Category calculus and algebra for multi-model databases

1
CATEGORICAL CALCULUS AND
ALGEBRA FOR MULTI-MODEL
DATA
Jiaheng Lu
Department of Computer Science
University of Helsinki, FInland

• Multi-model Databases
• Categorical Algebra and Calculus
• Algebraic Transformation Rules
• Conclusion
• Note: this talk will involve more in database knowledge, but only use basic
knowledge in category theory, such as limit and thin category.
2
OUTLINE

• Data come from different sources and have different formats
3
BIG DATA
Camera
Smart phone Social media
Acknowledgement: Icons created by Freepik - Flaticon

www.helsinki.fi
Variety challenge of big data
Customer
Social
Media
Gaming
Entertain
ment
Banking
Finance
Personal
Information
Purchase Customers
have different
types of data.
4
Customer-360-Degree-View

An example of different data and query
in databases
Marry (1)
John (2)
Friend
Friend
William (3)
{"Order_no":"0c6df508",
“Orderlines": [
{ "Product_no":"2724f”
“Product_Name":“Toy",
"Price":66 },
{ "Product_no":“3424g”,
"Product_Name":“Book",
"Price":40 } ]
}
Customer_ID Name Credit_limits
1 Mary 5,000
2 John 3,000
3 William 2,000
"1" -- > "34e5e759"
"2"-- > "0c6df508"
Persons made the order:
Social network
Table
Order information:

An example of different data and query
Marry (1)
John (2)
Friend
Friend
William (3)
“Orderlines": [
"Price":66 },
"Price":40 } ]
}
1 Mary 5,000
2 John 3,000
3 William 2,000
"1" -- > "34e5e759"
"2"-- > "0c6df508"
Persons made the order:
Social network
Table
Order information:

One application with different models of data
How to integrate those heterogenous data to provide a
unified service?
●Relational data: customer databases
●Graph data: social networks
●Hierarchical data: catalog, product
●Key-value data: orders by customers
7

Multi-Model Databases System
Tabular
RDF
XML
Spatial
Text
Multi-model database
systems
JSON
• One unified database system for multi-model data
8

What is DBMS?
• A Database Management System (DBMS) is software designed to
efficiently manage data, with traditional systems storing data in the form of
tables (RDBMS).
9
Student ID First name Last name Department
001 John Smith Biology
002 Emily Johnson Physics
003 Michael Brown History
004 Sarah Davis English
Students Relational Table

What is multi-model database management
system
● A multi-model database management system (MMDBMS) is
designed to support multiple data models against a single,
integrated backend.
● Document, graph, relational and key-value models are examples
of data models that may be supported by a multi-model database.
10

Four models of data
Marry (1)
John (2)
Friend
Friend
William (3)
“Orderlines": [
"Price":66 },
"Price":40 } ]
}
1 Mary 5,000
2 John 3,000
3 William 2,000
"1" -- > "34e5e759"
"2"-- > "0c6df508"
Key-value:
Graph:
Relation:
Document:

Advantages of MMDBMS over the traditional
relational database
• Handling diverse data types
• Handle various types of data, such as graph, relation, document and
key-value data and more models
• Enhanced query capabilities
• Support content-based search for multi-model data or spatial queries for
geospatial data.
• Flexible schema
• Greater flexibility in schema design and evolution. Relational DBMS has
the fixed database schema definitions.
12

Multi-model databases products
…

• Supporting graph, document, key/value and object models.

Marry (1)
John (2)
Friend
Friend
William (3)
“Orderlines": [
"Price":66 },
"Price":40 } ]
}
1 Mary 5,000
2 John 3,000
3 William 2,000
"1" -- > "34e5e759"
"2"-- > "0c6df508"
Query: Return all products which are ordered by a friend of a customer
whose credit_limit>4000
Answer: John is a friend of Mary (the credit_limit of Mary > 4000)

Query language of OrientDB:
SELECT
expand(out("Knows").Orders.orderlines.Product_no)
FROM Customers
WHERE CreditLimit > 4000
Recommendation query:
Return all products which are ordered by any
friend of a customer whose credit_limit>4000

Challenge: a new theory foundation
Research goal:
Call for a unified model and theory for multi-model data!
The theory of traditional relations is not adequate to
mathematically describe modern database systems.

One possible theory foundation:
Category Theory
● Introduced to mathematics world by Samuel Eilenberg
and Sauders MacLane in 1944
● Developed for a unified language of topology and algebra
Samuel Eilenberg Sauders MacLane

Set Category
● Databases are set categories:
○ Objects are sets and morphisms are functions
● We assume that it is a thin Category (or Posetal Category)
○ Given a pair of objects X and Y in a category C, and any two morphisms f ,
g: X → Y , we say that C is a thin category if and only if the morphisms f
and g are equal.
19

An example of Categorical Unification
20
Unified
Category
for three
types of
data

Relational algebra and relational calculus
● In the field of relational databases, relational algebra and
relational calculus are developed as two formal languages for
query databases.
● Similarly, categorical algebra and categorical calculus are
developed to query category databases.
21

Relational algebra
• Operators:
• Selection:  (sigma)
• Projection: 
• Union: 
• Intersection : 
• Difference: -
• Cartesian Product: 
• Derived operators:
• Joins (equi-join) ⋈

Categorical algebra
• Set operators:
• Unary operator:
• Map: f
• Selection: 
• Projection: 
• Binary operator:
• Division: ÷
• getParent(D1,D2) (tree data)
• getAncestor(D1,D2) (tree data)
• Tenary operator:
• getReach(S,T,E) (graph data)
• getNHop(S,T,E) (graph data)

