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Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a ..."
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Cited by 9 (8 self)
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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
On the Topology of Discrete Strategies
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2009
"... This paper explores a topological perspective of planning in the presence of uncertainty, focusing on tasks specified by goal states in discrete spaces. The paper introduces strategy complexes. A strategy complex is the collection of all plans for attaining all goals in a given space. Plans are like ..."
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This paper explores a topological perspective of planning in the presence of uncertainty, focusing on tasks specified by goal states in discrete spaces. The paper introduces strategy complexes. A strategy complex is the collection of all plans for attaining all goals in a given space. Plans are like jigsaw pieces. Understanding how the pieces fit together in a strategy complex reveals structure. That structure characterizes the inherent capabilities of an uncertain system. By adjusting the jigsaw pieces in a design loop, one can build systems with desired competencies. The paper draws on representations from combinatorial topology, Markov chains, and polyhedral cones. Triangulating between these three perspectives produces a topological language for describing concisely the capabilities of uncertain systems, analogous to concepts of reachability and controllability in other disciplines. The major nouns in this language are topological spaces. Three key theorems (numbered 1, 11, 20 in the paper) illustrate the sentences in this language: (a) Goal Attainability: There exists a strategy for attaining a particular goal
THE NATURE OF CONTEMPORARY CORE MATHEMATICS
, 2010
"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."
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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.
On the Topology of Discrete Planning with Uncertainty
, 2011
"... This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key ..."
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This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph.
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
AMS Short Course on Computational Topology, JMM2011, New Orleans. On the Topology of Discrete Planning with Uncertainty
, 2012
"... Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeter ..."
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Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph. 1. Planning with Uncertainty in Robotics The goal of Robotics is to animate the inanimate, so as to endow machines with the ability to act purposefully in the world. Roboticists, working in the subfield of planning, create software by which robots reason about future outcomes of potential actions. Using such planning software, robots combine individual actions into collections that together accomplish particular tasks in the world [29, 30]. Two fundamental and intertwined issues confound this seemingly straightforward approach. One is world complexity, the other is uncertainty.