257 research outputs found
The Rice-Shapiro theorem in Computable Topology
We provide requirements on effectively enumerable topological spaces which
guarantee that the Rice-Shapiro theorem holds for the computable elements of
these spaces. We show that the relaxation of these requirements leads to the
classes of effectively enumerable topological spaces where the Rice-Shapiro
theorem does not hold. We propose two constructions that generate effectively
enumerable topological spaces with particular properties from wn--families and
computable trees without computable infinite paths. Using them we propose
examples that give a flavor of this class
On the information carried by programs about the objects they compute
In computability theory and computable analysis, finite programs can compute
infinite objects. Presenting a computable object via any program for it,
provides at least as much information as presenting the object itself, written
on an infinite tape. What additional information do programs provide? We
characterize this additional information to be any upper bound on the
Kolmogorov complexity of the object. Hence we identify the exact relationship
between Markov-computability and Type-2-computability. We then use this
relationship to obtain several results characterizing the computational and
topological structure of Markov-semidecidable sets
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
First Steps in Synthetic Computability Theory
AbstractComputability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland's effective topos and Richman's formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Gödel encodings, but rather use familiar concepts from set theory and topology
Regularity Preserving but not Reflecting Encodings
Encodings, that is, injective functions from words to words, have been
studied extensively in several settings. In computability theory the notion of
encoding is crucial for defining computability on arbitrary domains, as well as
for comparing the power of models of computation. In language theory much
attention has been devoted to regularity preserving functions.
A natural question arising in these contexts is: Is there a bijective
encoding such that its image function preserves regularity of languages, but
its pre-image function does not? Our main result answers this question in the
affirmative: For every countable class C of languages there exists a bijective
encoding f such that for every language L in C its image f[L] is regular.
Our construction of such encodings has several noteworthy consequences.
Firstly, anomalies arise when models of computation are compared with respect
to a known concept of implementation that is based on encodings which are not
required to be computable: Every countable decision model can be implemented,
in this sense, by finite-state automata, even via bijective encodings. Hence
deterministic finite-state automata would be equally powerful as Turing machine
deciders.
A second consequence concerns the recognizability of sets of natural numbers
via number representations and finite automata. A set of numbers is said to be
recognizable with respect to a representation if an automaton accepts the
language of representations. Our result entails that there is one number
representation with respect to which every recursive set is recognizable
A generalization of Markov's approach to the continuity problem for Type 1 computable functions
We axiomatize and generalize Markov's approach to the continuity problem for
Type 1 computable functions, i.e. the problem of finding sufficient conditions
on a computable topological space to obtain a theorem of the form "computable
functions are (effectively) continuous". In a computable topological space, a
point is called effectively adherent to a set if there is an algorithm
that on input a neighborhood of produces a point of in that
neighborhood. We say that a space satisfies a Markov condition if, whenever
a point of is effectively adherent to a subset of , the
singleton is not a semi-decidable subset of . We show that
this condition prevents functions whose domain is from having effective
discontinuities, provided that their codomain is a space where points have
neighborhood bases of co-semi-decidable sets. We then show that results that
forbid effective discontinuities can be turned into (abstract) continuity
results on spaces where the closure and effective closure of semi-decidable
sets naturally agree -this happens for instance on spaces which admit a dense
and computable sequence. This work is motivated by the study of the space of
marked groups, for which the author has shown that most known continuity
results (of Ceitin, Moschovakis, Spreen) do not apply.Comment: 21 page
On the Information Carried by Programs about the Objects They Compute
In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets
A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem
Rice's Theorem states that every nontrivial language property of the
recursively enumerable sets is undecidable. Borchert and Stephan initiated the
search for complexity-theoretic analogs of Rice's Theorem. In particular, they
proved that every nontrivial counting property of circuits is UP-hard, and that
a number of closely related problems are SPP-hard.
The present paper studies whether their UP-hardness result itself can be
improved to SPP-hardness. We show that their UP-hardness result cannot be
strengthened to SPP-hardness unless unlikely complexity class containments
hold. Nonetheless, we prove that every P-constructibly bi-infinite counting
property of circuits is SPP-hard. We also raise their general lower bound from
unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc
On the Information Carried by Programs About the Objects they Compute
International audienceIn computability theory and computable analysis, finite programs can compute infinite objects. Such objects can then be represented by finite programs. Can one characterize the additional useful information contained in a program computing an object, as compared to having the object itself? Having a program immediately gives an upper bound on the Kolmogorov complexity of the object, by simply measuring the length of the program, and such an information cannot usually be derived from an infinite representation of the object. We prove that bounding the Kolmogorov complexity of the object is the only additional useful information. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets. This article is an extended version of [8], including complete proofs and a new result (Theorem 9)
Effectivity and Density in Domains A Survey
AbstractThis article surveys the main results on effectivity and totality in domain theory and its applications. A more abstract and informative proof of Normann's generalized density theorem for total functionals of finite type over the reals is presented
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