DS-2006-03:
Kupke, Clemens
(2006)
*Finitary coalgebraic logics.*
Doctoral thesis, University of Amsterdam.

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## 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.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2006-03 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2006 |

Subjects: | Language Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2050 |

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