Program Equivalence with Names

The nu-calculus of Pitts and Stark was introduced as a paradigmatic functional language with a very basic local-state effect: references of unit type. These were called names, and the motto of the new language went as follows. “Names are created with local scope, can be tested for equality, and are passed around via function application, but that is all.” Because of this limited framework, the hope was that fully abstract models and complete proof techniques could be obtained. However, it was soon realised that the behaviour of nu-calculus programs is quite intricate, and program equivalence in particular is surprisingly difficult to capture. Here we focus on the following “hard” equivalence, new x, y in λf.(fx = fy) ≡ λf. true where x, y are names, and f : Name → Bool. We examine attempts and proofs of the above, explain the advantages and disadvantages of the proof methods and discuss why program equivalence in this simple language remains to date a mystery.