15-814 Types and Programming Languages

Course Information

Time: Tue-Thu 12:00-1:20
Room: GHC 4303
Instructors: Karl Crary
Teaching Assistant: Carlo Angiuli
Office Hours
Crary: TBA GHC 9217
Angiuli: TBAGHC 9227


This is an introductory course on the foundations of programming languages. The central organizing principle of language design is the identification of language features with types. The theory of programming languages, therefore, reduces to the theory of types. Type theory is a comprehensive foundational theory of computation. Type theory has its origins in proof theory (the theory of human reason) and is closely related to category theory (the general theory of mathematical structures). The tripartite relationship between type theory, proof theory, and category theory is fundamental to the study of programming languages. We will place special emphasis on the propositions-as-types correspondence, in which programs of a type are identified with proofs of the corresponding proposition. (Similar correspondences hold between type theory and category theory (types are objects, programs are maps) and between category theory and proof theory (propositions are objects, proofs are maps), but we shall not consider these in this course.)



Robert Harper, Practical Foundations for Programming Languages , Spring 2012.


Jean-Yves Girard (with Yves Lafont and Paul Taylor), Proofs and Types, Cambridge University Press, 1990. (Out-of-print, available on-line as linked.)
Benjamin C. Pierce, Types and Programming Languages , MIT Press, 2002.
Benjamin C. Pierce, ed., Advanced Topics in Types and Programming Languages , MIT Press, 2005.


You are required to achieve a grade of B on each homework assignment. If you fail to turn in an assignment on time, or if your initial grade is F, then you will not have the opportunity to re-do the assignment, and will fail the course. If your initial grade is D or C, you must revise it to a B standard within one week. This policy ensures that the emphasis is properly on learning the course material, and not on "getting through it."

All homeworks are due at the beginning of lecture on the due date. Homeworks are to be submitted in PDF format (only!), and must be typeset. All homeworks are to be submitted by sending the PDF via e-mail to Carlo with "15-814 Homework" as the subject line. No late homeworks will be accepted. Any re-do's must be finished within one week of their being returned to you. Only one re-do is permitted per assignment.

Final Exam

There will be a 24-hour take-home final examination during the exam period at the end of the semester.

Academic Integrity

Unless explicitly instructed otherwise, all homework and exam work is to be solely your own, and may not be shared with or borrowed from any other person in the course. You are not permitted to draw upon assignments or solutions from previous instances of the course, nor to use course materials (such as assignments or programs) obtained from any web site or other external source in preparing your work.

You may discuss homework assignments with other students in the class, but you must adhere to the whiteboard policy. At the end of discussion the whiteboard must be erased, and you must not transcribe or take with you anything that has been written on the board during your discussion. You must be able to reproduce the results solely on your own after any such discussion.


CS PhD students are assigned a pass/fail grade in the University grading system, but are given an internal letter grade for Black Friday purposes. A final letter grade of B is required to pass this course. To achieve this, you must have (1) completed all homework assignments with a grade of B; and (2) earned a grade of B or better on the final exam.

Undergraduate students and students in other programs will be assigned letter grades according to the same policies used to assign internal letter grades for CS PhD students.

Schedule of Lectures

Date Topic Reading Homework
Sep 10 λ-Calculus PFPL 1-2,17
17 λ-Calculus HW1 out.
(handout [source] [macros])
24 λ-Calculus PFPL 3-6 (optional) HW1 due.
26 HW2 out.
(handout [source] [macros])
Oct 1 Type Safety PFPL 4-6
3 HW2 due.
8 PFPL 8-9
10 Gödel's T
15 PCF PFPL 10
17 Product and Sum Types PFPL 11-12 HW3 out.
(handout [source] [macros])
22 Recursive Types PFPL 15-16
24 Polymorphism PFPL 20
29 Parametricity and Predicativity PFPL 49 (optional)
31 Existential Types PFPL 21 HW3 due.
Nov 5 Observational Equivalence PFPL 47, 49
7 HW4 out.
(handout [source] [macros])
14 Abstraction Theorems
19 Existential Types PFPL 21
21 The Curry-Howard Isomorphism PFPL 30 HW4 due.
HW5 out.
(handout [source] [macros])
26 Mutable State PFPL 36
28 (Thanksgiving: no class)
Dec 3
5 Subtyping HW5 due.

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