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.
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.
The third part, titled `The EMILE Grammar Inducer', 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.