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Dissertation/Thesis Abstract

Complexity Theoretic Parallels Among Automata, Formal Languages and Real Variables Including Multi-Patterns, L-Systems and Cellular Automata
by Xie, Jingnan, Ph.D., State University of New York at Albany, 2017, 112; 10272502
Abstract (Summary)

In this dissertation, we emphasize productiveness not just undecidability since pro- ductiveness implies constructive incompleteness. Analogues of Rice’s Theorem for different classes of languages are investigated, refined and generalized. In particular, several sufficient but general conditions are presented for predicates to be as hard as some widely discussed predicates such as “= ∅” and “= {0,1}∗”. These conditions provide several general methods for proving complexity/productiveness results and apply to a large number of simple and natural predicates. As the first step in apply- ing these general methods, we investigate the complexity/productiveness of the pred- icates “= ∅”, “= {0,1}∗” and other predicates that can be useful sources of many- one reductions for different classes of languages. Then we use very efficient many- one reductions of these basic source predicates to prove many new non-polynomial complexity lower bounds and productiveness results. Moreover, we study the com- plexity/productiveness of predicates for easily recognizable subsets of instances with important semantic properties. Because of the efficiency of our reductions, intuitively these reductions can preserve many levels of complexity. We apply our general methods to pattern languages [1] and multi-pattern lan- guages [2]. Interrelations between multi-pattern languages (or pattern languages) and standard classes of languages such as context-free languages and regular languages are studied. A way to study the descriptional complexity of standard language descriptors (for examples, context-free grammars and regular expressions) and multi-patterns is illustrated. We apply our general methods to several generalizations of regular ex- pressions. A productiveness result for the predicate “= {0,1}∗” is established for synchronized regular expressions [3]. Because of this, many new productiveness re- sults for synchronized regular expressions follow easily. We also apply our general methods to several classes of Lindenmayer systems [4] and of cellular automata [5]. A way of studying the complexity/productiveness of the 0Lness problem is developed and many new results follow from it. For real time one-way cellular automata, we observe that the predicates “= ∅” and “= {0,1}∗” are both productive. Because vi of this, many more general results are presented. For two-way cellular automata, we prove a strong meta-theorem and give a complete characterization for testing containment of any fixed two-way cellular automaton language. Finally, we generalize our methods and apply them to the theory of functions of real variables. In rings, the equivalence to identically 0 function problem which is an analogue of “= ∅” is studied. We show that the equivalence to identically 0 function problem for some classes of elementary functions is productive for different domains including open and closed bounded intervals of real numbers. Two initial results for real fields are also presented.

Indexing (document details)
Advisor: Hunt, Harry B., III
Commitee: Hunt, III, Harry B., Narendran, Paliath, Stearns, Richard E.
School: State University of New York at Albany
Department: Computer Science
School Location: United States -- New York
Source: DAI-B 78/09(E), Dissertation Abstracts International
Subjects: Theoretical Mathematics, Computer science
Keywords: Cellular automata, Computational complexity theory, Lindenmayer systems, Pattern matching, Theory of computation, Theory of functions of real variables
Publication Number: 10272502
ISBN: 978-1-369-72182-9
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