The universality of polynomial time Turing equivalence

Marks, Andrew

doi:10.1017/s0960129516000232

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…In contrast, the set CBER of countable Borel equivalence relations (those with countable equivalence classes) has the greatest element E ∞ which is equivalent to many natural equivalence relations. For instance, it was recently shown in [71,72] that the equivalence relations on the Cantor space induced by some natural reducibilities in computability theory (e.g., polynomial-time many-one, polynomial-time Turing, and the arithmetical reducibilities) are Borel equivalent to E ∞ .…”

Section: Borel Reducibility Of Borel Equivalence Relationsmentioning

confidence: 99%

Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

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We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

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Section: Borel Reducibility Of Borel Equivalence Relationsmentioning

confidence: 99%

Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

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“…In the study of computability, P = ? NP is one of the very important and interesting open problems in mathematics and in theoretical computer science with deep ramifications in cryptography, 12,13 algorithms, 14–16 artificial intelligence, 14,17 game theory, 18 and economics, 19 to name a few. Intuitively, the P = NP conjecture is an investigation of whether every problem whose solution can be quickly verified in polynomial time can also be solved fairly quickly in polynomial time.…”

Section: Computational Complexitymentioning

confidence: 99%

Complexity and mission computability of adaptive computing systems

Dasari

Im²,

Geerhart

2019

Journal of Defense Modeling & Simulation

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There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic polynomial time complexity class called mission class, as many polynomial problems are not computable in mission time. By focusing on such subclass of languages in the context for successful military applications, we also discuss their computational and communicational constraints. We investigate feasible (non)linear models that will minimize energy and maximize memory, efficiency, and computational power, and also provide an approximate solution obtained within a pre-determined length of computation time using limited resources so that an optimal solution to a language could be determined.

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“…We refer to literature by Aaronson, 6 Arora-Barak, 7 Gritzmann-Sturmfels, 8 and Hartmanis-Stearns 9 for an extensive background on computational complexity in mathematics and in theoretical computer science. Computational complexity is the study of the difficulty of solving problems about finite combinatorial objects, with deep ramifications in cryptography, 10,11 algorithms, [12][13][14] game theory, 15 economics, 16 and artificial intelligence. 12,17 For example, given two integers a and b, are they relatively prime, or given a list of cities and a distances between any two cities, what is the shortest route so that one visits each city and returns to the original position?…”

Section: Computational Complexity Of Adaptive Mission Platformmentioning

confidence: 99%

Optimization problems with low SWAP tactical computing

Im¹,

Dasari

Beshaj³

et al. 2019

Disruptive Technologies in Information Sciences II

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In a resource-constrained, contested environment, computing resources need to be aware of possible size, weight, and power (SWaP) restrictions. SWaP-aware computational efficiency depends upon optimization of computational resources and intelligent time versus efficiency tradeoffs in decision making. In this paper we address the complexity of various optimization strategies related to low SWaP computing. Due to these restrictions, only a small subset of less complicated and fast computable algorithms can be used for tactical, adaptive computing.

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The universality of polynomial time Turing equivalence

Cited by 5 publications

References 20 publications

Well-Quasi Orders and Hierarchy Theory

Well-Quasi Orders and Hierarchy Theory

Complexity and mission computability of adaptive computing systems

Optimization problems with low SWAP tactical computing

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