on Coalgebras and Modal Logic
The course notes ( ps.gz,
ps) are available in a revised version from October 2001.
Sketch of the 5 Lectures:
Introduction to Systems. Behavioural equivalence,
bisimulation, coinduction, final coalgebras, examples, applications. (No
Coalgebra. Introduction to universal coalgebra and, as far
as needed, category theory. (In contrast to the notes, we will spend
quite some time to introduce the necessary categorical concepts.)
Introduction to Modal Logic. Just some basics on Kripke
models and frames, definability and bismulation.
Coalgebras and Modal Logic (coalgebras as dynamic systems
and modal logics as the appropriate specification languages; an overview
of some current approaches).
On the Duality of Modal and Equational Logic Using a
categorical formulation of the coalgebraic semantics of modal logic we
can explain the common ground of the different approaches discussed in
the previous lecture; moreover, the idea that `modal logic is dual to
equational logic' can now be formally justified. As an application we
prove the dual of the HSP-theorem (Birkhoff's variety theorem).
Recommended Literature (updated May 2002):
I plan to make the lecture self contained, so it will not be necessary
to read any of the following. If you want to prepare a bit nevertheless
read on modal logic and on category theory (see below).
The remaining references are for the curious ...
A good introduction to algebras and coalgebras presenting
(co)induction, duality, and examples is
B. Jacobs and
J.J.M.M. Rutten, A Tutorial on (Co)Algebras and (Co)Induction.
Bulletin of EATCS Vol. 62, 1997, pp. 222--259.
best for a first introduction.
The standard reference for a theory of universal coalgebra along
the lines of universal algebra is
J.J.M.M. Rutten, Universal coalgebra: a theory of
systems.Technical Report CS-R9652, CWI, Amsterdam, 1996. A
revised version appeared in Theoretical Computer Science 249(1), 2000,
This paper contains a thorough study of universal
coalgebra and contains background material for further studies.
Another more recent reference for universal coalgebra are the lecture
H. Peter Gumm, Elements of the general theory of coalgebras. LUATCS'99,
Rand Afrikaans University, Johannesburg, South Africa, 1999.
In case you are interested in universal algebra an excellent
introduction from the computer science point of view is
W. Wechler, Universal Algebra for Computer Scientists, Springer, 1991.
For background on modal logic I recommend
P. Blackburn, M. de Rijke, Y. Venema, Modal Logic. CUP,
My ESSLLI-lecture will present what is necessary to know
about modal logic in order to understand the relationship to
coalgebras. But to really appreciate this relationship (which has only
been discovered during the last few years) some additional background
on modal logic will be useful: For this read chapters 1.1 - 1.3, 2.1 -
2.3, and note theorem 2.65.
The discovery of the relation between coalgebra and modal logic can be
traced back to the book
J. Barwise, L. Moss, Vicious Circles,
CSLI, Stanford 1996.
A very readable and exciting book on
paradoxa, non-well founded set theory, coalgebras, and modal logic.
Concerning category theory, the course will be
self-contained. But if you have seen before the definitions of functor,
product/coproduct, epi/mono, and preferably also of limit/colimit,
natural transformation, and adjunction things will be easier (even if
you don't remember the definitions, seeing them again will make you feel
more comfortable when they appear in the course). You can consult
e.g. one of the following:
Lawvere, Schanuel, Conceptual Mathematics, Cambridge University
This book teaches to think categorically and avoids technicalities or
advanced examples. Take a look!
S. Mac Lane, Categories for the Working Mathematician, Springer
The classical reference.
J. Adamek, H. Herrlich, G. Strecker, Abstract and Concrete
Categories , John Wiley and Sons, 1990.
Presents a lot of material not available in Mac Lane's book.
G. Longo. Categories, Types and Structures, MIT Press,
For those who are interested in the relationship between
category theory and type theory. Electronically available.
M. Barr, C. Wells, Category Theory for Computing Science,
Prentice Hall, 1990.
A very readable introduction with many
examples familiar to computer scientists. A shorter introduction is
electronically available as the lecture notes of their ESSLLI'99
F. Borceux. Handbook of Categorial Algebra. Cambridge University
Perhaps not the best starter but most valuable for
learning category theory seriously.
B. Pierce, Basic Category Theory for Computer Scientist, MIT
A small book but still containing all the definitions
R.F.C. Walters, Categories and Computer Science, MIT Press,
An undergraduate text emphasising objects as data types and
arrows as programs.
Additional list of electronically available lecture notes on
M. Caccamo, J.M.E. Hyland, G. Winskel: Lecture Notes
in Category Theory. BRICS Lecture Series, 2001.
A Gentle Introduction to Category Theory - the calculational
approach. University of Utrecht, 1992.
Chris Hillman: A
Categorical Primer. August 2001.
This page contains an informal
introduction to category theory and, for example, a nice explanation
of the Yoneda Lemma.
Jaap van Oosten:
Basic Category Theory.
D. Turi: Category
Theory Lecture Notes. LFCS, Univeristy of Edinburgh, 2001.
Last modified: Wed May 15 10:49:01 CEST 2002