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Alternating Turing machines and the analytical hierarchy

10 pagesPublished: June 22, 2012


We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy.
The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

Keyphrases: alternating Turing machines, hyperarithmetical sets, the analytical hierarchy

In: Andrei Voronkov (editor). Turing-100. The Alan Turing Centenary, vol 10, pages 204--213

BibTeX entry
  author    = {Daniel Leivant},
  title     = {Alternating Turing machines and the analytical hierarchy},
  booktitle = {Turing-100. The Alan Turing Centenary},
  editor    = {Andrei Voronkov},
  series    = {EPiC Series in Computing},
  volume    = {10},
  pages     = {204--213},
  year      = {2012},
  publisher = {EasyChair},
  bibsource = {EasyChair,},
  issn      = {2398-7340},
  url       = {},
  doi       = {10.29007/t77g}}
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