Incompleteness and the Halting Problem

Studia Logica ◽  
2021 ◽  
Author(s):  
Cristian S. Calude
Keyword(s):  
Halting Problem

A proof of Beigel's cardinality conjecture

1992 ◽  
Vol 57 (2) ◽  
pp. 677-681 ◽  
Author(s):  
Martin Kummer
Keyword(s):  
Complexity Theory ◽  
Structural Complexity ◽  
Halting Problem ◽  
Parallel Queries

In 1986, Beigel [Be87] (see also [Od89, III.5.9]) proved the nonspeedup theorem: if A, B ⊆ ω, and as a function of 2n variables can be computed by an algorithm which makes at most n queries to B, then A is recursive (informally, 2n parallel queries to a nonrecursive oracle A cannot be answered by making n sequential (or “adaptive”) queries to an arbitrary oracle B). Here, 2n cannot be replaced by 2n − 1. In subsequent papers of Beigel, Gasarch, Gill, Hay, and Owings the theory of “bounded query classes” has been further developed (see, for example, [BGGOta], [BGH89], and [Ow89]). The topic has also been studied in the context of structural complexity theory (see, for example, [AG88], [Be90], and [JY90]).If A ⊆ ω and n ≥ 1, let . Beigel [Be87] stated the powerful “cardinality conjecture” (CC): if A, B ⊆ ω, and can be computed by an algorithm which makes at most n queries to B, then A is recursive. Owings [Ow89] verified CC for n = 1, and, for n 1, he proved that A is recursive in the halting problem. We prove that CC is true for all n.

The Halting Problem and Formal Proofs

1993 ◽  
pp. 121-138
Author(s):  
Jan L. A. Snepscheut
Keyword(s):  
Formal Proofs ◽  
Halting Problem

Formalization of the Undecidability of the Halting Problem for a Functional Language

2018 ◽  
pp. 196-209 ◽  
Author(s):  
Thiago Mendonça Ferreira Ramos ◽  
César Muñoz ◽  
Mauricio Ayala-Rincón ◽  
Mariano Moscato ◽  
Aaron Dutle ◽  
...  
Keyword(s):  
Functional Language ◽  
Halting Problem

A probabilistic anytime algorithm for the halting problem

Computability ◽  
2018 ◽  
Vol 7 (2-3) ◽  
pp. 259-271 ◽  
Author(s):  
Cristian S. Calude ◽  
Monica Dumitrescu
Keyword(s):  
Halting Problem ◽  
Anytime Algorithm

scholarly journals Undecidable, Unrecognizable, and Quantum Computing

2020 ◽  
Vol 2 (3) ◽  
pp. 337-342
Author(s):  
Michael Siomau
Keyword(s):  
Quantum Computing ◽  
Turing Machine ◽  
Quantum Computer ◽  
Practical Realization ◽  
Real Numbers ◽  
Theoretical Limit ◽  
Halting Problem ◽  
The Real ◽  
Computing Model ◽  
Successful Design

Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.

Promptness does not imply superlow cuppability

2009 ◽  
Vol 74 (4) ◽  
pp. 1264-1272 ◽  
Author(s):  
David Diamondstone
Keyword(s):  
Explicit Construction ◽  
Negative Answer ◽  
Related Question ◽  
Truth Table ◽  
Turing Degrees ◽  
Classical Theorem ◽  
Halting Problem ◽  
Simple Set ◽  
Complete Set

AbstractA classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅′. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.

scholarly journals On the strongly generic undecidability of the Halting Problem

2007 ◽  
Vol 377 (1-3) ◽  
pp. 268-270 ◽  
Author(s):  
Alexander Rybalov
Keyword(s):  
Halting Problem

The halting problem for Turning machines

1986 ◽  
Vol 18 (1) ◽  
pp. 52-52 ◽  
Author(s):  
D M Wilson
Keyword(s):  
Halting Problem
Sign in / Sign up

Export Citation Format

Share Document