Author:
Bonelli Eduardo,Kesner Delia,Viso Andrés
Abstract
When translating a term calculus into a graphical formalism many inessential
details are abstracted away. In the case of $\lambda$-calculus translated to
proof-nets, these inessential details are captured by a notion of equivalence
on $\lambda$-terms known as $\simeq_\sigma$-equivalence, in both the
intuitionistic (due to Regnier) and classical (due to Laurent) cases. The
purpose of this paper is to uncover a strong bisimulation behind
$\simeq_\sigma$-equivalence, as formulated by Laurent for Parigot's
$\lambda\mu$-calculus. This is achieved by introducing a relation $\simeq$,
defined over a revised presentation of $\lambda\mu$-calculus we dub $\Lambda
M$.
More precisely, we first identify the reasons behind Laurent's
$\simeq_\sigma$-equivalence on $\lambda\mu$-terms failing to be a strong
bisimulation. Inspired by Laurent's \emph{Polarized Proof-Nets}, this leads us
to distinguish multiplicative and exponential reduction steps on terms. Second,
we enrich the syntax of $\lambda\mu$ to allow us to track the exponential
operations. These technical ingredients pave the way towards a strong
bisimulation for the classical case. We introduce a calculus $\Lambda M$ and a
relation $\simeq$ that we show to be a strong bisimulation with respect to
reduction in $\Lambda M$, ie. two $\simeq$-equivalent terms have the exact same
reduction semantics, a result which fails for Regnier's
$\simeq_\sigma$-equivalence in $\lambda$-calculus as well as for Laurent's
$\simeq_\sigma$-equivalence in $\lambda\mu$. Although $\simeq$ is formulated
over an enriched syntax and hence is not strictly included in Laurent's
$\simeq_\sigma$, we show how it can be seen as a restriction of it.
Publisher
Centre pour la Communication Scientifique Directe (CCSD)