4,395 research outputs found
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 -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
-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 , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .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 -complete to determine whether
the omega-language of a given Turing machine is countably infinite, where
is the class of 2-differences of -sets. Secondly,
it is -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)
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