p\right)\cdot\chi_2(n)\right)\cdot G(n) \end{align*}$

(10)

$\begin{align*} = &(4\alpha^2-2p)\chi_2(n)\cdot G(n)+(2p+3)G(n)-(4p-2)\chi_2(n) -2(4\alpha^2-2p)\nonumber \end{align*}$

and

$\begin{align*} \left[G^2(n)-(2p-1)\right]^2 = \left[\chi_2(n)\cdot(4\alpha^2-2p)-2\chi_2(n)\cdot G(n)\right]^2, \end{align*}$

which implies that

$\begin{align*} G^4(n) = (4p+2)\cdot G^{2}(n)+8(p-2\alpha^2)\cdot G(n)+[(4\alpha^2-2p)^2-(2p-1)^2]. \end{align*}$

(11)

So for any integer $k\geq4$ , from (8)–(11), we have the fourth-order linear recurrence formula

$\begin{align*} G^k(n)& = G^{k-4}(n)\cdot G^4(n)\\ & = (4p+2)\cdot G^{k-2}(n)+8(p-2\alpha^2)\cdot G^{k-3}(n)+[(4\alpha^2-2p)^2-(2p-1)^2]\cdot G^{k-4}(n). \end{align*}$

This proves Lemma 4.

3. Proof of the theorem

In this section, we will complete the proof of our theorem.

Let $p$ be any prime with $p\equiv 1\; \rm mod\; 4$ , then we have

$\begin{align*} K_0(p) = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^0(n) = \frac{p-1}{p-1} = 1.\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \end{align*}$

(12)

$\begin{align*} K_1(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^1(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)\\& = 0, \end{align*}$

(13)

$\begin{align*} \; \; \; \; \; K_2(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^2(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)^2 \\ & = 2p+1,\end{align*}$

(14)

$\begin{align*} \; \; \; \; \; K_3(p)& = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^3(n)\nonumber\\ & = \frac{1}{p-1}\sum\limits_{n = 1}^{p-1}\left(-\chi_2(n)+\frac{1}{\sqrt{p}}\cdot(\chi_{4}(n)\tau^2(\bar{\chi_{4}})+\bar{\chi_{4}}(n)\tau^2(\chi_{4}))\right)^3 \\ & = -3(4\alpha^2-2p).\end{align*}$

(15)

It is clear that from Lemma 4, if $k\geq4$ , we have

$\begin{align*} \; \; \; \; \; K_k(p) = &\frac{1}{p-1}\sum\limits_{n = 1}^{p-1}G^k(n)\nonumber\\ = &(4p+2)\cdot K_{k-2}(p)-8\left(2\alpha^2-p\right)\cdot K_{k-3}(p) +(16\alpha^4-16p\alpha^2+4p-1)\cdot K_{k-4}(p). \end{align*}$

(16)

Now Theorem 1 follows (12)–(16). Obviously, using Theorem 1 to all negative integers, and that lead to Corollary 1.

This completes the proofs of our all results.

Some notes:

Note 1: In our theorem, know $n$ is an integer, and $(n, p) = 1$ . According to the properties of quadratic residual, $\chi_2(n) = \pm1$ , $\chi_4(n) = \pm1$ .

Note 2: In our theorem, we only discussed the case $p\equiv1\; \rm mod\; 8$ . If $p\equiv3\; \rm mod\; 4$ , then the result is trivial. In fact, in this case, for any integer $n$ with $(n, p) = 1$ , we have the identity

$\begin{align*} G(n)& = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^4}{p}\right)\cdot\left(\frac{a^4+n}{p}\right)\\ & = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a}{p}\right)\cdot\left(\frac{a+n}{p}\right) = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{a^2+na}{p}\right)\\ & = 1+\sum\limits_{a = 1}^{p-1}\left(\frac{1+n\bar{a}}{p}\right) = \sum\limits_{a = 0}^{p-1}\left(\frac{1+na}{p}\right) = 0. \end{align*}$

Thus, for all prime $p$ with $p\equiv3\; \rm mod\; 4$ and $k\geq1$ , we have $K_k(p) = 0$ .

4. Conclusions

The main result of this paper is Theorem 1. It gives an interesting computational formula for $K_k(p)$ with $p\equiv1\; \rm mod\; 4$ . That is, for any integer $k$ , we have the identity

$K_k(p) = (4p+2)\cdot K_{k-2}(p)-8\left(2\alpha^2-p\right)\cdot K_{k-3}(p)+(16\alpha^4-16p\alpha^2+4p-1)\cdot K_{k-4}(p).$

Thus, the problems of calculating a linear recurrence formula of one kind special character sums modulo a prime are given.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the anonymous referee for very helpful and detailed comments.

This work is supported by the N.S.F. (11971381, 12371007) of China and Shaanxi Fundamental Science Research Project for Mathematics and Physics (22JSY007).

