Expressivity of Coalgebraic Modal Languages
Raul Andres Leal Rodriguez
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.