Knowing One's Limits. Logical Analysis of Inductive Inference
Nina Gierasimczuk
Abstract:
The thesis links learning theory with logics of knowledge and belief.
Following the introduction and mathematical preliminaries, Chapter 3
contains a methodological analysis of both frameworks, in particular
it analyzes the basic learning-theoretic setting in terms of dynamic
epistemic logic.
In Chapter 4 we use learning theory to evaluate dynamic epistemic
logic-based belief-revision policies. We investigate them with respect
to their ability to converge to the true belief for sound and complete
streams of positive data, streams of positive and negative data, and
erroneous fair information. We show that some belief-revision methods
are universal on certain types of data, i.e., they have full learning
power.
Chapter 5 is concerned with expressing identification in the limit and
finite identifiability in the languages of modal and temporal logics
of epistemic and doxastic change. We characterize learnability by
formulas of various logics of knowledge and belief.
In Chapter 6 we investigate the notion of definite finite tell-tale
set in finite identifiability of languages, in particular the
computational complexity of finding various kinds of minimal
DFTTs. Assuming the computability of learning functions we show that
there are classes of languages that are finitely identifiable, but no
computable agent can always conclude it as soon as it is objectively
possible.
In Chapter 7 we analyze different levels of cooperativeness between
the learner and the teacher in a game of perfect information based on
sabotage games. We give formulas of sabotage modal logic that
characterize the existence of winning strategies in such games. We
show that non-cooperative case is PSPACE-complete, and that relaxing
the strict alternation of the moves of the two players does not
influence the winning conditions.
In Chapter 8 we generalize the Muddy Children puzzle, to account for
arbitrary quantifier announcements. We characterize the solvability of
the generalized version of the Muddy Children puzzle and we propose a
new representation of the epistemic situation of Muddy Children
scenarios. Our modeling is linear with respect to the number of
agents, and is more concise than the one used in the classical dynamic
epistemic approach.
Overall, we focus on building a connection between formal learning
theory and dynamic epistemic logic. We provide dynamic epistemic logic
with a uniform framework for considering iterated actions. On the
other hand, this leads to a logical view on inductive inference and to
syntactic characterizations of learnability in modal and temporal
logics. Further topics of the thesis, taken from the domains of
computability, games, and multi-agency, strengthen the connection by
providing additional computational, logical and philosophical insights
into the process of epistemic and doxastic change.