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	A. Hernando, E. Roanes-Lozano, L. M. Laita, A polynomial model for logics with a prime power number of truth values, J. Autom. Reasoning, 46 (2011), 205–221. https://doi.org/10.1007/s10817-010-9191-0 doi: 10.1007/s10817-010-9191-0
[2]	J. Hsiang, Refutational Theorem Proving using Term-Rewriting Systems, Artif. Intell., 25 (1985), 255–300.
[3]	D. Kapur, P. Narendran, An Equational Approach to Theorem Proving in First-Order Predicate Calculus, In: Proceedings of the 9th International Joint Conference on Artificial Intelligence (IJCAI-85), 2 (1985), 1146–1153.
[4]	J. Chazarain, A. Riscos, J. A. Alonso, E. Briales, Multivalued logic and Gröbner Bases with applications to modal logic, J. Symb. Comput., 11 (1991), 181–194. https://doi.org/10.1016/S0747-7171(08)80043-0 doi: 10.1016/S0747-7171(08)80043-0
[5]	J. A. Alonso, E. Briales, Lógicas Polivalentes y Bases de Gröbner, In: C. Martin, ed., Actas del V Congreso de Lenguajes Naturales y Lenguajes Formales, University of Seville, Seville, 1995,307–315.
[6]	E. Roanes-Lozano, L. M. Laita, E. Roanes-Macías, A polynomial model for multivalued logics with a touch of algebraic geometry and computer algebra, Math. Comput. Simul., 45 (1998), 83–99.
[7]	B. Buchberger, Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elementals of the residue class ring of a zero dimensional polynomial ideal, J. Symbol. Comput., 41 (2006), 475–511. https://doi.org/10.1016/j.jsc.2005.09.007 doi: 10.1016/j.jsc.2005.09.007
[8]	B. Buchberger, Applications of Gröbner Bases in Non-Linear Computational Geometry, In: J. R. Rice, ed., Mathematical Aspects of Scientific Software. Springer-Verlag, IMA Vol. 14, New York, (1988), 60–88.
[9]	J. Abbott, A. M. Bigatti, CoCoALib: a C++ library for doing Computations in Commutative Algebra, 2019. Available from: {http://cocoa.dima.unige.it/cocoalib}
[10]	L. M. Laita, E. Roanes-Lozano, V. Maojo, L. de Ledesma, L. Laita, An Expert System for Managing Medical Appropriateness Criteria Based on Computer Algebra Techniques, Comput. Math. Appl., 51 (2000), 473–481.
[11]	C. Pérez-Carretero, L. M. Laita, E. Roanes-Lozano, L. Lázaro, J. González-Cajal, L. Laita, A logic and computer algebra-based Expert System for diagnosis of anorexia, Math. Comput. Simul., 58 (2002), 183–202.
[12]	C. Rodríguez-Solano, L. M. Laita, E. Roanes Lozano, L. López Corral, L. Laita, A computational system for diagnosis of depressive situations, expert system with applications, 31 (2006), 47–55. https://doi.org/10.1016/j.eswa.2005.09.011
[13]	M. Lourdes Jimenez, J. M. Santamaría, R. Barchino, L. Laita, L. M. Laita, L. A. González, et al., Knowledge representation for diagnosis of care problems through an expert system: Model of the auto-care deficit situations, Expert Syst. Appl., 34 (2008), 2847–2857. https://doi.org/10.1016/j.eswa.2007.05.039 doi: 10.1016/j.eswa.2007.05.039
[14]	E. Roanes-Lozano, J. L. Galán-García, G. Aguilera-Venegas, A prototype of a RBES for personalized menus generation, Appl. Math. Comput., 315 (2017), 615–624. https://doi.org/10.1016/j.amc.2016.12.023 doi: 10.1016/j.amc.2016.12.023
[15]	G. Aguilera-Venegas, E. Roanes-Lozano, E. Rojo-Martínez, J. L. Galán-García, A proposal of a mixed diagnostic system based on decision trees and probabilistic experts rules, J. Comput. Appl. Math., 427 (2023). https://doi.org/10.1016/j.cam.2023.115130 doi: 10.1016/j.cam.2023.115130
[16]	G. Aguilera-Venegas, A. López-Molina, G. Rojo-Martínez, J. L. Galán-García, Comparing and tuning machine learning algorithms to predict type 2 diabetes mellitus, J. Comput. Appl. Math., 427 (2023). https://doi.org/10.1016/j.cam.2023.115115 doi: 10.1016/j.cam.2023.115115
[17]	E. Roanes-Lozano, E. A. Casella, F. Sánchez, A. Hernando, Diagnosis in Tennis Serving Technique, Algorithms, 13 (2020). https://doi.org/10.3390/a13050106 doi: 10.3390/a13050106
[18]	A. Roanes-Lozano, J. L. Galán-García, G. Aguilera-Venegas, A prototype of a functional approach to personalized menus generation using set operations, Adv. Comput. Math., 13 (2019), 1881–1895.
