Please file bugs with the course materials, including this website, here:

Virtual Course Structure

Course lectures will be delivered virtually through Canvas, preferring Zoom which offers better performance and bandwidth in my experience, but using Collaborate Ultra as a fallback. All lectures will be recorded, so asynchronous participation is possible, although I strongly encourage students to attend synchronously to ask questions and engage with in-class activities.

You are encouraged but not required to use video chat. Video chat will help everyone engage and connect with each other and make for a more engaging course. However, we know that this may not be possible for all those attending the course.

You are also encouraged but not required to attend lectures synchronously. Lectures will be recorded and provided to you in some way (most likely through Canvas).

The course staff will hold regular synchronous office hours (schedule can be found under Resources), and you can email us to schedule meetings if these time slots do not work for you.

We will heavily use Piazza (see below) to facilitate additional asynchonous activities and discussion.

The course will rely on an honour code and not make use of invasive invigilation software.

Course Description

Programming languages are a fundamental part of computer science. This course introduces the formal tools needed to describe precisely what a program means. These tools help us answer many useful questions about program analyses and transformations, such as:

Topics include:


This course is intended for graduate students in computer science. There are no formal course prerequisites, but you are expected to have the kind of mathematical maturity typical of one who has taken an undergraduate discrete math or theory of computation course. We will explicitly cover proof techniques in this course, so don't worry if you are rusty or not very familiar. Here are some resources I find helpful for refreshing or improving your skill at writing proofs:

Familiarity with a functional programming language (e.g., Scheme, Racket, ML, Haskell) is useful but is not required. We will use the Racket programming language at times in the course to help reinforce the connection between the mathematics and programs/programming. I will introduce any needed programming concepts in class.


To facilitate discussion among students in the class and myself, we are using the Piazza Q&A platform. The system allows you to ask questions, refine answers as a group, carry on followup discussions, and disseminate relevant information. Rather than emailing questions to me, I ask that you post your questions to Piazza. If you have any problems or feedback for the developers, email
Find our class page here .

This course has no required textbook. Material will primarily be provided in a set of notes and/or covered in class, as well as through some supplementary readings. The material we cover will draw from a variety of sources.

The following books are recommended but not required: Some other useful texts that provide a different perspective or more depth in some areas are:


There will be an in-class midterm exam (date TBD).

Final Presentation

Students will give a final presentation to the class on a topic of their own choosing related to Programming Language Principles.


There will be approximately 6 homework assignments during the course of the semester. Homeworks will I recommend that homeworks be typeset using the LaTeX document preparation system, but will not require it: you have the option to prepare your homework by hand, so long as you make sure that it is clearly legible by me. I plan to provide LaTeX templates for you, so this is a good chance to learn one of the more common tools for writing academic computer science papers, though the learning curve may be steep at first. I'm happy to give guidance on how to work with LaTeX (though I probably don't know all the latest tricks). You can turn in assignments electronically as PDFs either scanned or generated by LaTeX.

Assignments must be your individual work. You may discuss the homeworks with others, but you must write up and hand in your own solutions. In particular, follow the whiteboard policy: at the end of the discussion the "whiteboard" must be erased and you must not transcribe or take with you anything that has been written on the board (or elsewhere) during your discussion. You must be able to reproduce the results solely on your own after any such discussion.

Do not draw upon solutions to assignments (or in notes) from similar courses, nor use other such materials (e.g., programs) from any website or other external source in preparing your work.

Grading Policy

The final grade will be comprised of the following components, with the following plan for distribution of marks (subject to revision):


The following resources are to help you succeed in the class.

Course Schedule

The following is a draft course schedule, based on a prior offering of the course. The exact details (including some topics) will vary depending on the content covered in class and the interests and needs of the students (and myself).

I often update the notes as the term goes along. They are timestamped, so that you can tell when the most recent version was uploaded (note that the timestamp is distinct from the original date of creation).

Under Construction!
# Date Topics Notes Supplementary Readings
0 Wed, Sep 9 Course Introduction
1 Mon, Sep 14 Modeling Programming Languages; Set Theory and Logic Modeling Languages
Set Theory
Proof Techniques
2 Wed, Sep 16 More on Set Theory and Logic
3 Mon, Sep 21 B: A Language with many programs and few results
Inductive Definitions
Derivations as Data Structures
Inductive Definitions
Proof by Induction
Proving Something False
4 Wed, Sep 23 Principles of Induction
Derivations cont'd.
From Inversion to Parsing
Parsing as Proof Search; Parsers as Theorem Provers
Proof Techniques
Proving Something False
5 Mon, Sep 28 HW1 Review
6 Wed, Sep 30 Operational Semantics
  • Natural (Big-Step) Semantics
  • Structural Operational (Small-Step) Semantics (S.O.S.)
  • Reduction Semantics
  • Abstract Machines
IMP: An Imperative Programming Language
BS Notes
Video on Operational Semantics
SS Notes
RS Notes
AM Notes
7 Mon, Oct 5 Postcards from 509
Backwards Reasoning Principles
Induction, sure, but on which thing?!?
8 Wed, Oct 7 More Imp: From Inversion to Interpreters
A Taste of Divergence
Big Picture Review
Formal propositional logic and Formal English
9 Mon, Oct 12 Thanksgiving: No Class
10 Wed, Oct 14 HW2 Highlights
The Truth™ About Inductive Definitions and Induction Principles
Coinductive Definition: A Counterpoint to Inductive Definition
Modeling Divergence
Coinduction Notes
11 Mon, Oct 19 Proof by Coinduction
12 Wed, Oct 21 Lexical Variables 1 SS Notes
RS Notes
AM Notes
Lexical Binding Notes
13 Mon, Oct 26 Lexical Variables 2
Introduction to Procedures
Lexical Binding Notes Procedures
Using Procedures
14 Wed, Oct 28 Store-Passing Semantics and Mutable References
Mutable References
15 Mon, Nov 2 Compilers: Semantics-Preserving Cross-Language Translators
Variable Scoping and Environments
16 Wed, Nov 4 Lexical Scoping vs. Dynamic Scoping
Choose Your Own Induction Principle
Environments and Scoping Induction Unchained!
17 Mon, Nov 9 Choose Your Own Induction Principle
Induction Unchained!
18 Wed, Nov 11 Rememberance Day: No Class
19 Mon, Nov 16 Proper Tail Calls
Abstracting Abstract Syntax (Gödel Numbering)
Tail Calls
20 Wed, Nov 18 Type Systems
21 Mon, Nov 23 Type Systems 2
22 Wed, Nov 25 Braxton and Noa on Typescript and Gradual Typing
Alison and James on Church Encoding (sans rap-battle)
23 Mon, Nov 30 Katharine and Ramon on SIMD semanticsimplementing binding and alpha equivalence
Finn, John, and Arsh on Formal Logic and Coq
24 Wed, Dec 2 No class; Exam Assigned