# UnTiCk: Unsupervised Type-Aware Complex Logical Queries Reasoning over Knowledge Graphs

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## Abstract

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## 1. Introduction

- We propose UnTiCk, an embedding-based unsupervised type-aware complex logical queries reasoning model. It is a novel solution that extends unsupervised type constraints to multi-hop complex logical query embedding models.
- We designed four type compatibility measurement meta-operations that reflect good modularity and generalization. They capture the diversity of entity types in different relations and locations in complex logical queries.
- We conducted experiments on three popular benchmark datasets, combining our model with popular complex logical embedding models. With the same number of embedding dimensions, our models showed better results than the complex logical embedding models, which contain only entity structure information. We also demonstrate the effectiveness of our unsupervised type feature extraction with a visualization.

## 2. Related Work

#### 2.1. Logical Query Embedding Models for Multi-Hop Reasoning

#### 2.2. Unsupervised Type Information Embedding Models

## 3. Background and Problem Definition

**Definition 1**

**.**A knowledge graph $\mathcal{G}=(\mathcal{E},\mathcal{R},\mathcal{T})$ consists of entities $e\in \mathcal{E}$, relations $r\in \mathcal{R}$, and tuples (or triples) $(h,r,t)\in \mathcal{T}$. h represents the head entity of the triple, and t represents the tail entity of the triple connected by relation r. We consider $r(h,t)=\{1,0\}$ to be a binary function representing $(h,r,t)\in \mathcal{T}$ when $r(h,t)=1$ and $(h,r,t)\notin \mathcal{T}$ otherwise.

**Definition 2**

**.**A type constraint collection ${\mathcal{G}}^{\mathcal{T}}$ of a knowledge graph $\mathcal{G}$ is denoted as ${\mathcal{G}}^{\mathcal{T}}=({\mathcal{E}}^{\mathcal{T}},{\mathcal{R}}^{\mathcal{T}},{\mathcal{C}}^{\mathcal{E}},{\mathcal{C}}^{\mathcal{R}})$, where ${\mathcal{E}}^{\mathcal{T}}$ is the constraint distinct type set of $\mathcal{E}$ and ${\mathcal{R}}^{\mathcal{T}}$ is the distinct type set of $\mathcal{R}$, ${\mathcal{C}}^{\mathcal{E}}$ is a constraint function that ${\mathcal{C}}^{\mathcal{E}}\left(e\right)=\{{c}_{1},{c}_{2},...,{c}_{e}\}$ represents the distinct type set of entity e. ${\mathcal{C}}^{\mathcal{R}}$ is a constraint function that ${\mathcal{C}}^{\mathcal{R}}\left(r\right)=\{({c}_{{h}_{1}},{c}_{{t}_{1}}),({c}_{{h}_{2}},{c}_{{t}_{2}}),...,({c}_{{h}_{n}},{c}_{{t}_{n}})\}$ represents the head (domain) and tail (range) distinct type constraints tuple of relation r.

**Definition 3**

**.**We formally define an EPFO logical query as follows:

**Definition 4**

**.**A complex logical query q has a corresponding dependency graph denoted as $\mathcal{Q}=({\mathcal{E}}_{\mathcal{Q}},{\mathcal{R}}_{\mathcal{Q}},{\mathcal{L}}_{\mathcal{Q}})$. $\mathcal{Q}$ consists of entities ${\mathcal{E}}_{\mathcal{Q}}\subseteq \mathcal{E}$, which contain the anchor nodes’ set and the existentially quantified bound variables’ set. The relation between entities ${\mathcal{R}}_{\mathcal{Q}}\subseteq \mathcal{R}$ and $\mathcal{L}$ is the logical operator. In this paper, we considered the logical collection ${\mathcal{L}}_{\mathcal{Q}}\subseteq \mathcal{L}=\{\mathbb{P},\mathbb{I},\mathbb{U}\}$ (project, intersection, union operators). The computation graph represents the actual order of operations of our framework for reasoning about the dependency graph.

