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A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

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hdl:2117/119165

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Chen, Yijia
Muller, Moritz Martin
Yokoyama, Keita
Document typeConference report
Defense date2018
PublisherAssociation for Computing Machinery (ACM)
Rights accessOpen Access
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
ProjectAUTAR - A Unified Theory of Algorithmic Relaxations (EC-H2020-648276)
Abstract
The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy. On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.
CitationChen, Y., Müller, M., Yokoyama, K. A parameterized halting problem, the linear time hierarchy, and the MRDP theorem. A: Annual ACM/IEEE Symposium on Logic in Computer Science. "LICS '18: proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science". New York: Association for Computing Machinery (ACM), 2018, p. 235-244. 
URIhttp://hdl.handle.net/2117/119165
DOI10.1145/3209108.3209155
ISBN978-1-4503-5583-4
Publisher versionhttps://dl.acm.org/citation.cfm?doid=3209108.3209155
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