# Polynomial-Computable Representation of Neural Networks in Semantic Programming

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (1)
- $nil$: a constant that selects an empty list;
- (2)
- $hea{d}^{\left(1\right)}$: returns the last element of the list or is $nil$ otherwise;
- (3)
- $tai{l}^{\left(1\right)}$: returns a list without the last element or is $nil$ otherwise;
- (4)
- $getElemen{t}^{\left(2\right)}$: returns the ith element of the list or is $nil$ otherwise;
- (5)
- $NumElement{s}^{{}^{\left(1\right)}}$: returns the number of elements in the list;
- (6)
- $firs{t}^{\left(1\right)}$: returns the first element of the list or is $nil$ otherwise;
- (7)
- $secon{d}^{\left(1\right)}$: returns the second element of the list or is $nil$ otherwise;
- (8)
- ${\in}^{\left(2\right)}$: the predicate “to be an element of a list”;
- (9)
- ${\subseteq}^{\left(2\right)}$: the predicate“to be an initial segment of a list”.

- $Iteratio{n}_{g,\phi}({t}_{1},{t}_{2})$ is an L-program, where $g,{t}_{1},{t}_{2}$ are L-programs and $\phi $ is an L-formula;
- $Cond({t}_{1},{\phi}_{1},\cdots ,{t}_{n},{\phi}_{n},{t}_{n+1})$ is an L-program, where ${t}_{1},\cdots ,{t}_{n+1}$ are L-programs and ${\phi}_{1},{\phi}_{n}$ are L-formulas;
- $F({t}_{1},\cdots ,{t}_{n})$ is an L-program, where $F\in \sigma $ and ${t}_{1},\cdots ,{t}_{n}$ are L-programs

- ${t}_{1}={t}_{2}$ is an L-formula, where ${t}_{1},{t}_{2}$ are L-programs;
- $P({t}_{1},\cdots ,{t}_{n})$ is an L-formula, where $P\in \sigma $ and ${t}_{1},\cdots ,{t}_{n}$ are L-programs;
- $\mathsf{\Phi}\&\mathsf{\Psi}$, $\mathsf{\Phi}\vee \mathsf{\Psi}$, $\mathsf{\Phi}\to \mathsf{\Psi}$, $\neg \mathsf{\Phi}$ are L-formulas, where $\mathsf{\Phi}$, $\mathsf{\Psi}$ are L-formulas
- $\exists x\delta t\mathsf{\Phi}$, $\forall x\delta t\mathsf{\Phi}$ are L-formulas, where t is an L-program, $\mathsf{\Phi}$ is an L-formula and $\delta \in \{\in ,\subseteq ,\le \}$.

**Theorem**

**1**

- (1)
- Any L-program has polynomial computational complexity.
- (2)
- For any p-computable function, there is a suitable L-program that implements it.

- (1)
- $|{w}_{1}^{*}|\le |{w}_{1}|+C\xb7{\sum}_{i=2}^{n}{\left|{w}_{i}\right|}^{p}$;
- (2)
- $|{w}_{i}^{*}|\le |{w}_{i}|$, for all $i\in [2,\cdots ,n]$.

## 3. Neural Networks

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**2.**

## 4. Materials and Methods

## 5. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Goncharov, S.; Nechesov, A.
Polynomial-Computable Representation of Neural Networks in Semantic Programming. *J* **2023**, *6*, 48-57.
https://doi.org/10.3390/j6010004

**AMA Style**

Goncharov S, Nechesov A.
Polynomial-Computable Representation of Neural Networks in Semantic Programming. *J*. 2023; 6(1):48-57.
https://doi.org/10.3390/j6010004

**Chicago/Turabian Style**

Goncharov, Sergey, and Andrey Nechesov.
2023. "Polynomial-Computable Representation of Neural Networks in Semantic Programming" *J* 6, no. 1: 48-57.
https://doi.org/10.3390/j6010004