#### 3.1. Logical Query Embedding Operators

**Entity projection operator:**Given an entity embedding or an existential quantified variable box region and a relation box embedding $\mathbf{r}$, we model the entity projection operation as a linear transformation:

**Entity intersection operator:**Given a collection of entity sets, the entity intersection operation’s intuition is to find the overlap of multiple spaces. Since the importance of each collection element is not the same for the merged space, it uses the attention mechanism to find the entity centers and the $sigmoid$ function of $Deepsets$ [38] to shrink the offset:

**Entity union operator:**Given a collection of entity sets, the entity union operation’s intuition is to transform the query into a Disjunctive Normal Form (DNF) [39], instead of creating a new neural logical operator. In the final step, the box regions of all anchor nodes need to be united, and the shortest distance in the answer space will determine the confidence of the entity:

#### 3.2. Type-Aware Logical Query Operations

**Problem 1**

**.**Given a set of entities ${\mathcal{E}}_{q}\subseteq \mathcal{E}$, relation $r\in \mathcal{R}$, and their corresponding type constraint collection ${\mathcal{G}}_{q}^{\mathcal{T}}$, for ${e}_{h}\in {\mathcal{E}}_{q}$, we use a type-aware projection operation to obtain the answer collection ${\mathbb{P}}_{t}({e}_{h},r)=\left\{{e}_{t}\right|r({e}_{h},{e}_{t})=1,\exists ({c}_{h},{c}_{t})\in {\mathcal{C}}^{\mathcal{R}}\left(r\right),$${c}_{h}\in {\mathcal{C}}^{\mathcal{E}}\left({e}_{h}\right),{c}_{t}\in {\mathcal{C}}^{\mathcal{E}}\left({e}_{t}\right)\}.$

**Problem 2**

**.**Given a collection of entity sets $E=\{{\mathcal{E}}_{1},{\mathcal{E}}_{2},...,{\mathcal{E}}_{n}\}$, the corresponding entity type constraint set $C=\{{\mathcal{C}}_{1}^{\mathcal{E}},{\mathcal{C}}_{2}^{\mathcal{E}},...,{\mathcal{C}}_{n}^{\mathcal{E}}\}$. The prediction constraint set after the intersection is ${\mathcal{C}}_{\mathbb{I}}^{\mathcal{E}}$. Then, we use the type-aware intersection operator to obtain the answer collection ${\mathbb{I}}_{t}(E,C)={\cap}_{i=1}^{n}\{e\in {\mathcal{E}}_{i}|\exists {c}_{e}\in {\mathcal{C}}_{i}^{\mathcal{E}}\left(e\right),{c}_{e}\in {\mathcal{C}}_{\mathbb{I}}^{\mathcal{E}}\}$.

**Problem 3**

**.**Given a collection of entity sets $E=\{{\mathcal{E}}_{1},{\mathcal{E}}_{2},...,{\mathcal{E}}_{n}\}$, the corresponding entity type constraint set $C=\{{\mathcal{C}}_{1}^{\mathcal{E}},{\mathcal{C}}_{2}^{\mathcal{E}},...,{\mathcal{C}}_{n}^{\mathcal{E}}\}$. The prediction constraint set after the union is ${\mathcal{C}}_{\mathbb{U}}^{\mathcal{E}}$. Then, we use the type-aware union operator to obtain the answer collection ${\mathbb{U}}_{t}(E,C)={\cup}_{i=1}^{n}\{e\in {\mathcal{E}}_{i}|\forall {c}_{e}\in {\mathcal{C}}_{i}^{\mathcal{E}}\left(e\right),{c}_{e}\in {\mathcal{C}}_{\mathbb{U}}^{\mathcal{E}}\}$.

#### 3.3. UnTiCk Problem Definition

## 4. UnTiCk: Unsupervised Type-Aware Complex Logical Queries Reasoning Framework

#### 4.1. Type Compatibility Measurement Meta-Operators

**Example 1.**

**Jay Chou**, when it is in

**(Jay Chou, isComposerOf, The Swan)**, its entity type feature is expressed as

**Composer**. In

**(Jay Chou, isDirectorOf, Secret)**, its entity type feature is expressed as

**Director**. In

**(Nunchuncks, isSangBy, Jay Chou)**, its entity type feature is expressed as

**Singer**.

**Definition 5**

**.**Given an entity e and an outgoing relation r, $Ex(e,r)$ aims to obtain the domain type representation of this entity $Ex(e,r)=\left\{{c}_{h}\right|\exists ({c}_{h},{c}_{t})\in {\mathcal{C}}^{\mathcal{R}}\left(r\right),$${c}_{h}\in {\mathcal{C}}^{\mathcal{E}}\left(e\right)\}$ for this outgoing edge r.

**Example 2.**

**Jay Chou**and

**Vincent Fang**in the previous example, we should not show modeling specific intermediate nodes during reasoning, but we need to obtain intermediate nodes as range type constraint representations of

**isComposerOf**and

**isLyricistOf**.

**Definition 6**

**.**Given a domain entity type representation ${c}_{h}^{e}$ and an outgoing relation r, $Tr\left(r\right)$ aims to obtain the range type representation of this entity e for this outgoing edge r, denoted as $Tr({c}_{h}^{e},r)=\left\{{c}_{t}\right|({c}_{h}^{e},{c}_{t})\in {\mathcal{C}}^{\mathcal{R}}\left(r\right)\}$.

**Example 3.**

**isComposerOf**and

**isLyricistOf**in the query, we need to regress the virtual nodes’ representation from the range representation for each logical path. These nodes refer to the original features of the entity node composed by

**Jay Chou**and written lyrics by

**Vincent Fang**, and they will be used as the following Relation

**isSangBy**for type feature extraction.

**Definition 7**

**.**Given a relation r and a range entity type representation ${c}^{v}$ of the intermediate virtual node V for r, $Re({c}^{v},r)$ aims to obtain the original entity type representation of the intermediate virtual node V, $Re({c}^{v},r)\subseteq \left\{v\right|\exists ({c}_{h},{c}^{v})\in {\mathcal{C}}^{\mathcal{R}}\left(r\right),{c}^{v}\in {\mathcal{C}}^{\mathcal{E}}\left(v\right)\}$.

**Example 4.**

**Jay Chou**and

**Vincent Fang**, is located. We need to calculate the scores of each path and jointly find the target answers.

**Definition 8**

**.**Given the target node e, the list of incoming edges of the target node R, and the list of type constraints C for each logical path, $Fu(e,R,C)$ aims to obtain a composite score of the type constraints’ compatibility for multiple logical paths after feature intersection or union (R and C identified as intersection or union).

#### 4.2. UnTiCk Query Reasoning Process

Algorithm 1: UnTiCk query embedding generation |

**Type-aware projection operation:**In the entity structure information embedding module, we perform one entity projection operation on it. In the entity type information embedding module, a relation feature extraction operation and a type feature transformation operation are performed on it if our target entity is an anchor node. It will add a type feature regression meta-operation before this if our target entity is an intermediate node.

**Lemma 1.**

**Proof.**

**Type-aware intersection operation:**In the entity structure information embedding module, we perform one entity intersection operation on it. In the entity type information embedding module, each different incoming relation is a different constraint for the sink node of the intersection operation, and we should not ignore this information. Therefore, we record different path type compatibility separately, add the paths that need to be intersected with the intersection path list, and record the relevant information in the computation order list. We handle all this in the final path fusion operation.

**Lemma 2.**

**Proof.**

**Type-aware union operation:**Similar to the type-aware intersection operation, on the entity structure information embedding module, we perform one entity union operation on it. In the entity type information embedding module, we separately record different path type compatibility and add the paths that need to be united to the union path list and the relevant information in the computation order list. We deal with it during the final path fusion meta-operation.

**Lemma 3.**

**Proof.**

#### 4.3. Optimization Objective

## 5. Experiments

#### 5.1. Datasets

**FB15k:**FB15k [4] is a subset created from Freebase and is a frequently used standard dataset for embedding knowledge graphs. It includes knowledge base relation triples and textual references to Freebase entity pairs.**FB15k-237:**FB15k-237 [41] is a variant of the original FB15k dataset in which inverse relations have been eliminated because it was discovered that inverting triplets in the training set yielded many test triplets, which may cause test leakage.

**WN18:**WN18 [4] is a dataset commonly used for knowledge graph linkage prediction, deriving its name from it as a subset of WordNet containing 18 different relations. Its entities correspond to senses, and the relation types define the lexical relations between senses.**WN18RR:**WN18RR [43] is a subset created from WN18 in order to handle test leakage due to training set triplet inversion in WN18.**YAGO3-10:**YAGO3-10 is a publicly available and commonly used dataset, which is a subset of YAGO3 [44], which only contains entities with at least ten relations. Most triples are descriptive attributes of people. We processed it to fit our experiments, as described below.

#### 5.2. Evaluation Metrics

#### 5.3. Baselines and Hyperparameter Settings

- (1)
- (2)
**GQE-Double**, which has the same basic model settings as GQE, but uses double-embedding dimensionality to enable the model dimensionality to be on the same level as other models;- (3)
**Query2Box**[21] models the query as a box embedding, projection as a linear transformation, intersection as the center using the attention mechanism, offset using Deepsets, and the sigmoid function to shrink, and union uses the same DNF-query rewriting strategy.

#### 5.4. Experimental Results and Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**This is the graph of the complex logical query “Who are the singers of the songs composed by Jay Chou and lyrics written by Vincent Fang?” (

**a**) Dependency graph for the above query. (

**b**) Computation graph for the above query.

**Figure 3.**This is an example of the type knowledge graph used in this section. It shows the entities and relations and the different types of entities. V represents the virtual intermediate node. ${A}_{?}$ represents the target node.

**Figure 4.**This is a diagram of the nine logical query patterns. It contains five training logical query patterns and four unseen logical query patterns.

**p**represents projection,

**i**intersection, and

**u**union.

**Figure 5.**Stacked column chart of robustness of Hit@3 results on additional datasets: (

**a**) Stacked Hit@3 results on WN18. (

**b**) Stacked Hit@3 results on WN18RR. (

**c**) Stacked Hit@3 results on YAGO3-10.

**Figure 6.**The clustered column charts and stacked bar chart of the Hit@3 results for different parameters of UnTiCk (GQE-Double): (

**a**) Hit@3 results for different parameter on FB15k. (

**b**) Hit@3 results for different parameter on FB15k-237. (

**c**) Hit@3 results for different parameters on NELL-995. (

**d**) stacked Hit@3 results for different parameters on three datasets.

**Figure 7.**Visualization of the embedding of UnTiCk (Q2B) and original Query2Box: (

**a**) Result of entity type constraint embedding of UnTiCk (Q2B). (

**b**) Result of entity structure information embedding of UnTiCk (Q2B). (

**c**) Result of entity information embedding of the original Query2Box.

**Figure 8.**Visualization about type people and relation /people/person/profession: (

**a**) Result without type feature extraction meta-operation. (

**b**) Result after using type feature extraction meta-operation.

Fields | Model Name | Model Category | Type Constraint | ||
---|---|---|---|---|---|

Logical Query | Type Information | Unsupervised | Supervised | ||

Logical Query Embedding | GQE [14] | Point | – | – | – |

Query2Box [21] | Box region | – | – | – | |

NewLook [22] | Box region | – | – | – | |

BetaE [23] | Beta distribution | – | – | – | |

Type Information Embedding | TypeDM and TypeComplex [19] | – | Entity-relation matching | 🗸 | |

CooccurX [20] | – | Entity-relation matching | 🗸 | ||

ProtoE [17] | – | Entity-relation matching | 🗸 | ||

AutoETER [18] | – | Relation-specific extraction | 🗸 | ||

TEMP [24] | Plug-in module | Message passing | 🗸 |

Items | Statistics | Training | Validation | Test | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Dataset | Entities | Relations | Single ^{1} | Complex ^{2} | Triples | Single | Complex | Triples | Single | Complex | Triples |

FB15k | 14,951 | 1345 | 273,710 | 273,710 | 483,142 | 59,097 | 8000 | 50,000 | 67,016 | 8000 | 59,071 |

FB15k-237 | 14,505 | 237 | 149,689 | 149,689 | 272,115 | 20,101 | 5000 | 17,526 | 22,812 | 5000 | 20,438 |

NELL-995 | 63,361 | 200 | 107,982 | 107,982 | 114,213 | 16,927 | 4000 | 14,324 | 17,034 | 4000 | 14,267 |

WN18 | 40,943 | 18 | 171,254 | 171,254 | 141,442 | 9006 | 3000 | 5000 | 9028 | 3000 | 5000 |

WN18RR | 40,943 | 11 | 103,509 | 103,509 | 86,835 | 5202 | 2000 | 3034 | 5356 | 2000 | 3034 |

YAGO3-10 | 51,374 | 36 | 64,420 | 40,000 | 53,554 | 3998 | 1500 | 2250 | 4160 | 1500 | 2333 |

^{1}Single: 1p logical query pattern.

^{2}Complex: other logical query patterns.

