DS201008: Gheerbrant, Amélie (2010) FixedPoint Logics on Trees. Doctoral thesis, University of Amsterdam.
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Abstract
In this thesis, we study prooftheoretic and modeltheoretic aspects
of some widely used modal and quantified fixedpoint logics on trees.
Chapter 2 includes basics of modal logic, temporal logic, fixedpoint
logics, and some firstorder and higherorder logics of tree
structures.
In Chapter 3, we consider the class of finite nodelabelled
siblingordered trees. We present axiomatizations of its monadic
secondorder logic (MSO), monadic transitive closure logic (FO(TC1 ))
and monadic least fixedpoint logic (FO(LFP1 )) theories. Using
modeltheoretic 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 fixedpoint construct
to the language. Using techniques from abstract modeltheory, 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 fixedpoint operator µ (known as linear time µcalculus);
and µTL(U), the least fixedpoint extension of the "Untilonly"
fragment of LTL.
In Chapter 5, we focus on the logic µTL(U), that we identified in the
previous chapter as the stutterinvariant fragment of the lineartime
µcalculus 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 ωwords. 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
stutterinvariant 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 fixedpoint 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 fixedpoint logics of
trees. We then generalize our perspective on games to an exploration
of fixedpoint logics on finite trees that best fit gametheoretic
equilibria. We end with a broader program for merging computational
logics to the area of game theory.
Keywords:
Item Type:  Thesis (Doctoral) 

Report Nr:  DS201008 
Series Name:  ILLC Dissertation (DS) Series 
Year:  2010 
Subjects:  Computation Language Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  14 Jun 2022 15:16 
Last Modified:  14 Jun 2022 15:16 
URI:  https://eprints.illc.uva.nl/id/eprint/2091 
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