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Fuzzy Logic Based Logical Query Answering on Knowledge Graphs (2022)

huyi / December 2022

0. Abstract

task

Answering complex First-Order Logical (FOL) queries on large-scale incomplete knowledge graphs (KGs)

flaws of precvious work

• do not satisfy the axiomatic system of classical logic
• logical operators are parameterized –> need more training data

this work

• fuzzy logic to define logical operators in a principled and learning-free manner
• 只有 entity 和 relation 的 embedding 是要学的

performance

用 labeled FOL 去训

超sota不少

只用 link prediction 去训

comparable performance to those trained with extra data

1. Introduction

challenges of querying KG

• time complexity
  • grow with the query complexity
  • affected by the size of intermediate results

–> difficult to scale to modern KG,e.g. SPARQL

• incomplete KG

–> 无法直接做 searching

limitations of previous parametric methods

1. 逻辑操作符不规范
2. deep architectures (主要应该是指logical operaters也要训练)–> 难训练

3. Preliminaries

4. Methodology

4.1 Queries and Entities in Fuzzy Space

Query Embedding

$q$: query; $S_q$: answer set; $\textbf{S}_q\in[0,1]^d$: fuzzy vector

$\Omega$是所有entitiy的集合文章中说的elements目前理解是KG中entities的集合，${U_i}_{i=1}^{d}$是$\Omega$的一种划分，fuzzy vector $\textbf{S}_q$中的每个维度就是$U_i\subseteq S_q$的概率

U_i是事先划定的？

Entity Embedding

entity embedding: $\textbf{p}_e\in[0,1]^d$

每个维度代表这个entity属于$U_i$的概率

Score Function

有没有一种可能$e\in S_i,U_i not \subseteq S_q$

这里的$U_i$感觉上是给了一种'直观'，赋予每个S_q和p_e里的元素logic上的意义。应该指的是一些没有明确定义的fuzzy sets，第二个等号感觉还是从fuzzy set的角度去理解会比较容易，但还是有点怪，数学上有支撑吗？，倾向于不用U_i，直接理解

4.2 Relation Projection for Atomic Queries

embedding of Atomic queries: anchor entity embedding 过一个归一化的线性层，再过一个激活函数

4.3 Fuzzy Logic Based Logical Operators

product logic

Gödel logic

4.4 Model Learning and Inference

对比学习来使$\phi(q,e)$最大