scholarly journals Characterisations of variant transfinite computational models: Infinite time Turing, ordinal time Turing, and Blum–Shub–Smale machines

Computability ◽  
2021 ◽  
pp. 1-22
Author(s):  
Philip Welch
Keyword(s):  
Computational Models ◽  
Turing Machines ◽  
Infinite Time ◽  
Open Problems ◽  
The Subject ◽  
Time Machines ◽  
Space Requirements

We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, Σ 2 -codes and Π 3 -reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have On length tapes; 2) determine which admissible lengths of tapes for transfinite time machines with long tapes allow the machine to address each of their cells – a question raised by B. Rin; 3) characterise exactly the strength and behaviour of transfinitely acting Blum–Shub–Smale machines using a Liminf rule on their registers – thereby establishing there is a universal such machine. This is in contradistinction to the machine using a ‘continuity’ rule which fails to be universal.

Some Assumptions about Problem Solving Representation in Turing’s Model of Intelligence

1970 ◽  
Vol 4 (2) ◽  
pp. 136-142 ◽  
Author(s):  
Raymundo Morado ◽  
Francisco Hernández-Quiroz
Keyword(s):  
Problem Solving ◽  
Computational Models ◽  
Turing Machines

Turing machines as a model of intelligence can be motivated under some assumptions, both mathematical and philosophical. Some of these are about the possibility, the necessity, and the limits of representing problem solving by mechanical means. The assumptions about representation that we consider in this paper are related to information representability and availability, processing as solving, nonessentiality of complexity issues, and finiteness, discreteness and sequentiality of the representation. We discuss these assumptions and particularly something that might happen if they were to be rejected or weakened. Tinkering with these assumptions sheds light on the import of alternative computational models.

An Invitation to Combinatorics

2021 ◽  
Author(s):  
Shahriar Shahriari
Keyword(s):  
Stirling Numbers ◽  
Exclusion Principle ◽  
Open Problems ◽  
Warm Up ◽  
Calculus Sequence ◽  
Award Winning ◽  
Self Study ◽  
Conversational Style ◽  
The Subject ◽  
Partitions Of Integers

Active student engagement is key to this classroom-tested combinatorics text, boasting 1200+ carefully designed problems, ten mini-projects, section warm-up problems, and chapter opening problems. The author – an award-winning teacher – writes in a conversational style, keeping the reader in mind on every page. Students will stay motivated through glimpses into current research trends and open problems as well as the history and global origins of the subject. All essential topics are covered, including Ramsey theory, enumerative combinatorics including Stirling numbers, partitions of integers, the inclusion-exclusion principle, generating functions, introductory graph theory, and partially ordered sets. Some significant results are presented as sets of guided problems, leading readers to discover them on their own. More than 140 problems have complete solutions and over 250 have hints in the back, making this book ideal for self-study. Ideal for a one semester upper undergraduate course, prerequisites include the calculus sequence and familiarity with proofs.

scholarly journals P ≠ NP ∩ co-NP for Infinite Time Turing Machines

2005 ◽  
Vol 15 (5) ◽  
pp. 577-592 ◽  
Author(s):  
Vinay Deolalikar ◽  
Joel David Hamkins ◽  
Ralf Schindler
Keyword(s):  
Turing Machines ◽  
Infinite Time

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

Milnor's conjecture on monotonicity of topological entropy: Results and questions

2014 ◽  
Author(s):  
Sebastian van Strien
Keyword(s):  
Topological Entropy ◽  
State Of The Art ◽  
Rigidity Theorem ◽  
Real Polynomial ◽  
Open Problems ◽  
Polynomial Maps ◽  
History Of ◽  
The Subject ◽  
Quadratic Case ◽  
Real Polynomial Maps

This chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.

scholarly journals Processing Information in the Clouds

Proceedings ◽  
2020 ◽  
Vol 47 (1) ◽  
pp. 25
Author(s):  
Mark Burgin ◽  
Eugene Eberbach ◽  
Rao Mikkilineni
Keyword(s):  
Artificial Intelligence ◽  
Cloud Computing ◽  
Human Brain ◽  
Incremental Learning ◽  
Computational Models ◽  
Higher Order ◽  
Turing Machines ◽  
New Paradigm ◽  
New Model ◽  
Processing Information

Cloud computing makes the necessary resources available to the appropriate computation to improve scaling, resiliency, and the efficiency of computations. This makes cloud computing a new paradigm for computation by upgrading its artificial intelligence (AI) to a higher order. To explore cloud computing using theoretical tools, we use cloud automata as a new model for computation. Higher-level AI requires infusing features of the human brain into AI systems such as incremental learning all the time. Consequently, we propose computational models that exhibit incremental learning without stopping (sentience). These features are inherent in reflexive Turing machines, inductive Turing machines, and limit Turing machines.

Local Algorithms for Topology Control in Ad-Hoc Networks

2010 ◽  
pp. 51-58
Author(s):  
Evangelos Kranakis ◽  
Jorge Urrutia
Keyword(s):  
Topology Control ◽  
Ad Hoc ◽  
Edge Coloring ◽  
Dominating Sets ◽  
Open Problems ◽  
Local Algorithms ◽  
Trade Offs ◽  
The Subject ◽  
Hoc Networks ◽  
Local Topology

In this chapter, we present a survey of recent techniques for local topology control in location aware Unit Disk Graphs including local algorithms for Routing, Traversal, Planar Spanners, Dominating and Connected Dominating Sets, and Vertex and Edge Coloring. In addition to investigating trade-offs for these problems, we discuss open problems that will play an important role in the future development of the subject.

The formal ball model for -categories

2010 ◽  
Vol 21 (1) ◽  
pp. 41-64 ◽  
Author(s):  
MATEUSZ KOSTANEK ◽  
PAWEŁ WASZKIEWICZ
Keyword(s):  
Metric Space ◽  
Computational Models ◽  
Metric Spaces ◽  
List Type ◽  
Type Number ◽  
Open Problems

We generalise the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann in order to obtain computational models for separated-categories. We fully describe-categories that are(a)Yoneda complete(b)continuous Yoneda completevia their formal ball models. Our results yield solutions to two open problems in the theory of quasi-metric spaces by showing that:(a)a quasi-metric spaceXis Yoneda complete if and only if its formal ball model is a dcpo, and(b)a quasi-metric spaceXis continuous and Yoneda complete if and only if its formal ball modelBXis a domain that admits a simple characterisation of approximation.

scholarly journals Space and time complexity for infinite time Turing machines

2020 ◽  
Vol 30 (6) ◽  
pp. 1239-1255
Author(s):  
Merlin Carl
Keyword(s):  
Time Complexity ◽  
Space Complexity ◽  
Complexity Classes ◽  
Turing Machines ◽  
Infinite Time ◽  
Space And Time ◽  
Open Questions ◽  
Separation Results ◽  
Ordinal Computability

Abstract We consider notions of space by Winter [21, 22]. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in Section 7.2.2 of Carl (2019, Ordinal Computability: An Introduction to Infinitary Machines), which appear here as Lemma $6$ up to Corollary $9$.

Sign in / Sign up

Export Citation Format

Share Document