Modal Fixpoint Logic: Some Model Theoretic Questions
Gaëlle Fontaine

Abstract:
This thesis is a study into some model-theoretic aspects of the modal
\mu-calculus, the extension of modal logic with least and greatest
fixpoint operators. We explore these aspects through a
`fine-structure' approach to the \mu-calculus.  That is, we
concentrate on special classes of structures and particular fragments
of the language.  The methods we use also illustrate the fruitful
interaction between the \mu-calculus and other methods from automata
theory, game theory and model theory.
 
Chapter 3 establishes a completeness result for the \mu-calculus on
finite trees. The proof of the completeness of the \mu-calculus on
arbitrary structures is well-known for its difficulty, but it turns
out that on finite trees, we can provide a much simpler argument. The
technique we use consists in combining an Henkin-type semantics for
the \mu-calculus together with model theoretic methods (inspired by
the work of Kees Doets).

In Chapter 4, we study the expressive power of the \mu-calculus at the
level of frames. The expressive power of the \mu-calculus on the level
of models (labeled graphs) is well understood, while nothing is known
on the level of frames (graphs without labeling). In the setting of
frames, the proposition letters correspond to second-order variables
over which we quantify universally. Our main result is a
characterization of those monadic second-order formulas that are
equivalent on trees (seen as frames) to a formula of the \mu-calculus.

In Chapter 5, we provide characterizations of particular fragments of
the \mu-calculus, the main ones being the Scott continuous fragment
and the completely additive fragment. An interesting feature of the
continuous formulas is that they are constructive, that is, their
least fixpoints can be calculated in at most \omega steps.  We also
give an alternative proof of the characterization of the completely
additive fragment obtained by Marco Hollenberg, following the lines of
the proof for the characterization of the continuous fragment.

In the next chapter, we investigate the expressive power of a fragment
of CoreXPath. XPath is a navigation language for XML documents and
CoreXpath has been introduced to capture the logical core of XPath. In
Chapter 6, we exploit the tight connection between CoreXPath and modal
logic: by combining well-known results concerning the \mu-calculus
(one of them appearing in Chapter 5), we establish a characterization
of an important fragment of CoreXPath.

Finally, in Chapter 7, we develop automata-theoretic tools for
coalgebraic fixpoint logics, viz. generalizations of the \mu-calculus
to the abstraction level of coalgebras. Coalgebras provide an abstract
way of representing evolving systems. We use those tools to show the
decidability of the satisfiability problem as well as a small model
property for coalgebraic fixpoint logics in a general setting. We also
obtain a double exponential upper bound on the complexity of the
satisfiability problem.