Step-Indexed Biorthogonality: a Tutorial Example

The purpose of this note is to illustrate the use of step-indexing [2] combined with biorthogonality [10, 9] to construct syntactical logical relations. It walks through the details of a syntactically simple, yet non-trivial example: a proof of the “CIU Theorem” for contextual equivalence in the untyped call-by-value λ-calculus with recursively defined functions. I took as inspiration two works: Ahmed’s step-indexed syntactic logical relations for recursive types [1] and Benton & Hur’s work on compiler correctness that combines biorthogonality with step-indexing [4]. The logical relation constructed here will come as no surprise to those familiar with these works. However, compared with Ahmed, we do not regard biorthogonality as “complex machinery” to be avoided—in my view it simplifies matters; and compared with Benton & Hur, I work entirely with operational semantics and with a high-level language. Both things are true of the recent work by Dreyer et al [7]; indeed I believe everything in this note can be deduced from their logical relation for call-by-value System F extended with recursive types, references and continuations. Nevertheless, it seems useful, for tutorial purposes, to extract a specific example of what the combination of step-indexing and biorthogonality can achieve, in as simple yet non-trivial a setting as possible. Of course there are other ways to prove the CIU theorem for untyped call-by-value λcalculus; for example, by using Howe’s method (see [12]). However, two points about the logical relation constructed here are of interest. First, and the main point of the technique as far as I am concerned, is the way step-indexing is used to break the vicious circle in the mixed-variance specification of the logical relation—see definition (10). Second is the fact that, unlike for some other forms of syntactical logical relation (see [11] for example), no compactness property (also known as an “unwinding theorem”) is needed to deal with recursively defined functions—see the proof of Lemma 4.3(ii).