MoL-2007-19:
Rodriguez, Raul Andres Leal
(2007)
*Expressivity of Coalgebraic Modal Languages.*
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## Abstract

Through history logic has been used for several purposes, for example

as foundation of mathematics or in philosophy to provide an controlled

environment of argumentation. In this thesis we are interested in

using logic to describe mathematical structures. For example, it is

well know that algebraic logic can be used to describe algebraic

structures. Birkhoff Theorem clarifies under what conditions a class

of algebraic structures can be characterized using algebraic logic.

The dual concept of algebraic structure is that of coalgebraic

structure. In the last decade there has been a development of

different logical languages to describe coalgebraic structures. There

is no agreement in which of these language is the most appropriate to

describe coalgebras. This disagreement is partially based on the fact

that there is not much work comparing these languages. In this thesis

we will do some steps on this direction comparing two languages to

describe coalgebraic structures.

Coalgebras and algebras are formally dual concepts, i.e. given a

functor T, a coalgebra is a function \alpha : A -> TA and an algebra

is a function \alpha : TA -> A. Peter Gumm claims that coalgebras as

direct duals of algebras have been in scene for more than 30 years,

but did not receive much attention primarily due to the lack of vital

examples. The vital examples came from computer science. Various kind

of transitions systems, automata and functional programming languages

are naturally represented as coalgebraic structures. These new

examples demonstrated that a better intuition to understand universal

coalgebra is to conceive it as a general and uniform theory to

describe dynamic systems.

Our starting point is that of basic modal logic and Kripke structures,

frames and models. It is well known that modal logic is an expressive

language to talk about Kripke structures or relational

structures. Using modal logic we can provide an internal local

perspective on relational structures. Now Kripke frames and Kripke

models can be naturally represented in coalgebraic terms. For example,

a Kripke frame (A,R) is represented by a function R : A -> PA, where P

is the covariant power set functor and R(s) is the the successors of

s. Here, we can see that modal logic is a language to talk about

coalgebraic structures. This immediately rises interesting

questions. Are there other formal languages like basic modal logic for

other coalgebras? Is the modal language an isolated language or is it

an example of a more general concept? Since there are many languages

to talk about coalgebras what is the relation between them?

The first question can be answered through abstract model theory. A

language for coalgebras is a set with a class of satisfaction

relations. The satisfaction of a modal formula on some point of a

Kripke frame can be seen as a process executed over some Kripke

structure. Here, we can say that from the point of modal logic Kripke

structures are dynamic systems, therefore coalgebras are a

generalization of Kripke frames, but there is no natural

interpretation of basic modal formulas over arbitrary

coalgebras. Pattinson and Schroeder solved this problem showing that

there is a direct generalization of modal logic, see Chapter 4 here,

using the concept of relation lifting; we call those languages

coalgebraic modal languages. These language can be uniformly defined

for a large class of functors. Hence modal logic is a particular

instance of a more general phenomenon. Historically, coalgebraic modal

languages were not the first languages invented to talk about

coalgebras in a uniform framework and as a generalization of modal

logic. The first language with this features was invented by Lawerence

S. Moss; we call this language Moss' language. The presence of at

least two different languages to talk about coalgebras explain the

third question. What is the relation between them? We have no general

answer for this question. Out here we do answer the third question in

the particular case of Moss' language and coalgebraic modal languages.

To return to our starting point, it is usually claimed that in the

particular case of the covariant power set functor, Moss' language and

the coalgebraic modal language are equally expressive. Unfortunately

in the literature used for this thesis, there is no much material

explaining what it means for two languages to be equally

expressive. This thesis will not provide such general theory. Instead

we will compare Moss' language and coalgebraic modal languages at a

semantical level and at a syntactical level.

One way to define the expressiveness of a language is using its

semantics. A language L1 is said to be more expressive than a language

L2 if L1 can express differences between coalgebras that the language

L2 cannot. Using this criterion, we will then show that Moss' language

and coalgebraic modal languages are equally expressive.

Another way to compare the expressiveness of two languages is trough

translations. Using the notation from the previous paragraph, this

means that each formula in the language L2 is translated into an

equivalent formula in L1. Using this criterion, we will show that in

the case of Kripke polynomial functors every predicate lifting can be

translated into Moss' language.

The structure of the thesis is as follows: In the proceeding chapter

we introduce the formal context in which this thesis is located, we

discuss some basics of category theory and universal coalgebra. We

also establish the background related to languages and translations,

including expressive languages. In Chapter 3 we define Moss' language,

provide examples in the case of Kripke polynomial functors, and

finally we show how to extend Moss' language with disjunctions and

negations. In Chapter 4 we show how to generalize modal logic to

coalgebraic modal languages, provide examples in the case of Kripke

polynomial functors, and show, in an original work, that coalgebraic

modal languages can be represented as initial algebras. With these

preliminaries out of the way, we are ready to compare Moss' language

and coalgebraic modal languages. In Chapter 5 we demonstrate that the

existence of expressive languages is equivalent to the existence of a

final object. We present a new elementary proof developed by the

author and Clemens Kupke. Using this result, we define non

constructive translations between Moss' Language and coalgebraic modal

languages. In the final chapter, we refine such translations, defying

constructive finitary translations for the particular case of Kripke

polynomial functors.

Item Type: | Report |
---|---|

Report Nr: | MoL-2007-19 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2007 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/791 |

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