Decidability of All Minimal Models (Revised Version - 2012)

This unpublished note is an alternate, shorter (and hopefully more readable) proof of the decidability of all minimal models. The decidability follows a proof of the existence of a cellular term in each observational equivalence class of a minimal model. The first proof I gave of the decidability of all minimal models [3] was far from being easy to understand. I was only an inexperienced student at the time, struggling to solve the problem of the decidability of Higher Order Matching. I was trying to generalise to order five my decidability result at order four ([4], [5]) when I realized that every solution of an atomic matching problem (a problem whose right-members are constants of ground type) could be transformed into a cellular term (the so-called “transferring” terms in [3]), a term of very simplified structure. The decidability of atomic matching followed immediately from this key-result. A few months later, at the open-problem session of TLCA 1995, Ralph Loader pointed out that another immediate consequence of this decidability result was the existence of a computable selector for the observational equivalence classes of the minimal models of simply-typed lambda-calculus. Because I was so immersed in the Matching Problem and wanted to prove the decidability of atomic matching at each order, it seemed very natural to prove the existence of cellular representatives by induction on the order of terms. Unfortunately this choice was probably the worst I could make, and resulted in a long, tedious and obfuscated proof. Two years later, using the same techniques, Ralph Loader proved the decidability of Unary PCF [2]. Loader’s proof was then drastically simplified by Manfred Schmidt-Schauß [6] who gave a clever, simple and beautiful algorithm to compute a selector for Unary PCF – a fortiori for every minimal model. This is the point where I realized that something was probably wrong with my own proof : even if the decidability of all minimal models followed from the existence of cellular representatives at each order, the latter property was actually independant from the first, and clearly required a proof by induction on the length of terms. This unpublished note – written a few years ago – presents a short and simple proof of the existence for each term of an observationally equivalent cellular term, followed by a proof of decidability of each minimal model. The proof considers only one ground type, but can be easily extended to finitely many ground types (if you feel it is really necessary, you can try to read [3], or even [4] if you can read French).