Finitary coalgebraic logics
Clemens Kupke

Abstract:
Finitary coalgebraic logics
Clemens Kupke

The aim of this thesis is to deepen our understanding of
the close connection between modal logic  
and coalgebras. This connection becomes not only
manifest in the fact that Kripke frames are a 
special type of coalgebras,
but more generally in the fact that the relationship
between modal logic and coalgebras 
can be seen to be the categorical dual 
of the fruitful and well-studied 
relationship between equational logic and algebras.

Various types of modal languages have been
proposed in the literature for reasoning about coalgebras.
In this thesis we consider the following three approaches:
The inductively defined languages for Kripke polynomial functors,
which were developed in successive papers by Kurz, Roessiger
and Jacobs, Pattinson's coalgebraic modal languages that 
are given by predicate liftings, and finitary 
coalgebraic fixed point logics, which were introduced by Venema
as a modification of Moss' infinitary coalgebraic logics.

Throughout the thesis we hold the view that 
a useful logical language for reasoning about coalgebras
should have finitary syntax. 
Therefore all languages that we discuss are finitary. 
Languages with a finitary syntax, however, generally lack the
Hennessy-Milner property.

It is therefore a natural question whether one can find a
class of coalgebras that still allows for logics
with finitary syntax that have the Hennessy-Milner property. 
We propose to resolve this issue by
generalizing a well-known concept from modal logic, namely the 
concept of a descriptive general frame.
These descriptive general frames can be represented as coalgebras
for the Vietoris functor on the category of Stone spaces. Hence
Stone coalgebras, i.e. coalgebras 
for functors over the category of Stone spaces, 
are a natural generalization of this concept. 

One way of increasing the expressivity of a modal language
is the use of so-called fixed point operators. 
Venema's coalgebraic fixed point logics
have a finitary syntax and offer the possibility 
to reason about infinite, ongoing
behaviour. These logics can be seen as a generalization of the modal
mu-calculus and they allow, similar to the modal mu-calculus,  
for an automata-theoretic interpretation: there is a 
one-to-one correspondence between formulas of coalgebraic
fixed point logic and the so-called coalgebra automata.

In this thesis we prove certain closure properties of coalgebra automata 
and show that the non-emptiness problem of coalgebra automata is in many 
cases decidable. Our results can be looked at from two perspectives:
Firstly they generalize known results about automata on infinite objects, 
such as automata on infinite words, trees and graphs. Secondly our results 
have logical corollaries: we show that coalgebraic fixed point logics all 
enjoy the finite model property. This yields in particular a proof of the finite 
model property of the modal mu-calculus. As another consequence we obtain decidability
for a large class of coalgebraic fixed point logics. Furthermore we prove the soundness 
of a certain distributive law for the modal operator of coalgebraic logic. 

The thesis is structured as follows: After the 
Introduction in Chapter 1, we give an overview over
three types of modal languages which are discussed in this thesis.

Chapter 3 contains a first application of the idea of considering
coalgebras over Stone spaces. We look at inductively defined logics for Kripke 
polynomial functors: for every Kripke polynomial functor we define
a corresponding functor on the category of Stone spaces and obtain
what we call the class of Vietoris polynomial functors. For each of these
functors we obtain the final coalgebra using a modified canonical model construction. 
This construction yields, in particular, that the languages associated with Vietoris 
polynomial functors have the Hennessy-Milner property. Furthermore we prove that
for every Vietoris polynomial functor F and the logic associated to it, 
there exists an adjunction between the algebraic semantics of the logic,
defined as a category of many-sorted algebras, and the category of F-coalgebras. 
Finally we give a characterization of those many-sorted algebras for which 
this adjunction turns into an equivalence of categories.

In Chapter 4 we turn to coalgebraic modal logics given in terms of
a set of predicate liftings and a set of axioms with modal depth 1. 
Given an endofunctor F on the category of sets or the category 
of Stone spaces and a logic for F we devise a functor L on the category 
of Boolean algebras. The category of algebras for this functor constitutes 
the algebraic semantics for the logic. We use this algebraic semantics to
give a categorical analysis of conditions for the logic to be sound and complete
with respect to the coalgebraic semantics. This is done by relating 
soundness and completeness of the logic to properties of a natural transformation
that connects the functors L and F. For the case that F is a functor on Stone spaces 
we obtain the following result: the logic is sound, complete and 
has the Hennessy-Milner property if L is dual to F.

In Chapter 5 we prove closure properties of coalgebra automata
and we show how one can effectively solve the non-emptiness
problem for a large class of coalgebra automata.
The main result of this chapter is the proof that for every
coalgebra automaton we can construct an equivalent 
non-deterministic coalgebra automaton. 
The proof works uniformly for all types of 
coalgebra automata, in the special case of tree automata it 
implies Rabin's complementation lemma.