# Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Graph Query Framework

#### 3.1. Preliminaries

**Definition**

**1.**

- V and E are sets, called, respectively, the set of nodes and set of edges of G.
- μ associates each node/edge in the graph with a set of properties $\mu :(V\cup E)\times R\to J$, where R represents the set of keys for properties, and J represents the set of values.

**Definition**

**2.**

**Definition**

**3.**

- 1.
- If $e\in E$ and $u,\phantom{\rule{4pt}{0ex}}v\in \gamma \left(e\right)$ with $u{\le}_{e}v$, then $\rho =u\stackrel{e}{\to}v\in {\mathcal{P}}_{G}$. We will say that ρ connects the nodes u and v of G, and we will denote it by $u\stackrel{\rho}{\u21dd}v$.
- 2.
- If ${\rho}_{1},\phantom{\rule{4pt}{0ex}}{\rho}_{2}\in {\mathcal{P}}_{G}$, with $u\stackrel{{\rho}_{1}}{\u21dd}v$ and $v\stackrel{{\rho}_{2}}{\u21dd}w$ then ${\rho}_{1}\xb7{\rho}_{2}\in {\mathcal{P}}_{G}$, with $u\stackrel{{\rho}_{1}\xb7{\rho}_{2}}{\u21dd}w$.

- If $u\stackrel{\rho}{\u21dd}v$, then we write ${\rho}^{o}=u$ and ${\rho}^{i}=v$.
- We denote the paths through u, starting inu, and ending in u, respectively, by:$${\mathcal{P}}_{u}\left(G\right)=\{\rho \in \mathcal{P}\left(G\right):\phantom{\rule{4pt}{0ex}}u\in \rho \},$$$${\mathcal{P}}_{u}^{o}\left(G\right)=\{\rho \in \mathcal{P}\left(G\right):\phantom{\rule{4pt}{0ex}}{\rho}^{o}=u\},$$$${\mathcal{P}}_{u}^{i}\left(G\right)=\{\rho \in \mathcal{P}\left(G\right):\phantom{\rule{4pt}{0ex}}{\rho}^{i}=u\}.$$

**Definition**

**4.**

#### 3.2. Graph Queries

**Definition**

**5.**

- $\alpha :{V}_{Q}\cup {E}_{Q}\to \{+,-\}$.
- $\theta :{V}_{Q}\cup {E}_{Q}\to For{m}^{2}\left(L\right)$.

**Definition**

**6.**

#### 3.3. Refinement Sets

**Definition**

**7.**

- 1.
- ${Q}_{1}$ refines ${Q}_{2}$ in G (${Q}_{1}{\u2aaf}_{G}{Q}_{2}$) if: $\forall S\subseteq G\phantom{\rule{4pt}{0ex}}(S\vDash {Q}_{1}\Rightarrow S\vDash {Q}_{2})$.
- 2.
- They are equivalent in G (${Q}_{1}{\equiv}_{G}{Q}_{2}$) if: ${Q}_{1}{\u2aaf}_{G}{Q}_{2}$ and ${Q}_{2}{\u2aaf}_{G}{Q}_{1}$.

**Theorem**

**1.**

- 1.
- ${Q}_{1}{\u2aaf}_{G}{Q}_{1}$.
- 2.
- ${Q}_{1}{\u2aaf}_{G}{Q}_{2}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}{Q}_{2}{\u2aaf}_{G}{Q}_{1}\Rightarrow {Q}_{1}{\equiv}_{G}{Q}_{2}$.
- 3.
- ${Q}_{1}{\u2aaf}_{G}{Q}_{2}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}{Q}_{2}{\u2aaf}_{G}{Q}_{3}\Rightarrow {Q}_{1}{\u2aaf}_{G}{Q}_{3}$.

**Definition**

**8.**

- 1.
- ${Q}_{2}\subseteq {Q}_{1}$.
- 2.
- $\forall n\in {V}_{{Q}_{2}}^{-}\phantom{\rule{0.166667em}{0ex}}\forall e\in {\gamma}_{{Q}_{1}}\left(n\right)\phantom{\rule{0.166667em}{0ex}}\exists {e}^{\prime}\in {\gamma}_{{Q}_{2}}\left(n\right)\phantom{\rule{4pt}{0ex}}({Q}_{e}\equiv {Q}_{{e}^{\prime}})$.