[19]	M. Villalba-Orero, E. Roanes-Lozano, A prototype of a decision support system for equine cardiovascular diseases diagnosis and management, Mathematics, 20 (2021). https://doi.org/10.3390/math9202580 doi: 10.3390/math9202580
[20]	A. Hernando, A new algebraic model for implementing expert systems represented under the 'Concept-Attribute-Value' paradigm, Math. Comput. Simul., 82 (2011), 29–43. https://doi.org/10.1016/j.matcom.2010.06.020 doi: 10.1016/j.matcom.2010.06.020
[21]	A. Hernando, R. Maestre-Martínez, E. Roanes-Lozano, A natural language for implementing algebracially expert systems, Math. Comput. Simul., 129 (2016), 31–49. https://doi.org/10.1016/j.matcom.2016.04.006 doi: 10.1016/j.matcom.2016.04.006
[22]	M. Brickenstein, A. DreyerPolyBoRi: Polybori: A framework for Gröbner-basis computations with Boolean polynomials, Journal of Symbolic Computation, 44 (2009), 1326–1345. https://doi.org/10.1016/j.jsc.2008.02.017 doi: 10.1016/j.jsc.2008.02.017
[23]	E. Roanes-Lozano, L. M. Laita, An applicable topology-independent model for railway interlocking systems, Math. Comput. Simul., 45 (1998), 175–183. https://doi.org/10.1016/S0378-4754(97)00093-1 doi: 10.1016/S0378-4754(97)00093-1
[24]	E. Roanes-Lozano, L. M. Laita, E. Roanes-Macías, An application of an AI methodology to railway interlocking systems using computer algebra, in Tasks and Methods in Applied Artificial Intelligence, (1998), 687–696. https://doi.org/10.1007/3-540-64574-8_455
[25]	E. Roanes-Lozano, E. Roanes-Macías, L. Laita, Railway interlocking systems and Gröbner bases, Math. Comput. Simul., 51 (2000), 473–481. https://doi.org/10.1016/S0378-4754(99)00137-8 doi: 10.1016/S0378-4754(99)00137-8
[26]	E. Roanes-Lozano, A. Hernando, J. A. Alonso, L. M. Laita, A logic approach to decision taking in a railway interlocking system using Maple, Math. Comput. Simul., 82 (2011), 15–28. https://doi.org/10.1016/j.matcom.2010.05.024 doi: 10.1016/j.matcom.2010.05.024
[27]	A. Hernando, E. Roanes-Lozano, R. Maestre-Martínez, J. Tejedor, A logic-algebraic approach to decision taking in a railway interlocking system, Ann. Math. Artif. Intell., 65 (2012), 317–328. https://doi.org/10.1007/s10472-012-9321-y doi: 10.1007/s10472-012-9321-y
[28]	A. Hernando, R. Maestre, E. Roanes-Lozano, A new algebraic approach to decision making in a railway interlocking system based on preprocess, Math. Probl. Eng., 2018 (2018), 4982974. https://doi.org/10.1155/2018/4982974 doi: 10.1155/2018/4982974
[29]	A. Hernando, E. Roanes-Lozano, J. L. Galán-García, G. Aguilera-Venegas, Decision making in railway interlocking systems based on calculating the remainder of dividing a polynomial by a set of polynomials, Electron. Res. Arch., 31 (2023), 6160–6196. https://doi.org/10.3934/era.2023313 doi: 10.3934/era.2023313

This article has been cited by:

1.	Mounia Mouy, Hamid Boulares, Saleh Alshammari, Mohammad Alshammari, Yamina Laskri, Wael W. Mohammed, On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation, 2022, 7, 2504-3110, 31, 10.3390/fractalfract7010031
2.	Sabbavarapu Nageswara Rao, Manoj Singh, Ahmed Hussein Msmali, Abdullah Ali H. Ahmadini, Existence of Positive Solutions for a Coupled System of p-Laplacian Semipositone Hadmard Fractional BVP, 2023, 7, 2504-3110, 499, 10.3390/fractalfract7070499

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.1

Metrics

Article views(1519) PDF downloads(78) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

A novel way to build expert systems with infinite-valued attributes

Related Papers:

Abstract

1. Introduction

2. Some lemmas

3. Proof of the theorem

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Some lemmas

3. Proof of the theorem

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

AIMS Mathematics

A novel way to build expert systems with infinite-valued attributes

Related Papers:

Abstract

1. Introduction

2. Some lemmas

3. Proof of the theorem

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Some lemmas

3. Proof of the theorem

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References