Computability and Complexity Properties of Automatic Structures and their Applications

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Continuing a tradition of studying computability in mathematics, we examine automatic structures, mathematical objects which can be represented by automata, and apply resulting observations to computer science. We measure the complexity of automatic structures via well-established concepts from model theory, topology, and set theory. We prove the following results. The ordinal height of any automatic well-founded partial order is bounded by \omega^\omega. The ordinal heights of automatic well-founded relations are unbounded below the first uncomputable ordinal. For any computable ordinal, there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first uncomputable ordinal and the successor of the first uncomputable ordinal. For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal. Next, we study infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations have finite degree and can be described by finite automata over a one-letter alphabet. We investigate algorithmic properties of such graphs in terms of their finite presentations. In particular, we ask how hard it is to check whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms answering each of these questions. For a fixed input graph, the algorithm for infinite component membership works in constant time and reachability is decided uniformly by a single automaton. Hence, we improve on previous work, in which nonelementary or nonuniform algorithms were found. We turn our attention to automata techniques for deciding first-order logical theories. These techniques are useful in Integer Linear Programming and Mixed Integer Linear Programming, which in turn have applications in diverse areas of computer science and engineering. We extend known work to address the enumeration problem for linear programming solutions. Then, we apply a similar paradigm to give an automata theoretic decision procedure for the p-adic valued ring under addition and for formal Laurent series over a finite field with valuation and addition

Disciplines and Styles in Pure Mathematics, 1800-2000

This workshop addressed issues of discipline and style in number theory, algebra, geometry, topology, analysis, and mathematical physics. Most speakers presented case studies, but some offered global surveys of how stylistic shifts informed the transition and transformation of special research fields. Older traditions in established research communities were considered alongside newer trends, including changing views regarding the role of proof

Ideas and Explorations : Brouwer's Road to Intuitionism

This dissertation is about the initial period of Brouwer's role in the foundational debate in mathematics, which took place during the first decades of the twentieth century. His intuitionistic and constructivistic attitude was a reaction to logicism (Russell, Couturat) and to Hilbert's formalism. Brouwer's own dissertation (1907) is a first introduction to his intuitionism, which was the third movement in the foundational debate. This intuitionism reached maturity from 1918 onwards, but one of my aims is to show that there are demonstrable traces of this new development of mathematics as early as 1907 and even before, viz. in his personal notes, which are composed of his numerous ideas in the field of mathematics and philosophy. To mention some important ones: 1. Mathematics is entirely independent of language. Mathematics is created by the individual mind (Brouwer certainly is a solipsist) and the role of language is limited to that of communication a mathematical content to others and is also useful for one's own memory. 2. The ur-intuition of the 'move of time', that is, the experience that two events are not coinciding, is the most fundamental basis of all mathematics. A separate space intuition (Kant) is not needed. 3. A strict constructivism. Only that what is constructed by the individual mind counts as a mathematical object. 4. Logic only describes the structure of the language of mathemqatics. Hence logic comes after mathematics, instead of being its basis. 5. An axiomatic foundation is rejected by Brouwer. Axioms only serve the purpose of describing concisely the properties of a mathematical construction. These five items have far-reaching consequences for Brouwer's mathematical building. To mention the most relevant ones: - The only possible cardinalities for sets are: finite, denumerably infinite, denumerably infinite unfinished and the continuum. - The continuum is not composed of points (Aristotle already said so), but is given to us in its entirety in the ur-intuition. It can be turned into an everywhere dense measurable continuum by constructing a rational scale on it. - Cantor's second number class does not exist as a finished totality for Brouwer, since there is no conceivable closure for the elements of this class. - The continuum problem is a trivial one: Every well-defined subset of the continuum is finite, denumerably infinite, or has the cardinality of the continuum. Finally, in my dissertation the sixth chapter is devoted to Brouwer's view on the application of mathematics to the human evironment and on his outlook on man and on human society in general (chapter 2 of Brouwer's dissertation). His opinion about humanity turns out to be a pessimistic one: All man's effort, when applying mathematics to the surrounding world, is aimed at a domination over his environment and over his fellow men