Abstract
In this talk we will survey some results on extensions of modal logic
and first order logic with fixed points.
First, we will consider the mu-calculus which is an extension of the
modal system K with fixed point operators. We will briefly survey the
properties of the mu-calculus and its connections to monadic
second-order logic. In particular we will discuss the connections
between fixed point and quantifier hierarchies.
Next, we will consider guarded fragment (GF) of first-order logic and
its extension with fixed point operators. GF is defined by
syntactically restricting the use of quantifiers. It contains more
than just formulas resulting from the translation of modal
formulas. For example, in GF there is no restriction on the arity of
relations. We will show that despite the increase in expressive power
many good properties of the mu-calculus still hold for the extension
of GF with fixed points. We will also discuss the relation of this
extension with fragments of second-order logic.