Datasets | Models | Avg | 1p | 2p | 3p | 2i | 3i | ip | pi | 2u | up |
---|---|---|---|---|---|---|---|---|---|---|---|

FB15k | GQE | 0.3979 | 0.6448 | 0.3508 | 0.2543 | 0.5434 | 0.6567 | 0.1491 | 0.3191 | 0.3865 | 0.2764 |

GQE-Double | 0.4058 | 0.6498 | 0.3578 | 0.2596 | 0.5566 | 0.6732 | 0.1564 | 0.3349 | 0.3828 | 0.2812 | |

Query2Box | 0.4872 | 0.7883 | 0.4175 | 0.3087 | 0.5908 | 0.7119 | 0.2111 | 0.4124 | 0.6122 | 0.3317 | |

UnTiCk (GQE) | 0.4484 | 0.6924 | 0.4071 | 0.3191 | 0.5843 | 0.6869 | 0.1846 | 0.3759 | 0.4576 | 0.3275 | |

UnTiCk (GQE-D) | 0.4577 | 0.6977 | 0.4124 | 0.3287 | 0.6016 | 0.7068 | 0.1937 | 0.3909 | 0.4571 | 0.3300 | |

UnTiCk (Q2B) | 0.5010 | 0.7913 | 0.4420 | 0.3547 | 0.6178 | 0.7310 | 0.2286 | 0.4413 | 0.6199 | 0.3631 | |

FB15k-237 | GQE | 0.2305 | 0.4044 | 0.2141 | 0.1557 | 0.2993 | 0.4179 | 0.0859 | 0.1728 | 0.1634 | 0.1613 |

GQE-Double | 0.2388 | 0.4100 | 0.2190 | 0.1577 | 0.3206 | 0.4374 | 0.0877 | 0.1851 | 0.1662 | 0.1656 | |

Query2Box | 0.2702 | 0.4692 | 0.2504 | 0.1893 | 0.3208 | 0.4486 | 0.1091 | 0.2087 | 0.2453 | 0.1902 | |

UnTiCk (GQE) | 0.2473 | 0.4286 | 0.2465 | 0.1932 | 0.2829 | 0.3999 | 0.0961 | 0.1795 | 0.2080 | 0.1914 | |

UnTiCk (GQE-D) | 0.2565 | 0.4378 | 0.2507 | 0.1943 | 0.3049 | 0.4258 | 0.1006 | 0.1906 | 0.2120 | 0.1922 | |

UnTiCk (Q2B) | 0.2753 | 0.4715 | 0.2619 | 0.2082 | 0.3056 | 0.4471 | 0.1122 | 0.2061 | 0.2562 | 0.2087 | |

NELL-995 | GQE | 0.2514 | 0.4262 | 0.2295 | 0.2060 | 0.3205 | 0.4585 | 0.0788 | 0.1840 | 0.2120 | 0.1468 |

GQE-Double | 0.2588 | 0.4282 | 0.2368 | 0.2110 | 0.3369 | 0.4821 | 0.0814 | 0.1930 | 0.2113 | 0.1481 | |

Query2Box | 0.3078 | 0.5549 | 0.2652 | 0.2354 | 0.3492 | 0.4822 | 0.1328 | 0.2113 | 0.3695 | 0.1693 | |

UnTiCk (GQE) | 0.2770 | 0.4165 | 0.2685 | 0.2659 | 0.3205 | 0.4585 | 0.0910 | 0.1972 | 0.2745 | 0.2000 | |

UnTiCk (GQE-D) | 0.2823 | 0.4166 | 0.2793 | 0.2731 | 0.3286 | 0.4678 | 0.0948 | 0.2020 | 0.2684 | 0.2104 | |

UnTiCk (Q2B) | 0.3189 | 0.5457 | 0.2936 | 0.2800 | 0.3431 | 0.4804 | 0.1349 | 0.2123 | 0.3781 | 0.2017 |

Datasets | Models | Avg | 1p | 2p | 3p | 2i | 3i | ip | pi | 2u | up |
---|---|---|---|---|---|---|---|---|---|---|---|

FB15k | GQE | 0.3371 | 0.5114 | 0.3056 | 0.2241 | 0.4614 | 0.5626 | 0.1375 | 0.2769 | 0.3066 | 0.2479 |

GQE-Double | 0.3440 | 0.5067 | 0.3113 | 0.2274 | 0.4775 | 0.5840 | 0.1449 | 0.2910 | 0.3024 | 0.2509 | |

Query2Box | 0.4153 | 0.6604 | 0.3795 | 0.2778 | 0.4934 | 0.6021 | 0.1933 | 0.3499 | 0.4762 | 0.3053 | |

