Some of the hotels are listed below.

**Alexandru Baltag**

**Title:**
Universal Set Theory is the Canonical Model for Infinitary Modal Logic

**Abstract:**
In this talk, I will present some older work on modeling and axiomatizing a non-classical, non-wellfounded set theory with a universal set and very strong comprehension and topological closure properties, by using infinitary modal logic. The model is essentially given by the canonical model construction for a modal language with conjunctions of size less than kappa, where kappa is a weakly compact cardinal. The axiomatization is based on a natural extension of Aczel's Antifoundation Axiom, whose most direct formulation uses infinitary modal logic. The notion of observational equivalence, given by elementary equivalence with respect to infinitary modal logic (and close, but not equivalent, to the standard notion of bisimulation) plays a central role. The resulting model is a kind of Cauchy completion of Aczel's non-wellfounded universem and, in certain assumptions, is isomorphic with the ``topological set theory" model of Forti and Honsell, first proposed as a model for the so-called ``Positive Comprehension Axiom". I show how this theory ``solves" the classical paradoxes
and can be used to formalize a ``reflexive" category of all categories, as well as prove fixed point theorems relating set-theoretic recursion and corecursion. I speculate about the model-theoretic, topological and coalgebraic consequences of this work.

**Corina Cirstea**

(Joint work with M. Sadrzadeh)

**Title:**
Towards a Modular Coalgebraic Approach to Model-based
Verification

**Abstract:**
I will consider a variation of the notion of logical F-automata
operating on F-coalgebras, as introduced by Venema (2004), and present
some initial results towards using such automata for model-checking
coalgebraic models.

**Martin Escardo**

**Title:**
Stably Compact Spaces

**Abstract:**
Stably compact spaces were discovered independently by a
number of people in different guises, with more guises uncovered after
their initial identification. These spaces can be traced back to the
work of M. Stone (1937) on the representation of distributive lattices
by spectral spaces, of L. Nachbin (1950) on compact ordered
topological spaces, and of H. Priestley (1972) on representation of
distributive lattices by ordered Stone spaces. This talk will be a
tour of a dozen pieces of work on stably compact spaces. In
particular, I'll present many equivalent characterizations of the
notion/category of stably compact spaces.

**Ichiro Hasuo**

**Title:**
Coalgebras in Kleisli Categories

**Abstract:**
The theory of coalgebras has been a big success as a mathematical
framework to reason about state-based reactive systems. As a base category, the category Sets of sets and functions is conventionally taken:
then the coalgebraic notions capture semantics of systems modulo
bisimilarity. Namely, in Sets,

- a coalgebra is a system, - a morphism of coalgebras is a map which preserves behavior modulo bisimulation, that is, a functional bisimulation,

- via coinduction we obtain a semantics of systems modulo bisimilarity.

Then we may ask the following questions:

- If we take a base category different from Sets, what computational
meaning do the coalgebraic notions (morphisms, coinduction, etc.) have?

- If we are interested in a semantics of systems which is different from
(i.e. coarser than) bisimilarity, how can we capture it
coalgebraically?

In this talk we shall give one answer to these questions, proposing a Kleisli category for a monad as an alternative base category. There the coalgebraic notions capture trace semantics of systems. More precisely, in a Kleisli category,

- a coalgebra is a system,

- a lax/oplax morphism of coalgebras is a forward/backward simulation
between systems, and,

- via coinduction we obtain the trace semantics of systems.

By replacing monads we can handle both non-deterministic and
probabilistic branching in a uniform manner.

Hence we identify a new application field of coalgebras.
This also demonstrates the power of categorical--in particular
coalgebraic--methods in computer science: the theory of coalgebras
is robust against change of base categories, i.e. change of semantics
we want to capture.

References:

[1] Ichiro Hasuo, Bart Jacobs and Ana Sokolova.

Generic Trace Theory. In CMCS 2006.

[2] Ichiro Hasuo.

Generic Forward and Backward Simulations.
In CONCUR 2006.

**Clemens Kupke**

**Title:**
The Finite Model Property of Coalgebraic Fixed Point Logics

(Joint work with Yde Venema)

**Abstract:**
A coalgebra automaton is a finite automaton that may accept
or reject a given pointed coalgebra. Automata that operate on infinite
words, trees and graphs can be seen as examples of coalgebra automata.

In my talk, I will sketch a solution to the non-emptiness problem
of coalgebra automata: every automaton that accepts some
pointed coalgebra accepts a pointed coalgebra that has a finite carrier set.
This implies that coalgebraic fixed point logics, and in particular
the modal mu calculus, have the finite model property.

**Frank Wolter**

**Title:**
Conservative Extensions in Modal Logic

**Abstract:**
A theory *T* is said to be a conservative extension of a theory
*T′* if any consequence of *T*, which only uses symbols from
*T′*,
is a consequence of *T′* as well. This notion plays an important role in
mathematical logic and the foundations of mathematics.

Rather surprisingly, in modal logic the notion of conservative extension has
hardly been investigated. Indeed, modal theories - similarly to first-order
theories - have become fundamental tools for representing various domains.
For example, in epistemic logic, modal theories represent the (possibly
introspective) knowledge of an agent; in temporal logic, theories serve as
specifications of concurrent systems; in description logic, theories (called
TBoxes) are ontologies used to fix the terminology of an application domain,
etc. In all these examples, the notion of a conservative extension can be
used to compare different theories and derive important information about
their relation to each other. For instance, a description logic ontology *T′*
can be regarded as a `safe' extension of another description logic ontology
*T* if, and only if, *T′* is a conservative extension of *T*.

One of the main reasons for using modal logic instead of full first-order
logic in the applications above is that reasoning in modal logic is often
decidable. To employ the notion of conservative extension for modal logics,
it is therefore crucial to analyse the algorithmic problem of deciding
whether one modal theory is a conservative extension of another modal theory.

In this talk, we investigate the notion of conservative extension for a
number of basic modal logics and, in particular, determine the computational
complexity of the conservativeness problem for these logics. We will
also discuss the relation to uniform interpolation and bisimulation
quantifiers.

We begin by showing that deciding non-conservativeness is NExpTime-hard for
all modal logics which have rooted frames with more than N
successors of the root, for any finite N. This result covers almost all
standard modal logics, for example, K, S4, and S5. Next, we consider the
modal logic S5 and show that in this case non-conservativeness is
NExpTime-complete by proving that one can construct a uniform interpolant of
exponential size in exponential time (in the size of a given formula) and
using the fact that S5-satisfiability is decidable in NP.
A slightly different technique is used to prove that for K
non-conservativeness is NExpTime-complete. Finally, we consider the
conservativeness problem for S4 and establish an ExpSpace upper bound.

The talk is based on joint work with Silvio Ghilardi, Carsten Lutz, and
Michael Zakharyaschev.