Categorical algebra
• Category operators:
• Sets and Functions to Category:
• Cat(S1,...,Sn, f1 : Si1 → Sj1 ,..., fm : Sim → Sjm )
• This operator, called Categorification, constructs a category using a
given set of objects and morphisms.
• Category to Set
• Limit which converts a category into a relational object (set).
• Lim(Cat(S1,...,Sn, f1 : Si1 → Sj1 ,..., fm : Sim → Sjm ))

Example of categorical algebra:
Selection
Query: Find all courses taken by ”Smith”
S1:=  student · Last_name=“Smith” (Registration)
S2:= S1 ·Course ·Title
Return S2

Example of categorical algebra: Division
Query: Find the titles of courses taken by all students
S1:= Registration[Student]÷Student
S2:= S1 · Course · Title
Return S2

28
Two examples of query plan with
categorical algebra

Categorical calculus
• Categorical calculus, a declarative language for describing results in the
category; Categorical algebra, a procedural language for listing operations
in the category.
• The formulae of the Categorical Calculus

Reachable
predicate from
node x1 to x2

Ancestor predicate
to determine the
relationship
between two
nodes in trees.

Unsafe
variables refer
to a variable
that has
possibly infinite
number of
values or is
unbounded.

Categorical calculus and categorical
algebra are equivalent (I)
Query: Find all courses taken by ”Smith”
S1:=  student · Last_name=“Smith” (Registration)
S2:= S1 ·Course ·Title
Equivalent calculus:
{ 𝑥 | 𝑥 ∈ 𝑇𝑖𝑡𝑙𝑒, ∃𝑦 ∈ 𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛, 𝑦 ∙ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 ∙ 𝐿𝑎𝑠𝑡_𝑁𝑎𝑚𝑒 =
"Smith"˄ 𝑦 ∙ Course ∙ Title= 𝑥}

Query: Find the titles of courses taken by all students
S1:= Registration[Student]÷Student
S2:= S1 · Course · Title
Equivalent calculus:
{ 𝑥 | 𝑥 ∈ 𝑇𝑖𝑡𝑙𝑒, ∀𝑦 ∈ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡, ∃𝑟 ∈ 𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛, 𝑟 ∙ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 = 𝑦 ˄ 𝑟 ∙
Course ∙ Title= 𝑥}
Categorical calculus and categorical
algebra are equivalent (II)

• Multi-model Databases
• Categorical Algebra and Calculus
• Algebraic Transformation Rules (Query optimization)
• Conclusion
35
OUTLINE

Algebraic transformation rules (I)
• Rewrite the algebraic operators for query optimization.
• Limit and Projection:
• Pushing σ to one or multiple objects in lim

Algebraic transformation rules (II)
• Pushing σ to one or multiple objects in getReach:
• Commuting function mapping with the product operator.
• Commuting 𝜋 with the Lim operation.

Algebraic transformation rules (III)
• Commuting g with the Lim operation..
• The following diagram holds:
then the two operators g and Lim can be commuted as follows:

Algebraic transformation rules (III)
• Commuting g with the Lim operation..
• The following diagram holds:
then the two operators g and limit can be commuted as follows:

40
An optimization query plan with
algebraic operators transformation

• Theorem 13. Categorical calculus and categorical algebra can
express all of the following:
• Relational calculus and algebra queries;
• Graph pattern matching and graph reachability queries;
• XML twig pattern queries.
41
EXPRESSIBILITY POWER

• Previous works use category theory on relational databases, but
this paper focuses on multi-model databases.
• Libkin and Wong (1997) showcase the connection between database
operations and the categorical notion of a monad.
• Schultz and Spivak (2016) introduce a categorical query language
that serves as a data integration scripting language
• …..
• There are existing algebra and calculus for relational data, graph
data, and object-oriented data, but not multi-model data.
43
RELATED WORK

• Model multi-model databases as thin set category
• Define categorical algebra and calculus, and their equivalence
• Develop the algebraic transformation rules for query optimization
44
MAIN CONTRIBUTION AND CONCLUSION
Applied category theory (ACT) here can contribute
to practical query processing and optimization
of multi-model databases.

• Jeremy Gibbons, Fritz Henglein, Ralf Hinze & Nicolas Wu (2018): Relational algebra
by way of adjunctions. Proc. ACM Program. Lang. 2(ICFP), pp. 86:1–86:28.
• Leonid Libkin & Limsoon Wong (1997): Query Languages for Bags and Aggregate
Functions. J. Comput. Syst. Sci. 55(2), pp. 241–272.
• Patrick Schultz, David I. Spivak, Christina Vasilakopoulou & Ryan Wisnesky (2016):
Algebraic Databases. arXiv:1602.03501.
• Allen Van Gelder & Rodney W. Topor (1991): Safety and translation of relational
calculus. ACM Trans Database Syst. 16(2), p. 235–278, doi:10.1145/114325.103712.
Available at https://doi.org/10.1145/114325.103712.
• Jiaheng Lu, Irena Holubová: Multi-model Databases: A New Journey to Handle the
Variety of Data. ACM Comput. Surv. 52(3): 55:1-55:38 (2019)
• Jiaheng Lu: A Categorical Unification for Multi-Model Data: Part I Categorical Model
and Normal Forms. CoRR abs/2502.19131 (2025)
45
REFERENCES

Category calculus and algebra for multi-model databases

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