UnTiCk (GQE) | 0.3909 | 0.5816 | 0.3721 | 0.2933 | 0.5025 | 0.6030 | 0.1703 | 0.3258 | 0.3697 | 0.3000 | |

UnTiCk (GQE-D) | 0.3987 | 0.5764 | 0.3773 | 0.2991 | 0.5201 | 0.6239 | 0.1782 | 0.3420 | 0.3649 | 0.3065 | |

UnTiCk (Q2B) | 0.4396 | 0.6760 | 0.4048 | 0.3208 | 0.5209 | 0.6266 | 0.2081 | 0.3775 | 0.4882 | 0.3334 | |

FB15k-237 | GQE | 0.2047 | 0.3469 | 0.1941 | 0.1430 | 0.2566 | 0.3631 | 0.0850 | 0.1583 | 0.1441 | 0.1509 |

GQE-Double | 0.2127 | 0.3503 | 0.1965 | 0.1477 | 0.2776 | 0.3860 | 0.0893 | 0.1690 | 0.1467 | 0.1512 | |

Query2Box | 0.2369 | 0.4033 | 0.2276 | 0.1760 | 0.2737 | 0.3769 | 0.1060 | 0.1847 | 0.2050 | 0.1793 | |

UnTiCk (GQE) | 0.2218 | 0.3754 | 0.2242 | 0.1794 | 0.2458 | 0.3541 | 0.0953 | 0.1663 | 0.1768 | 0.1751 | |

UnTiCk (GQE-D) | 0.2286 | 0.3772 | 0.2279 | 0.1815 | 0.2657 | 0.3774 | 0.0973 | 0.1748 | 0.1766 | 0.1792 | |

UnTiCk (Q2B) | 0.2414 | 0.4100 | 0.2397 | 0.1917 | 0.2654 | 0.3765 | 0.1061 | 0.1821 | 0.2080 | 0.1935 | |

NELL-995 | GQE | 0.2133 | 0.3161 | 0.1952 | 0.1769 | 0.2799 | 0.4072 | 0.0780 | 0.1667 | 0.1647 | 0.1349 |

GQE-Double | 0.2188 | 0.3189 | 0.1999 | 0.1797 | 0.2901 | 0.4274 | 0.0791 | 0.1744 | 0.1643 | 0.1352 | |

Query2Box | 0.2560 | 0.4145 | 0.2297 | 0.2106 | 0.2915 | 0.4183 | 0.1251 | 0.1918 | 0.2657 | 0.1566 | |

UnTiCk (GQE) | 0.2364 | 0.3228 | 0.2329 | 0.2371 | 0.2766 | 0.4039 | 0.0871 | 0.1815 | 0.2073 | 0.1781 | |

UnTiCk (GQE-D) | 0.2419 | 0.3253 | 0.2409 | 0.2408 | 0.2856 | 0.4157 | 0.0899 | 0.1846 | 0.2110 | 0.1837 | |

UnTiCk (Q2B) | 0.2658 | 0.4106 | 0.2494 | 0.2513 | 0.2879 | 0.4156 | 0.1253 | 0.1935 | 0.2766 | 0.1816 |

Datasets | Models | Avg | 1p | 2p | 3p | 2i | 3i | ip | pi | 2u | up |
---|---|---|---|---|---|---|---|---|---|---|---|

FB15k | GQE | 0.2185 | 0.3310 | 0.2009 | 0.1400 | 0.3213 | 0.4214 | 0.0786 | 0.1690 | 0.1492 | 0.1554 |

GQE-Double | 0.2234 | 0.3171 | 0.2032 | 0.1391 | 0.3392 | 0.4499 | 0.0826 | 0.1829 | 0.1411 | 0.1555 | |

Query2Box | 0.2904 | 0.5043 | 0.2790 | 0.1867 | 0.3442 | 0.4553 | 0.1183 | 0.2271 | 0.2879 | 0.2107 | |

UnTiCk (GQE) | 0.2758 | 0.4291 | 0.2759 | 0.2039 | 0.3696 | 0.4753 | 0.1006 | 0.2097 | 0.2116 | 0.2068 | |

UnTiCk (GQE-D) | 0.2829 | 0.4147 | 0.2804 | 0.2077 | 0.3895 | 0.5012 | 0.1072 | 0.2285 | 0.2029 | 0.2136 | |

