Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Decision Problems For Turing Machines

We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(\Sigma_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(\Sigma_1^1)$ is the class of 2-differences of $\Sigma_1^1$-sets. Secondly, it is $\Sigma_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.Comment: To appear in Information Processing Letter

Revising Type-2 Computation and Degrees of Discontinuity

By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0

On Martin's Pointed Tree Theorem

We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT)