I discuss some examples posted on my blog, QA9, which show that executables produced by ghc (the main implementation of Haskell) can exhibit non-optimal beta-reduction.  Thanks to Victor Maia for major help with these.

In this episode I talk about how to represent lambda terms as graphs with duplicator nodes for splitting edges corresponding to bound variables.  I also start discussing the beginning of Lampings' abstract algorithm for optimal beta-reduction, in particular how we need to push duplicators inside lambda abstractions to initiate a lazy duplication.

We discussed last time how with a graph-sharing implementation of untyped lambda calculus, it can happen that you are forced to break sharing and copy a lambda abstraction.  We discuss in this episode the central issue with doing that, namely copying redexes and copying applications which could turn into redexes following other beta reductions.  The high-level idea of the proposed solution is also discussed, namely lazy graph duplication.

Some background on optimal beta reduction: Levy, Lamping.  The main problem to overcome is duplicating a lambda abstraction that is used in two different places in your term.  The solution is to try to duplicate it incrementally.

Many termination checkers support lexicographic (structural) recursion.  The lexicographic combination of orderings on sets A and B is an ordering on A x B where a pair decreases if the A component does (and then the B component can increase unboundedly) or else the A component stays the same and the B component decreases.  Connections with nested recursion and ordinals discussed.

Another type-based approach to termination-checking for recursive functions over inductive datatypes is to use so-called Mendler-style iteration.  On this approach, we write recursive functions by coding against a certain interface that features an abstract type R, which abstracts the datatype over which we are recursing; and a function from R to the result type of the recursion.  Subdata of the input data are available at type R only, not at the original datatype.  This allows us to make explicit recursive calls, but only on subdata.

Well-founded recursion is a technique to turn recursion which decreases along a well-founded ordering into a structural recursion.

Discussion of a compositional method of termination checking using so-called sized types.  Datatypes are indexed by sizes, and recursive calls can only be made on data of strictly smaller size than the data as input to the recursion.  Since the method is type-based, it is compositional: we can break out helper functions from a recursive function and not upset the termination checker.  A readable and interesting tutorial on the subject is here.

Review of need for termination analysis for recursive functions on inductive datatypes.  Discussion of a serious problem with syntactic termination checks, namely noncompositionality.  A function may pass the syntactic termination check, but abstracting part of it out into a helper function may result in code which no longer passes the check.  So we need a compositional termination check, which will be discussed in subsequent episodes.

Start of Chapter 8 of the podcast, on termination checking in type theory, and strong functional programming.  Discussion of a little history of adding datatypes to the original pure type theory of Coq (called the Calculus of Constructions).  Considering the most basic form of termination checking, which is checking syntactically that recursive calls are only made on subdata of the data input to the recursive function.