UnTiCk (Q2B) | 0.3175 | 0.5326 | 0.3068 | 0.2235 | 0.3762 | 0.4871 | 0.1316 | 0.2537 | 0.3095 | 0.2367 | |

FB15k-237 | GQE | 0.1203 | 0.2277 | 0.1196 | 0.0818 | 0.1433 | 0.2455 | 0.0430 | 0.0868 | 0.0540 | 0.0806 |

GQE-Double | 0.1277 | 0.2295 | 0.1198 | 0.0861 | 0.1653 | 0.2719 | 0.0472 | 0.0963 | 0.0543 | 0.0785 | |

Query2Box | 0.1432 | 0.2791 | 0.1457 | 0.1077 | 0.1541 | 0.2472 | 0.0559 | 0.1025 | 0.0917 | 0.1049 | |

UnTiCk (GQE) | 0.1354 | 0.2614 | 0.1447 | 0.1104 | 0.1403 | 0.2418 | 0.0500 | 0.0944 | 0.0771 | 0.0988 | |

UnTiCk (GQE-D) | 0.1415 | 0.2587 | 0.1467 | 0.1114 | 0.1595 | 0.2681 | 0.0503 | 0.1027 | 0.0730 | 0.1032 | |

UnTiCk (Q2B) | 0.1495 | 0.2903 | 0.1604 | 0.1206 | 0.1467 | 0.2477 | 0.0554 | 0.1051 | 0.1017 | 0.1176 | |

NELL-995 | GQE | 0.1140 | 0.1481 | 0.1026 | 0.0952 | 0.1598 | 0.2852 | 0.0363 | 0.0960 | 0.0436 | 0.0594 |

GQE-Double | 0.1186 | 0.1513 | 0.1055 | 0.0963 | 0.1678 | 0.3082 | 0.0363 | 0.1012 | 0.0435 | 0.0569 | |

Query2Box | 0.1466 | 0.2309 | 0.1336 | 0.1298 | 0.1655 | 0.2883 | 0.0727 | 0.1164 | 0.1036 | 0.0786 | |

UnTiCk (GQE) | 0.1348 | 0.1719 | 0.1371 | 0.1492 | 0.1587 | 0.2850 | 0.0400 | 0.1090 | 0.0696 | 0.0925 | |

UnTiCk (GQE-D) | 0.1404 | 0.1742 | 0.1469 | 0.1520 | 0.1679 | 0.2990 | 0.0408 | 0.1115 | 0.0740 | 0.0970 | |

UnTiCk (Q2B) | 0.1563 | 0.2329 | 0.1522 | 0.1671 | 0.1651 | 0.2882 | 0.0712 | 0.1177 | 0.1147 | 0.0972 |

Datasets | Models | Avg | 1p | 2p | 3p | 2i | 3i | ip | pi | 2u | up |
---|---|---|---|---|---|---|---|---|---|---|---|

FB15k | GQE | 0.5574 | 0.8136 | 0.5046 | 0.3825 | 0.7212 | 0.8151 | 0.2485 | 0.4876 | 0.6125 | 0.4314 |

GQE-Double | 0.5687 | 0.8220 | 0.5142 | 0.3901 | 0.7362 | 0.8276 | 0.2643 | 0.5034 | 0.6196 | 0.4410 | |

Query2Box | 0.6422 | 0.9060 | 0.5799 | 0.4536 | 0.7564 | 0.8518 | 0.3384 | 0.5812 | 0.8112 | 0.5013 | |

UnTiCk (GQE) | 0.6090 | 0.8379 | 0.5621 | 0.4752 | 0.7474 | 0.8371 | 0.3039 | 0.5531 | 0.6749 | 0.4896 | |

UnTiCk (GQE-D) | 0.6173 | 0.8425 | 0.5729 | 0.4809 | 0.7586 | 0.8472 | 0.3137 | 0.5669 | 0.6723 | 0.5009 | |

UnTiCk (Q2B) | 0.6611 | 0.9013 | 0.6042 | 0.5096 | 0.7748 | 0.8646 | 0.3544 | 0.6108 | 0.7993 | 0.5312 | |

FB15k-237 | GQE | 0.3711 | 0.5741 | 0.3349 | 0.2625 | 0.4881 | 0.5976 | 0.1618 | 0.2965 | 0.3337 | 0.2909 |

GQE-Double | 0.3802 | 0.5773 | 0.3428 | 0.2704 | 0.5047 | 0.6140 | 0.1661 | 0.3101 | 0.3417 | 0.2950 | |

