9 research outputs found
Infinitary lambda calculus and discrimination of Berarducci trees
AbstractWe propose an extension of lambda calculus for which the Berarducci trees equality coincides with observational equivalence, when we observe rootstable or rootactive behavior of terms. In one direction the proof is an adaptation of the classical Böhm out technique. In the other direction the proof is based on confluence for strongly converging reductions in this extension
Discriminating Lambda-Terms Using Clocked Boehm Trees
As observed by Intrigila, there are hardly techniques available in the
lambda-calculus to prove that two lambda-terms are not beta-convertible.
Techniques employing the usual Boehm Trees are inadequate when we deal with
terms having the same Boehm 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, is BY = BYS, an equation valid in
the classical model P-omega of lambda-calculus, and hence valid with respect to
BT-equality but nevertheless the terms are beta-inconvertible. To prove such
beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey
information of the tempo in which the data in the BT are produced. Boehm Trees
are thus enriched with an intrinsic clock behaviour, leading to a refined
discrimination method for lambda-terms. The corresponding equality is strictly
intermediate between beta-convertibility and Boehm Tree equality, the equality
in the model P-omega. An analogous approach pertains to Levy-Longo and
Berarducci Trees. Our refined Boehm Trees find in particular an application in
beta-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 Boehm, 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'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257
Bohm-Like Trees for Rewriting
Klop, J.W. [Promotor]Raamsdonk, F. [Copromotor]Vrijer, R.C. de [Copromotor