Alternating Automata and the Temporal Logic of Ordinals
Rohde, Gareth Scott
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https://hdl.handle.net/2142/86954
Description
Title
Alternating Automata and the Temporal Logic of Ordinals
Author(s)
Rohde, Gareth Scott
Issue Date
1997
Doctoral Committee Chair(s)
Schupp, Paul E.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Language
eng
Abstract
In their 1995 paper, Muller and Schupp define the concept of an alternating automaton, a sort of completion of the notion of a non-deterministic automaton, and show how the notion may be used to prove a number of major results in the theory of automata on infinite inputs in a unified way. In this thesis, we extend the notion of an alternating automaton to accommodate linear inputs of arbitrary ordinal length and then use these automata to prove a number of results about linear propositional temporal logic, a form of modal logic. First, we show that there is a natural interpretation of automaton inputs as structures for the logic and that under this interpretation, alternating automata and temporal logic are equally powerful. We then use this fact to investigate the satisfiability problem for temporal logic. In particular, not only do we look at the question of whether a given formula has a model, but we look at this question when the lengths of models is restricted to a specific ordinal. We show that the temporal logic of an arbitrary (but fixed) ordinal is decidable in exponential time, generalizing the known result that the temporal logic of $\omega$ is decidable in exponential time. We also look at which classes of ordinals are definable by temporal logic formulas (more precisely, which classes of ordinals result from projecting the class of models of a formula onto their respective lengths), and give a characterization of such classes.
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