On the structures inside truth-table degrees

2001 ◽  
Vol 66 (2) ◽  
pp. 731-770 ◽  
Author(s):  
Frank Stephan
Keyword(s):  
Finite Number ◽  
Open Problem ◽  
Affirmative Answer ◽  
Truth Table ◽  
Turing Degrees ◽  
Positive Degree

AbstractThe following theorems on the structure inside nonrecursive truth-table degrees are established: Dëgtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and therefore do not contain any objective set. Furthermore, a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enumerable semirecursive sets, one of coenumerable semirecursive sets and one of sets, which are neither enumerable nor coenumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integer as for example nineteen, but it is never an even finite number.

The theory of the recursively enumerable weak truth-table degrees is undecidable

1992 ◽  
Vol 57 (3) ◽  
pp. 864-874 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
André Nies ◽  
Richard A. Shore
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Undecidable Theory ◽  
Weak Truth Table Degrees

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

scholarly journals PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

2018 ◽  
Vol 24 (2) ◽  
pp. 165-174
Author(s):  
BJØRN KJOS-HANSSEN
Keyword(s):  
Open Problem ◽  
Main Idea ◽  
Computability Theory ◽  
Turing Degrees ◽  
Nontrivial Automorphism

AbstractIs there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

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.

OPERATING FUNCTIONS ON MODULATION AND WIENER AMALGAM SPACES

2017 ◽  
Vol 230 ◽  
pp. 72-82 ◽  
Author(s):  
MASAHARU KOBAYASHI ◽  
ENJI SATO
Keyword(s):  
Open Problem ◽  
Composition Operators ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Amalgam Spaces ◽  
Funct Anal

The goal of this paper is to characterize the operating functions on modulation spaces$M^{p,1}(\mathbb{R})$and Wiener amalgam spaces$W^{p,1}(\mathbb{R})$. This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal.270(2016), pp. 621–648).

A test for the existence of tautologies according to many-valued truth-tables

1950 ◽  
Vol 15 (3) ◽  
pp. 182-184 ◽  
Author(s):  
Jan Kalicki
Keyword(s):  
Finite Number ◽  
Truth Table ◽  
Effective Procedure ◽  
Single Variable ◽  
Truth Values ◽  
Contraction Process ◽  
The One ◽  
Image Position ◽  
Truth Tables

Theorem. There is an effective procedure to decide whether the set of tautologies determined by a given truth-table with a finite number of elements is empty or not.Proof. Let W(P) be a w.f.f. with a single variable P and n a given n-valued truth-table with elements (values)Substitute 1, 2, 3, …, n in succession for P. By the usual contraction process let W(P) assume the truth-values w1, w2, w3, …, wn respectively. The sequencewill be called the value sequence of W(P).Value sequences consisting of designated elements of exclusively will be called designated; others will be called undesignated.All the W(P)'s will be classified in the following way:(a) to the first class CL1 of W(P)'s there belongs the one element P,(b) to the (t + 1)th class CLt + 1 belong all the w.f.f. which can be built up by means of one generating connective from constituent w.f.f. of which one is an element of CLt and all the others (if any) are elements of CLn ≤ t.For example, if N and C are the connectives described by a truth-table etc.Let ∣CLn∣ stand for the set of value sequences of the elements of CLn.

Contiguity and distributivity in the enumerable Turing degrees

1997 ◽  
Vol 62 (4) ◽  
pp. 1215-1240 ◽  
Author(s):  
Rodney G. Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Turing Degrees ◽  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree

AbstractWe prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.

scholarly journals Words for maximal Subgroups of Fi24‘

2019 ◽  
Vol 17 (1) ◽  
pp. 1491-1500
Author(s):  
Faisal Yasin ◽  
Adeel Farooq ◽  
Chahn Yong Jung
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Finite Number ◽  
Physical Properties ◽  
Open Problem ◽  
Bond Order ◽  
Energy Levels ◽  
Maximal Subgroups ◽  
Mathematical Application ◽  
A Finite Group

AbstractGroup Theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties. The symmetry of a molecule provides us with the various information, such as - orbitals energy levels, orbitals symmetries, type of transitions than can occur between energy levels, even bond order, all that without rigorous calculations. The fact that so many important physical aspects can be derived from symmetry is a very profound statement and this is what makes group theory so powerful. In group theory, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite. The Fischer groups Fi22, Fi23 and Fi24‘ are introduced by Bernd Fischer and there are 25 maximal subgroups of Fi24‘. It is an open problem to find the generators of maximal subgroups of Fi24‘. In this paper we provide the generators of 10 maximal subgroups of Fi24‘.

scholarly journals Full rainbow matchings in graphs and hypergraphs

2021 ◽  
pp. 1-19
Author(s):  
Pu Gao ◽  
Reshma Ramadurai ◽  
Ian M. Wanless ◽  
Nick Wormald
Keyword(s):  
Open Problem ◽  
Affirmative Answer ◽  
Simple Graph ◽  
Rainbow Matching ◽  
Special Case

Abstract Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\] , where \[c > 9/10\] , and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from M i for each \[1 \leqslant i \leqslant m\] . This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

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