Generic complexity of undecidable problems

Abstract

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.