Parametricity and Modular Reasoning

Instructor: Derek Dreyer

Meeting time: Tue, Thu @ 2:00-4:00 PM

Place: Campus E1.5, Room 029 (videocast to KL)

Abstract

Abstract data types (ADTs) and other facilities for information hiding in programming languages (e.g. private fields, local variables) are widely considered to be crucial for supporting data abstraction and modularity, but what does that actually buy us in terms of reasoning about our code? As it turns out, it buys us a great deal, but formalizing what it buys us, especially in the context of modern programming languages, is quite tricky.

The formal essence of data abstraction was first characterized by John Reynolds in a landmark 1983 paper, "Types, Abstraction and Parametric Polymorphism", in which he introduced the concept of "relational parametricity" via his "abstraction theorem". The abstraction theorem formally establishes that the behavior of clients of an ADT must be unaffected by changes to the internal representation of the ADT that are preserved by its operations. However, Reynolds's original work only concerned pure System F, the polymorphic lambda-calculus, and there have since been decades of work on extending and generalizing his results to richer, more realistic languages supporting a host of computational effects.

In this course, we will start with Reynolds's work and build progressively toward semantic models of modern languages, such as Kripke logical relations and bisimulations models, which support very subtle and sophisticated forms of modular reasoning. To keep the formal material of the course in a unified framework, we will focus on models of data abstraction based on *operational* semantics, in the tradition of the work of Andrew Pitts.

As a basic prerequisite, students should be familiar with standard operational techniques, such as proofs by induction over operational semantics and type systems, which are covered in Pierce's TAPL book and Harper's PFPL book, among other sources. The grade will be based on homework assignments, student presentations on assigned papers, and class participation.

Along the way, we will explore a number of the following topics, possibly among others:

Schedule

Date and time

Topic

Presenter

Scribe

Other Files

Tue, 2012-10-16

Introduction

Derek

Dave

Thu, 2012-10-18

System F; Girard's method for proving termination

Derek

Dave

Girard, Lafont, Taylor (1990)
Gallier (1990)

Tue, 2012-10-23

Unary parametricity (applications of Girard's method)

Derek

Dave

Dave's writeup of Derek's FTLR proof

Thu, 2012-10-25

Class cancelled (Derek out of town)

Tue, 2012-10-30

Relational parametricity (Reynolds)

Derek

Dave

Reynolds (1983)
Homework #1

Thu, 2012-11-1

No class (All Saints' Day)

Tue, 2012-11-6

Definability of types by Church encodings

Derek

Dave

Plotkin, Abadi (1993)
Birkedal, M√łgelberg (2005)
Homework #2

Thu, 2012-11-8

Free theorems; short cut fusion

Derek

Dave

Wadler (1989)
Gill, Launchbury, Peyton Jones (1993)
Johann (2003)
Johann (2004)

Tue, 2012-11-13

Representation independence

Derek

Dave

Mitchell, Plotkin (1988)
Mitchell (1986)
Pitts' ATTAPL chapter (2005)

Thu, 2012-11-15

Recursion and admissibility

Derek

Dave

Pitts' lecture notes on denotational semantics

Tue, 2012-11-20

TT-closure

Derek

Dave

Pitts' ATTAPL chapter (2005)
Homework #3

Thu, 2012-11-22

TT-closure (continued)

Derek

Dave

Tue, 2012-11-27

TT-closure (continued); Completeness and CIU-equivalence

Derek

Dave

Homework #4

Thu, 2012-11-29

Unwinding Theorem; Contextual equivalence at existential types

Derek

Dave

Tue, 2012-12-4

Positive recursive types

Derek

Dave

Thu, 2012-12-6

First-class continuations

Derek

Dreyer, Neis, Birkedal (2010-12)

Tue, 2012-12-11

General recursive types, step-indexing

Derek

Appel, McAllester (2001)
Ahmed (2006)
Homework #5

Thu, 2012-12-13

General recursive types, step-indexing (continued)

Derek