**Theorem**

**2.**

**Proof.**

- If $n\in {V}_{{Q}_{2}}^{-}$, since ${Q}_{2}{\subseteq}^{-}{Q}_{1}$, then ${{Q}_{1}}_{n}^{-}={{Q}_{2}}_{n}^{-}$.
- If $n\in {V}_{{Q}_{2}}^{+}$, then ${{Q}_{1}}_{n}^{+}\to {{Q}_{2}}_{n}^{+}$, because (${\gamma}_{1}$, ${\gamma}_{2}$ are the incidence functions of ${Q}_{1}$ and ${Q}_{2}$, respectively):$$\begin{array}{cc}{{Q}_{1}}_{n}^{+}\hfill & =\exists v\in V\phantom{\rule{4pt}{0ex}}\left(\underset{e\in {\gamma}_{1}\left(n\right)}{\bigwedge}{{Q}_{1}}_{e}^{\alpha \left(e\right)}\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\exists v\in V\left(\underset{e\in {\gamma}_{1}\left(n\right)\cap {E}_{{Q}_{2}}}{\bigwedge}{{Q}_{1}}_{e}^{\alpha \left(e\right)}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}\underset{e\in {\gamma}_{1}\left(n\right)\setminus {E}_{{Q}_{2}}}{\bigwedge}{{Q}_{1}}_{e}^{\alpha \left(e\right)}\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\exists v\in V\left(\underset{e\in {\gamma}_{2}\left(n\right)\cap {E}_{{Q}_{2}}}{\bigwedge}{{Q}_{2}}_{e}^{\alpha \left(e\right)}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}\underset{e\in {\gamma}_{1}\left(n\right)\setminus {E}_{{Q}_{2}}}{\bigwedge}{{Q}_{1}}_{e}^{\alpha \left(e\right)}\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \to {{Q}_{2}}_{n}^{+}\hfill \end{array}$$

**Definition**

**9.**

**Theorem**

**3.**

**Proof.**

**Definition**

**10.**

- 1.
- $\forall \phantom{\rule{4pt}{0ex}}{Q}^{\prime}\in R\phantom{\rule{4pt}{0ex}}\left({Q}^{\prime}{\u2aaf}_{G}Q\right).$
- 2.
- $\forall \phantom{\rule{4pt}{0ex}}S\subseteq G\phantom{\rule{4pt}{0ex}}(S\vDash Q\Rightarrow \exists !\phantom{\rule{4pt}{0ex}}{Q}^{\prime}\in R\phantom{\rule{4pt}{0ex}}(S\vDash {Q}^{\prime})).$

**Theorem**

**4.**

**Proof.**

- Since $Q{\subseteq}^{-}{Q}_{1}$ and $Q{\subseteq}^{-}{Q}_{2}$, thus ${Q}_{1}\u2aafQ$ and ${Q}_{2}\u2aafQ$.
- Given $S\subseteq G$ such that $S\vDash Q$. Then:$$\begin{array}{cc}{Q}_{1}\hfill & =Q\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}{Q}_{m},\hfill \\ {Q}_{2}\hfill & =Q\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}\neg {Q}_{m}\hfill \end{array}$$If $G\ne \varnothing $, then $S\vDash {Q}_{1}$ and $S\u22ad{Q}_{2}$.If $G=\varnothing $, then $S\u22ad{Q}_{1}$ and $S\vDash {Q}_{2}$.

**Theorem**

**5.**

**Proof.**

- Since ${Q}^{\prime}$ is a clone of Q, then $Q\equiv {Q}^{\prime}$. In addition, ${Q}^{\prime}{\subseteq}^{-}{Q}_{1},{Q}_{2},{Q}_{3},{Q}_{4}$, thus ${Q}_{1},{Q}_{2},{Q}_{3},{Q}_{4}\u2aaf{Q}^{\prime}\equiv Q$.
- Let us consider the predicates:$$\begin{array}{cc}{P}_{n}\hfill & =\exists v\in V\phantom{\rule{4pt}{0ex}}\left(\underset{a\in \gamma \left(n\right)}{\bigwedge}{Q}_{a}^{\alpha \left(a\right)}\phantom{\rule{4pt}{0ex}}\wedge {Q}_{{e}^{o}}^{\alpha \left(e\right)}\right),\hfill \\ {P}_{m}\hfill & =\exists v\in V\phantom{\rule{4pt}{0ex}}\left(\underset{a\in \gamma \left(m\right)}{\bigwedge}{Q}_{a}^{\alpha \left(a\right)}\phantom{\rule{4pt}{0ex}}\wedge {Q}_{{e}^{i}}^{\alpha \left(e\right)}\right).\hfill \end{array}$$If $S\vDash {Q}_{n}$ and $S\vDash {Q}_{m}$, then we have four mutually complementary options:
- $S\vDash {P}_{n}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}S\vDash {P}_{m}\Rightarrow S\vDash {Q}_{1}$
- $S\vDash {P}_{n}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}S\u22ad{P}_{m}\Rightarrow S\vDash {Q}_{2}$
- $S\u22ad{P}_{n}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}S\vDash {P}_{m}\Rightarrow S\vDash {Q}_{3}$
- $S\u22ad{P}_{n}\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}S\u22ad{P}_{m}\Rightarrow S\vDash {Q}_{4}$

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

- The root node is labeled with ${Q}_{0}$ (some initial query).
- If a node on the tree is labeled with Q, and $R=\{{Q}_{1},\dots ,{Q}_{n}\}$ is a set that refines Q, then the child nodes will be labeled with the elements of R.

#### 3.4. Simplified Refinement Sets

**Definition**

**11.**

**Theorem**

**8.**

- $\alpha \left(n\right)=\alpha \left(m\right)$
- ${\theta}_{n}\equiv {\theta}_{m}$
- For each $e\in \gamma \left(n\right)$, exists ${e}^{\prime}\in \gamma \left(m\right)$, with $\alpha \left(e\right)=\alpha \left({e}^{\prime}\right)$, ${\theta}_{e}\equiv {\theta}_{{e}^{\prime}}$ and $\gamma \left(e\right)\setminus \left\{n\right\}=\gamma \left({e}^{\prime}\right)\setminus \left\{m\right\}$

**Proof.**

**Theorem**

**9.**

**Proof.**

#### 3.5. Graph Query Examples

`Yoda`,

`Luke`, and their

`TEACHES`relationship satisfies this query.

`FRIENDS`relationships. It can be verified on any subgraph containing three characters who are friends with each other (for example, the subgraph formed by

`Han Solo`,

`Chewbacca`,

`Princess Leia`, and the

`FRIENDS`relationships in Figure 9).

`FRIENDS`and

`TEACHES`relationships. Additionally, an auxiliary function, $g{r}_{s}\left(v\right)\in L$, is utilized to reference the outgoing degree of node v. This query will be validated by any subgraph that contains

`Luke`or

`Obi-Wan Kenobi.`

#### 3.6. Computational Complexity

## 4. Relational Machine Learning

#### 4.1. Information-Gain Pattern Mining

- A subset of the training set: ${\mathcal{L}}_{n}\subseteq \mathcal{L}$;
- A query ${Q}_{n}$, such that: $\forall S\in {\mathcal{L}}_{n}(S\vDash {Q}_{n})$.

#### 4.2. Relational Tree Learning Examples

`User A`,

`User B`, or

`Item`) to all nodes in the graph by exploiting relational information from the network. Furthermore, on the leaves of the tree, distinctive patterns are acquired for each node type, which can be used to directly assess nodes and clarify future classifications.

`species`property as a training dataset, the relational decision tree shown in Figure 15 categorizes and explains each character’s species in the graph. The leaf patterns of the tree characterize each species: human characters are born friends of

`Luke`, droids are unborn friends of

`Luke`, Wookiees are those born in

`Kashyyyk`, etc.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Almagro-Blanco, P.; Sancho-Caparrini, F.; Borrego-Díaz, J.
Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning. *Mathematics* **2023**, *11*, 4672.
https://doi.org/10.3390/math11224672

**AMA Style**

Almagro-Blanco P, Sancho-Caparrini F, Borrego-Díaz J.
Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning. *Mathematics*. 2023; 11(22):4672.
https://doi.org/10.3390/math11224672

**Chicago/Turabian Style**

Almagro-Blanco, Pedro, Fernando Sancho-Caparrini, and Joaquín Borrego-Díaz.
2023. "Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning" *Mathematics* 11, no. 22: 4672.
https://doi.org/10.3390/math11224672