Source: J. Symbolic Logic
Volume 73, Issue 2
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.
S. I. Adjan and V. G. Durnev, Decision problems for groups and semigroups, Russian Mathematical Surveys, vol. 55 (2000), no. 2, pp. 207--296.
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. 705--731.
--------, Algorithmic stratification of the conjugacy problem in Miller's groups, International Journal of Algebra and Computation, to appear.
A. Borovik, A. Myasnikov, and V. Shpilrain, Measuring sets in infinite groups, Computational and Statistical Group Theory, Contemporary Mathematics, vol. 298, 2002, pp. 21--42.
S. B. Cooper, Computability Ttheory, Chapman and Hall/CRC, 2003.
R. Gilman, A. D. Miasnikov, A. G. Myasnikov, and A. Ushakov, Generic complexity, preprint.
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. 515--524.
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. 665--694.
--------, Average-case complexity for the word and membership problems in group theory, Advances in Mathematics, vol. 190 (2005), no. 2, pp. 343--359.
D. E. Knuth, The Art of Computer Programming. Vol. 3: Sorting and Searching, Addison-Wesley, 1998.
Mathematical Reviews (MathSciNet): MR445948
A. A. Markov, On the impossibility of certain algorithms in the theory of associative systems, Doklady Akademii Nauk SSSR, vol. 55 (1947), pp. 587--590.
Mathematical Reviews (MathSciNet): MR20528
Y. V. Matiyasevich, Simple examples of undecidable associative calculi, Doklady Akademii Nauk SSSR, vol. 173 (1967), pp. 1264--1266.
Mathematical Reviews (MathSciNet): MR216955
E. Mendelson, Introduction to Mathematical Logic, Chapman and Hall/CRC, 1997.
A. Miasnikov, A. Ushakov, and D. W. Won, Generic complexity of the word problem in finitely presented semigroups, 2006, preprint.
E. L. Post, Recursive unsolvability of a problem of Thue, Journal of Symbolic Logic, vol. 12 (1947), no. 1, pp. 1--11.
Mathematical Reviews (MathSciNet): MR20527
A. Rybalov, On the strongly generic undecidability of the halting problem, Theoretical Computer Science, vol. 377 (2007), no. 1--3, pp. 268--270.
J. E. Savage, The Complexity of Computing, John Wiley and Sons Inc., 1977.
Mathematical Reviews (MathSciNet): MR495205
G. S. Tseitin, An associative system with undecidable equivalence problem, Reports of Mathematical Institute of Soviet Academy of Science, vol. 52 (1958), pp. 172--189.
Mathematical Reviews (MathSciNet): MR99922