Games, Walks and Grammars: Problems I've Worked On
Marco Vervoort
Abstract:
This dissertation consists of three disjunct parts.
The first part, titled `Blackwell Games', is about the problem of
determinacy of Blackwell games, a class of infinite games of imperfect
information, where both players simultaneously select moves from a
finite set, infinitely many rounds are played, and payoff is
determined by a Borel measurable function $f$ on the set of possible
resulting sequences of moves. We give elementary proofs of
determinacy for Blackwell games whose payoff function is an indicator
function of a Borel set up to complexity $G_{\delta\sigma}$. For
general Borel payoff functions, we give a reduction, found by
D.A. Martin, to the known result of determinacy of Borel perfect
information games. We also consider Blackwell games whose payoff
function is not Borel measurable, and formulate an analogue of the
Axiom of Determinacy for these games, Finally, we compare some of the
consequences of this `Axiom of Blackwell Determinacy' with those of
the original Axiom of Determinacy.
Keywords: determinacy, blackwell games
In the second part, titled `Random Walks', we consider recurrence in
reinforced random walks, where edges in a graph are traversed with
probabilities that may be different (reinforced) at second, third
etc. traversals. We focus on the case where the probability for any
edge only changes once, after its first traversal. As a special case,
we show that the once-reinforced random walk on the infinite ladder is
almost surely recurrent if reinforcement is small (extending a result
by T. Sellke, as well as when reinforcement is sufficiently large.
For the last result, we use an application of nonstandard analysis to
graph theory.
Keywords: random walks, nonstandard analysis, reinforcement
The third part, titled `EMILE', is about the EMILE program, a program
that reads in a text, and without prior knowledge attempts to
determine the grammatical structure of the language. The basic
concepts and algorithms underlying the program are discussed, as well
as the results of this approach, both in theory and in practice. It
is argued that natural languages satisfy the condition of shallowness,
and that this implies that the EMILE program will work well for
natural languages. In a separate appendix, explicit pseudo-code for
each of the sub-algorithms of EMILE is given.
Keywords: grammar induction, shallowness, separability