Fixed-Point Logics on Trees
Amélie Gheerbrant
Abstract:
In this thesis, we study proof-theoretic and model-theoretic aspects
of some widely used modal and quantified fixed-point logics on trees.
Chapter 2 includes basics of modal logic, temporal logic, fixed-point
logics, and some first-order and higher-order logics of tree
structures.
In Chapter 3, we consider the class of finite node-labelled
sibling-ordered trees. We present axiomatizations of its monadic
second-order logic (MSO), monadic transitive closure logic (FO(TC1 ))
and monadic least fixed-point logic (FO(LFP1 )) theories. Using
model-theoretic techniques, we show by a uniform argument that these
axiomatizations are complete, i.e., each formula which is valid on all
finite trees is provable using our axioms.
In Chapter 4 we consider various fragments and extensions of
propositional linear temporal logic (LTL), obtained by restricting the
set of temporal connectives or by adding a least fixed-point construct
to the language. Using techniques from abstract model-theory, for each
of these logics we identify its smallest extension that has Craig
interpolation. Depending on the underlying set of temporal operators,
this framework turns out to be one of the following three logics: the
fragment of LTL having only the Next operator; the extension of LTL
with a least fixed-point operator µ (known as linear time µ-calculus);
and µTL(U), the least fixed-point extension of the "Until-only"
fragment of LTL.
In Chapter 5, we focus on the logic µTL(U), that we identified in the
previous chapter as the stutter-invariant fragment of the linear-time
µ-calculus µTL. We also identified this logic as one of the three only
temporal fragments of µTL that satisfy Craig interpolation. Complete
axiom systems were known for the two other fragments, but this was not
the case for µTL(U). We provide complete axiomatizations of µTL(U) on
the class of finite words and on the class of ω-words. For this
purpose, we introduce a new logic µTL(♦_Γ), a variation of µTL where
the "Next time" operator is replaced by the family of its
stutter-invariant counterparts. This logic has exactly the same
expressive power as µTL(U). Using known results for µTL, we first
prove completeness for µTL(♦_Γ), which then allows us to obtain
completeness for µTL(U).
Finally, in Chapter 6 we take our style of analysis via modal and
temporal fixed-point logics to games. Current methods for solving games
embody a form of "procedural rationality" that invites logical
analysis in its own right. This chapter is a case study of Backward
Induction for extensive games. We consider a number of analyses from
recent years in terms of knowledge and belief update in logics that
also involve preference structure, and we prove that they are all
mathematically equivalent in the perspective of fixed-point logics of
trees. We then generalize our perspective on games to an exploration
of fixed-point logics on finite trees that best fit game-theoretic
equilibria. We end with a broader program for merging computational
logics to the area of game theory.
Keywords: