by Lutz Schröder
Abstract:
Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.
Reference:
Lutz Schröder: Expressivity of Coalgebraic Modal Logic: The Limits and Beyond, In Vladimiro Sassone, ed.: Foundations of Software Science And Computation Structures, Lecture Notes in Computer Science, vol. 3441, pp. 440–454, Springer; Berlin; http://www.springer.de, 2005. [preprint]
Bibtex Entry:
@InProceedings{Schroder05,
author = {Lutz Schr{\"o}der},
title = {Expressivity of Coalgebraic Modal Logic: The Limits and Beyond},
year = {2005},
editor = {Vladimiro Sassone},
booktitle = {Foundations of Software Science And Computation Structures},
publisher = {Springer; Berlin; http://www.springer.de},
series = {Lecture Notes in Computer Science},
volume = {3441},
pages = {440-454},
keywords = {coalgebra expressivity modal logic predicate lifting natural relation},
url = {http://www.springerlink.com/openurl.asp?genre=article&issn=0302-9743&volume=3441&spage=440},
comment = { <a href = "http://www8.informatik.uni-erlangen.de/~schroeder/papers/coalgml.pdf"> [preprint] </a>},
psurl = {http://www8.informatik.uni-erlangen.de/~schroeder/papers/coalgml.ps},
abstract = {Modal logic has a good claim to being the logic of choice for describing
the reactive behaviour of systems modeled as coalgebras. Logics with
modal operators obtained from so-called predicate liftings have been
shown to be invariant under behavioral equivalence. Expressivity results
stating that, conversely, logically indistinguishable states are
behaviorally equivalent depend on the existence of separating sets of
predicate liftings for the signature functor at hand. Here, we provide a
classification result for predicate liftings which leads to an easy
criterion for the existence of such separating sets, and we give simple
examples of functors that fail to admit expressive normal or monotone
modal logics, respectively, or in fact an expressive (unary) modal logic
at all. We then move on to polyadic modal logic, where modal operators
may take more than one argument formula. We show that every accessible
functor admits an expressive polyadic modal logic. Moreover, expressive
polyadic modal logics are, unlike unary modal logics, compositional.
},
}