Lambda the Ultimate

Andreas Abel and Brigitte Pientka's Well-Founded Recursion with Copatterns; a Unified Approach to Termination and Productivity is one of my highlights of the just-finished ICFP 2013, but it makes sense to focus on the first paper on this work, published at POPL back in January.

Copatterns: Programming Infinite Structures by Observations
Andreas Abel, Brigitte Pientka, David Thibodeau, Anton Setzer
2013

Inductive datatypes provide mechanisms to define finite data such as finite lists and trees via constructors and allow programmers to analyze and manipulate finite data via pattern matching. In this paper, we develop a dual approach for working with infinite data structures such as streams. Infinite data inhabits coinductive datatypes which denote greatest fixpoints. Unlike finite data which is defined by constructors we define infinite data by observations. Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data. This leads to a symmetric language design where pattern matching on finite and infinite data can be mixed. We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns. Our language naturally supports both call-by-name and call-by-value interpretations and can be seamlessly integrated into existing languages like Haskell and ML. We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.

Codata has been often discussed here and elsewhere. See notably the discussion on Turner's Total Functional Programming (historical note: this 2004 beautification of the original 1995 paper which had much of the same ideas), and on the category-theory-inspired Charity language. Given those precedents, it would be easy for the quick reader to "meh" on the novelty of putting "observation" first (elimination rather than introduction rules) when talking about codata; yet the above paper is the first concrete, usable presentation of an observation in a practical setting that feels right, and it solves long-standing problem that current dependently-typed languages (Coq and Agda) have.

Coinduction has an even more prominent role, due to its massive use to define program equivalence in concurrent process calculi; the relevant LtU discussion being about Davide Sangiorgi's On the origins of Bisimulation, Coinduction, and Fixed Points. The POPL'13 paper doesn't really tell us how coinduction should be seen with copatterns. It does not adress the question of termination, which is the topic of the more recent ICFP'13 paper, but I would say that the answer on that point feels less definitive.