Source: J. Symbolic Logic Volume 73, Issue 2
(2008), 656-673.
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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