Turing Computability

Soare, Robert I.

doi:10.1007/978-3-642-31933-4

Cited by 111 publications

(49 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Some formal axiomatic systems, such as Turing machines [39,45], describe computation and thus set the ultimate basis for machine intelligence. In this setting, the complexity of any other system is defined as the size of the smallest procedure, with respect to some reference machine model, that reproduces the data observed from that system [35].…”

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

“…A bigger system might be able to provide a solution, but we don't in general have the luxury of stepping out to it and peering into the phenomena with which we are concerned from the outside. The basic example of this is that we are bound to compute things from within the limitations incumbent in the models of computation, which are all known to be equivalent and absolute, and under the Church-Turing thesis are not surmountable by any other realisable system either [39,45].…”

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

“…The statement ⊥ says that the system is consistent. 18 Cantor's Diagonalisation Argument was first used to prove that there are infinite sets that cannot be put into one-one correspondence with the natural numbers, and later to prove that the real numbers are uncountable, Russell's Paradox whereby attempted formulations of set theory prior to Zermelo set theory are inconsistent, and Gödel's First Incompleteness Theorem, as well as the unsolvability of Turing's Halting Problem [39,45]. ently regular transient state, and thus must be dealing with some effectiveness with K-system uncertainty.…”

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

“…Since the time of Clausewitz, there has been a burgeoning of interest in uncertainty, from a number of distinct perspectives. In particular, I wish herein to briefly draw on the work on uncertainty and ignorance in economics, which particularly originates in its more philosophically oriented branches, and some of the studies of incompleteness and uncomputability phenomena in mathematics and computer science [19,35,39,45], and to relate the two together.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Autonomy Interrogative

Reid¹

2018

Foundations of Trusted Autonomy

View full text Add to dashboard Cite

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

Section: Uncertainty and Incompletenessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Autonomy Interrogative

Reid¹

2018

Foundations of Trusted Autonomy

View full text Add to dashboard Cite

“…We use standard material from the computability theory (see [17]). A function is subrecursive if it admits a computable total upper bound.…”

Section: Computabilitymentioning

confidence: 99%

Karol Borsuk 1905–1982

Duda¹

2016

Portrety Uczonych. Profesorowie Uniwersytetu Warszawskiego Po 1945, A−K

View full text Add to dashboard Cite

In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. The main results of the paper concern some new directions in this approach. In the case of computable groups we study decidability of amenability of finitely generated subgroups, complexity of the set of all effective Følner sequences and effective paradoxical decomposition.

show abstract

Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

View full text Add to dashboard Cite

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂Pfalse(ωfalse)$S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set νfalse(kfalse)$\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$. The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of normalΣ20$\Sigma ^0_2$‐computable families. We show that every Rogers semilattice of a normalΣ20$\Sigma ^0_2$‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$. In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset Rlm(S)$R_{lm}(S)$ is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all normalΣ20$\Sigma ^0_2$‐computable numberings. We prove that inside this class, the index set of l.m. numberings is normalΣ40$\Sigma ^0_4$‐complete.

show abstract

Turing Computability

Cited by 111 publications

References 30 publications

An Autonomy Interrogative

An Autonomy Interrogative

Karol Borsuk 1905–1982

Rogers semilattices of limitwise monotonic numberings

Contact Info

Product

Resources

About