Query2Box | 0.4207 | 0.6402 | 0.3887 | 0.3128 | 0.5119 | 0.6225 | 0.2014 | 0.3432 | 0.4396 | 0.3257 | |

UnTiCk (GQE) | 0.3921 | 0.5945 | 0.3824 | 0.3192 | 0.4593 | 0.5758 | 0.1801 | 0.3062 | 0.3837 | 0.3275 | |

UnTiCk (GQE-D) | 0.4004 | 0.6023 | 0.3869 | 0.3228 | 0.4826 | 0.5906 | 0.1843 | 0.3147 | 0.3871 | 0.3327 | |

UnTiCk (Q2B) | 0.4251 | 0.6410 | 0.4042 | 0.3399 | 0.4925 | 0.6161 | 0.2016 | 0.3362 | 0.4413 | 0.3527 | |

NELL-995 | GQE | 0.4076 | 0.6047 | 0.3767 | 0.3252 | 0.5386 | 0.6546 | 0.1526 | 0.3044 | 0.4151 | 0.2967 |

GQE-Double | 0.4151 | 0.6111 | 0.3829 | 0.3341 | 0.5461 | 0.6666 | 0.1581 | 0.3151 | 0.4190 | 0.3033 | |

Query2Box | 0.4673 | 0.7116 | 0.4244 | 0.3652 | 0.5534 | 0.6757 | 0.2265 | 0.3417 | 0.5784 | 0.3289 | |

UnTiCk (GQE) | 0.4392 | 0.5912 | 0.4313 | 0.4095 | 0.5219 | 0.6429 | 0.1762 | 0.3239 | 0.4903 | 0.3660 | |

UnTiCk (GQE-D) | 0.4426 | 0.5915 | 0.4331 | 0.4088 | 0.5280 | 0.6502 | 0.1845 | 0.3263 | 0.4955 | 0.3659 | |

UnTiCk (Q2B) | 0.4765 | 0.6983 | 0.4419 | 0.4176 | 0.5448 | 0.6686 | 0.2275 | 0.3386 | 0.5913 | 0.3600 |

Anchor Entities | Relation | Target Entities | Query2Box Predication | UnTiCk (Q2B) Predication |
---|---|---|---|---|

The Last King of Scotland (film) | (1) films_in_this_ genre_reverse, (2) titles, (3) production_ companies | (1) Channel 4 (organization), (2) SONY (organization) | (1) Walt Disney Animation Studios (organization), (2) Walt Disney Studios Motion Pictures (organization), (3) drama film (film_genre), (4) Pixar (organization), (5) historical drama (film_genre) | (1) Walt Disney Animation Studios (organization), (2) Magnolia Pictures (organization), (3) Walt Disney Studios MotionPictures (organization), (4) Pixar (organization), (5) BBC (organization) |

(1) Chester County (location), (2) Latin Grammy Award for Album of the Year (award_category) | (1) contains, (2) award_reverse, (3) people_with_ this_profession_ reverse | (1) model (profession), (2) audio engineer (profession), (3) songwriter (profession) | (1) songwriter (profession), (2) Chester County (location), (3) composer (profession), (4) actor (profession), (5) artist (profession) | (1) songwriter (profession), (2) artist (profession), (3) actor (profession), (4) composer (profession), (5) guitarist (profession) |

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## Share and Cite

**MDPI and ACS Style**

Chen, D.; Li, Q.; Gu, J. UnTiCk: Unsupervised Type-Aware Complex Logical Queries Reasoning over Knowledge Graphs. *Electronics* **2023**, *12*, 1445.
https://doi.org/10.3390/electronics12061445

**AMA Style**

Chen D, Li Q, Gu J. UnTiCk: Unsupervised Type-Aware Complex Logical Queries Reasoning over Knowledge Graphs. *Electronics*. 2023; 12(6):1445.
https://doi.org/10.3390/electronics12061445

**Chicago/Turabian Style**

Chen, Deyu, Qiyuan Li, and Jinguang Gu. 2023. "UnTiCk: Unsupervised Type-Aware Complex Logical Queries Reasoning over Knowledge Graphs" *Electronics* 12, no. 6: 1445.
https://doi.org/10.3390/electronics12061445