Abstract
For a regular cardinal kappa, a formula of the modal mucalculus is kappacontinuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of kappadirected sets. We define the fragment C1 (x) of the modal mucalculus and prove that all the formulas in this fragment are aleph_1continuous. For each formula phi(x) of the modal mucalculus, we construct a formula psi(x) in C1 (x) such that phi(x) is kappacontinuous, for some kappa, if and only if psi(x) is equivalent to phi(x). Consequently, we prove that (i) the problem whether a formula is kappacontinuous for some kappa is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment C0(x) studied by Fontaine and the fragment C1 (x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal mucalculus. An ordinal alpha is the closure ordinal of a formula phi(x) if its interpretation on every model converges to its least fixedpoint in at most alpha steps and if there is a model where the convergence occurs exactly in alpha steps. We prove that omega_1, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, omega, omega_1 by using the binary operator symbol + gives rise to a closure ordinal.
BibTeX  Entry
@InProceedings{gouveia_et_al:LIPIcs:2017:7692,
author = {Maria Jo{\~a}o Gouveia and Luigi Santocanale},
title = {{Aleph1 and the Modal muCalculus}},
booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
pages = {38:138:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770453},
ISSN = {18688969},
year = {2017},
volume = {82},
editor = {Valentin Goranko and Mads Dam},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7692},
URN = {urn:nbn:de:0030drops76926},
doi = {10.4230/LIPIcs.CSL.2017.38},
annote = {Keywords: Modal mucalculus, regular cardinal, continuous function, aleph1, omega1, closure ordinal, ordinal sum}
}
Keywords: 

Modal mucalculus, regular cardinal, continuous function, aleph1, omega1, closure ordinal, ordinal sum 
Collection: 

26th EACSL Annual Conference on Computer Science Logic (CSL 2017) 
Issue Date: 

2017 
Date of publication: 

16.08.2017 