Analysing the complexity of games on graphs
MSc thesis, Samson Tikitu de Jager
=Abstract=
This thesis does not have a thesis, in the sense of a central
statement that is developed and argued for. Instead, it explores three
different ways of describing complexity in simple games.
The three approaches are drawn from set theory, from automata theory,
and from theory of algorithms and computer science. These three areas,
while obviously related, generally use different notation and
different approaches even while ostensibly talking about the same
things: `games'. The progression of ideas through this thesis could be
seen as a movement from the mopst abstract of these representations to
the concrete.
The first complexity analysis is drawn from descriptive set theory and
topology; a `game' here is an explicitly infinitary concept and one
that (for sufficiently complicated cases) can be shown not to yield to
constructive methods of proof. Chapter 1 presents the topological
notion of complexity from descriptive set theory, and some abstract
solution concepts utilising infinitary procedures on infinitary
objects.
In Chapter 2 we move to automata theory, restricting ourselves to a
class of games that are played on finite structures (the games
themselves remain infinite, but can be given finite
representations). The focus here is on the finite computational
resources needed to implement a strategy, which in the descriptive set
theoretic context might easily be infinite.
Chapter 3 takes an algorithmic approach to the process of
*constructing* a strategy for an automaton game, and measures the
complexity of the game by the time complexity of the algorithm to
solve it. This is the most concrete of the three approaches, in fact a
Java application implementing some of these algorithms accompanies the
written text of this thesis [1]. Not only are the games restricted to
finite representations (automata games, on finite structures) and the
strategies implementable with finite computational resources, but the
process of producing a strategy is also finitary and algorithmic.
The final chapter contains information concerning the software, and a
short user manual. There is also a short Appendix, applying the
techniques of Chapters 1, 2 and 3 to more fine-grained analysis of one
of the complexity classes considered in Chapter 1.
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[1] Download the software from
http://www.illc.uva.nl/Publications/ResearchReports/MoL-2005-07.software.jar
Bugfixes and updates will be posted to my homepage, at
http://tikitu.dejager.net.nz/software/GameGraph/