Simon Fraser University
Engaging the World

Course Outlines

Summer 2026 - CMPT 981 G100

Special Topics in Theoretical Computing Science (3)

Decompositions&Orders -Algebra, Automata,Logic

Class Number: 4214

Delivery Method: In Person

Overview

  • Course Times + Location:

    May 11 – Aug 10, 2026: Tue, 2:30–4:20 p.m.
    Burnaby

    May 11 – Aug 10, 2026: Thu, 2:30–3:20 p.m.
    Burnaby

Description

COURSE DETAILS:

Course Title: Decompositions & Orders - Algebra, Automata, Logic

This course explores the interplay between algebra, automata, and logic, focusing on decompositions and order structures. The idea is to look at computational processes from different perspectives, hopefully, reconciling them at the end.

We will discuss regular and context-free languages, Myhill–Nerode equivalence, syntactic monoids, and minimal automata, and connect them to algebraic and order structures such as lattices, ideals, and Boolean/Heyting algebras.

We will study the abstract version of the Chinese Remainder Theorem (CRT), and will attempt to put it in the context of monoid factorization and algebraic semantics of logic. We will also present an algebraic version of the Compactness theorem. It is a seminar-style course cross-listed with CMPT 409.

Topics Include:

  • Partially ordered sets, Hasse diagrams
    meet/join operations, lattices, examples of lattices: subset, divisibility
  • Regular languages and automata (DFA, NFA), context-free languages and grammars
  • Myhill–Nerode equivalence theorem,
    minimal DFA construction
  • Syntactic (two-sided) equivalence, syntactic monoids, transition monoids and how they are connected
  • Chinese Reminder Theorem analogy motivation: independent monoid factors correspond to “coprime” components, Myhill–Nerode classes as products of factor classes,
    Connection to lattice of ideals/congruences
  • Compactness theorem for propositional logic using ultrafilters, Filters/ultrafilters as ``logical congruences'', Connection to MN equivalences and syntactic monoids
  • Interpretation of logical formulas in algebraic structures, MN classes as lattice atoms,
    Boolean/Heyting algebra semantics for logical formulas, logical decomposition via filters/ultrafilters
There are no formal prerequisites. 

Recommended pre-requisites: general mathematical maturity; some knowledge of the following topics will be especially useful:
- Formal languages and automata (DFA, NFA, regular languages)
- Introductory algebra (groups, rings)
- Basic logic and set theory

COURSE-LEVEL EDUCATIONAL GOALS:

Learning Objectives:

  • Compute Myhill–Nerode equivalence classes and minimal automata.
  • Understand syntactic monoids and their factorization.
  • Explore lattices of ideals/congruences and their role in decomposition,
  • Understand Boolean and Heyting algebras as algebraic logic semantics,
  • Integrate automata, algebra, and logic in examples and proofs.

Grading

NOTES:

Combination of theory, examples, exercises.
- Assessment: problem sets, quiz(s), midterm, final project.
Student presentations might also be included. Details will be discussed in the first week of classes.

Materials

RECOMMENDED READING:

Davey & Priestley, Introduction to Lattices and Order

Pin, Mathematical Foundations of Automata Theory

Kozen, Automata and Computability

Pin (ed.), Handbook of Automata Theory

Birkhoff, Lattice Theory

Almeida, Finite Semigroups and Universal Algebra

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

Department Graduate Notes:

The following are default policies in the School of Computing Science. Please check your course syllabus whether the instructor has chosen a different policy for your class, otherwise the following policies apply.
 
  • Students must attain an overall passing grade on the weighted average of exams in the course in order to get a C- or higher.
  • All student requests for accommodations for their religious practices must be made in writing by the end of the first week of classes, or no later than one week after a student adds a course. After considering a request, an instructor may provide a concession or may decline to do so. Students requiring accommodations as a result of a disability can contact the Centre for Accessible Learning (caladmin@sfu.ca).

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

At SFU, you are expected to act honestly and responsibly in all your academic work. Cheating, plagiarism, or any other form of academic dishonesty harms your own learning, undermines the efforts of your classmates who pursue their studies honestly, and goes against the core values of the university.

To learn more about the academic disciplinary process and relevant academic supports, visit: 


RELIGIOUS ACCOMMODATION

Students with a faith background who may need accommodations during the term are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion.