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Generic complexity of undecidable problems

  • Alexei G. Myasnikov (a1) and Alexander N. Rybalov (a2)

In this paper we study generic complexity of undecidable problems. It turns out that some classical undecidable problems are, in fact, strongly undecidable, i.e., they are undecidable on every strongly generic subset of inputs. For instance, the classical Halting Problem is strongly undecidable. Moreover, we prove an analog of the Rice theorem for strongly undecidable problems, which provides plenty of examples of strongly undecidable problems. Then we show that there are natural super-undecidable problems, i.e., problem which are undecidable on every generic (not only strongly generic) subset of inputs. In particular, there are finitely presented semigroups with super-undecidable word problem. To construct strongly- and super-undecidable problems we introduce a method of generic amplification (an analog of the amplification in complexity theory). Finally, we construct absolutely undecidable problems, which stay undecidable on every non-negligible set of inputs. Their construction rests on generic immune sets.

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[2] A. Borovik , A. Myasnikov , and V. N. Remeslennikov , Multiplicative measures on free groups, International Journal of Algebra and Compututation, vol. 13 (2003), no. 6, pp. 705731.

[4] A. Borovik , A. Myasnikov , and V. Shpilrain , Measuring sets in infinite groups, Computational and Statistical Group Theory, Contemporary Mathematics, vol. 298, 2002, pp. 2142.

[7] J. D. Hamkins and A. D. Miasnikov , The halting problem is decidable on a set of asymptotic probability one, Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 4, pp. 515524.

[8] I. Kapovich , A. Myasnikov , P. Schupp , and V. Shpilrain , Generic-case complexity and decision problems in group theory. Journal of Algebra, vol. 264 (2003), pp. 665694.

[9] I. Kapovich , A. Myasnikov , P. Schupp , and V. Shpilrain , Average-case complexity for the word and membership problems in group theory, Advances in Mathematics, vol. 190 (2005), no. 2, pp. 343359.

[16] A. Rybalov , On the strongly generic undecidability of the halting problem, Theoretical Computer Science, vol. 377 (2007), no. 1-3, pp. 268270.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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