Discriminating Lambda-Terms Using Clocked Boehm Trees

As observed by Intrigila [16], there are hardly techniques available in the λ-calculus to prove that two λ-terms are not β-convertible. Techniques employing the usual Böhm Trees are inadequate when we deal with terms having the same Böhm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott [24], is BY = BYS, an equation valid in the classical model Pω of λ-calculus, and hence valid with respect to BT-equality =BT, but nevertheless the terms are β-inconvertible. To prove such β-inconvertibilities, we employ ‘clocked’ BT’s, with annotations that convey information of the tempo in which the data in the BT are produced. Böhm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for λ-terms. The corresponding equality is strictly intermediate between =β and =BT, the equality in the model Pω. An analogous approach pertains to Lévy–Longo and Berarducci Trees. Our refined Böhm Trees find in particular an application in β-discriminating fixed point combinators (fpc’s). It turns out that Scott’s equation BY = BYS is the key to unlocking a plethora of fpc’s, generated by a variety of production schemes of which the simplest was found by Böhm, stating that new fpc’s are obtained by postfixing the term SI, also known as Smullyan’s Owl. We prove that all these newly generated fpc’s are indeed new, by considering their clocked BT’s. Even so, not all pairs of new fpc’s can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call ‘atomic clock’.