TY - THES
A1 - Zetzsche, Georg
T1 - Monoids as Storage Mechanisms
N2 - Automata theory has given rise to a variety of automata models that consist
of a finite-state control and an infinite-state storage mechanism. The aim
of this work is to provide insights into how the structure of the storage
mechanism influences the expressiveness and the analyzability of the
resulting model. To this end, it presents generalizations of results about
individual storage mechanisms to larger classes. These generalizations
characterize those storage mechanisms for which the given result remains
true and for which it fails.
In order to speak of classes of storage mechanisms, we need an overarching
framework that accommodates each of the concrete storage mechanisms we wish
to address. Such a framework is provided by the model of valence automata,
in which the storage mechanism is represented by a monoid. Since the monoid
serves as a parameter to specifying the storage mechanism, our aim
translates into the question: For which monoids does the given
(automata-theoretic) result hold?
As a first result, we present an algebraic characterization of those monoids
over which valence automata accept only regular languages. In addition, it
turns out that for each monoid, this is the case if and only if valence
grammars, an analogous grammar model, can generate only context-free
languages.
Furthermore, we are concerned with closure properties: We study which
monoids result in a Boolean closed language class. For every language class
that is closed under rational transductions (in particular, those induced by
valence automata), we show: If the class is Boolean closed and contains any
non-regular language, then it already includes the whole arithmetical
hierarchy.
This work also introduces the class of graph monoids, which are defined by
finite graphs. By choosing appropriate graphs, one can realize a number of
prominent storage mechanisms, but also combinations and variants thereof.
Examples are pushdowns, counters, and Turing tapes. We can therefore relate
the structure of the graphs to computational properties of the resulting
storage mechanisms.
In the case of graph monoids, we study (i) the decidability of the emptiness
problem, (ii) which storage mechanisms guarantee semilinear Parikh images,
(iii) when silent transitions (i.e. those that read no input) can be
avoided, and (iv) which storage mechanisms permit the computation of
downward closures.
Y1 - 2016
UR - https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4400
UR - https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-44003
ER -