A parameterized halting problem, the linear time hierarchy, and the MRDP theorem
Document typeConference report
PublisherAssociation for Computing Machinery (ACM)
Rights accessOpen Access
European Commission's projectAUTAR - A Unified Theory of Algorithmic Relaxations (EC-H2020-648276)
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 . 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.
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