scholarly journals Degrees of monotone complexity

2006 ◽  
Vol 71 (4) ◽  
pp. 1327-1341 ◽  
Author(s):  
William C. Calhoun
Keyword(s):  
Kolmogorov Complexity ◽  
Minimal Pair ◽  
Partial Ordering ◽  
Binary String ◽  
Linear Ordering ◽  
Relative Complexity ◽  
Computably Enumerable

AbstractLevin and Schnorr (independently) introduced the monotone complexity, Km (α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α | n) ≤ Km(β | n)+ c for all n. The monotone degree of α is the set of all β such that α Km β and β Km α. We show the monotone degrees contain an antichain of size , a countable dense linear ordering (of degrees of cardinality ), and a minimal pair.Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalban. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We defineareal to be (Km,K)-trivial if for some constant c, Km(α | n) ≤ K(n) + c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

A General Framework for Priority Arguments

1995 ◽  
Vol 1 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Steffen Lempp ◽  
Manuel Lerman
Keyword(s):  
Information Content ◽  
Equivalence Relation ◽  
Decision Problem ◽  
General Framework ◽  
Equivalence Classes ◽  
Partial Ordering ◽  
Decision Problems ◽  
Natural Numbers ◽  
Relative Complexity ◽  
Degrees Of Unsolvability

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure theinformation contentof sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question is equivalent to: Given a computer with access to an oracle which will answer membership questions about a setA, can a program (allowing questions to the oracle) be written which will correctly compute the answers to all membership questions about a setB? If the answer is yes, then we say thatBisTuring reducibletoAand writeB≤TA. We say thatB≡TAifB≤TAandA≤TB. ≡Tis an equivalence relation, and ≤Tinduces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called thedegrees of unsolvability, and elements of this poset are calleddegrees.Post was particularly interested in computability from sets which are partially generated by a computer, namely, those for which the elements of the set can be enumerated by a computer.

Prime models of computably enumerable degree

2008 ◽  
Vol 73 (4) ◽  
pp. 1373-1388
Author(s):  
Rachel Epstein
Keyword(s):  
Minimal Pair ◽  
Prime Model ◽  
Enumerable Degree ◽  
Density Result ◽  
Homogeneous Models ◽  
Computably Enumerable Degree ◽  
Computably Enumerable

AbstractWe examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low2 (c″ > 0″), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 < c < 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.

scholarly journals On the existence of a strong minimal pair

2015 ◽  
Vol 15 (01) ◽  
pp. 1550003 ◽  
Author(s):  
George Barmpalias ◽  
Mingzhong Cai ◽  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Minimal Pair ◽  
Turing Degrees ◽  
Computably Enumerable

We show that there is a strong minimal pair in the computably enumerable (c.e.) Turing degrees, i.e. a pair of nonzero c.e. degrees a and b such that a∩b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a.

Two-dimensional partial orderings: Recursive model theory

1980 ◽  
Vol 45 (1) ◽  
pp. 121-132 ◽  
Author(s):  
Alfred B. Manaster ◽  
Joseph G. Rosenstein
Keyword(s):  
Model Theory ◽  
Companion Paper ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Two Dimensional ◽  
Recursive Model ◽  
Linear Orderings ◽  
Rational Plane ◽  
Partial Orderings

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈A, R〉 to be n-dimensional if there are n linear orderings of A, 〈A, L1〉, 〈A, L2〉 …, 〈A, Ln〉 such that R = L1 ∩ L2 ∩ … ∩ Ln. Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈x1, y1〉 ≤Q×Q 〈x2, y2, if and only if x1 ≤ x2 and y1 ≤ y2) is 2-dimensional, since ≤Q×Q is the intersection of the two lexicographic orderings of Q × Q. In fact, as shown by Dushnik and Miller, a countable partial ordering is n-dimensional if and only if it can be embedded as a subordering of Qn.Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

scholarly journals ON THE COMPLEXITY OF THE SUCCESSIVITY RELATION IN COMPUTABLE LINEAR ORDERINGS

2010 ◽  
Vol 10 (01n02) ◽  
pp. 83-99 ◽  
Author(s):  
ROD DOWNEY ◽  
STEFFEN LEMPP ◽  
GUOHUA WU
Keyword(s):  
New Method ◽  
Independent Interest ◽  
Linear Ordering ◽  
Turing Degrees ◽  
Linear Orderings ◽  
Open Question ◽  
Computably Enumerable

In this paper, we solve a long-standing open question (see, e.g. Downey [6, Sec. 7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.

The independence of the Prime Ideal Theorem from the Order-Extension Principle

1999 ◽  
Vol 64 (1) ◽  
pp. 199-215 ◽  
Author(s):  
U. Felgner ◽  
J. K. Truss
Keyword(s):  
Boolean Algebra ◽  
Prime Ideal ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Extension Principle ◽  
Generic Extension ◽  
Direct Verification ◽  
The Family ◽  
Order Extension ◽  
Prime Ideal Theorem

AbstractIt is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel–Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a ‘generic’ extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure.

Indicators, Chains, Antichains, Ramsey Property

2014 ◽  
Vol 57 (3) ◽  
pp. 631-639
Author(s):  
Miodrag Sokić
Keyword(s):  
Partial Ordering ◽  
Linear Ordering ◽  
Ramsey Property ◽  
Relational Structures ◽  
Weak Ordering ◽  
Finite Structures ◽  
Maximal Chains

AbstractWe introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form , where ≤ is a total ordering on A and ⪯ is a linear ordering on the set fa 2 A : Ii (a)g. The second class contains structures of the form a , where (A,≤) is a weak ordering and ⪯ is a linear ordering on A such that A is partitioned by into maximal chains in the partial ordering ≤ and each is an interval with respect to .

NONSTANDARD APPROACH TO POSSIBILITY MEASURES

1996 ◽  
Vol 04 (03) ◽  
pp. 275-301 ◽  
Author(s):  
IVAN KRAMOSIL
Keyword(s):  
Random Variable ◽  
Unit Interval ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Possibility Measure ◽  
Real Numbers ◽  
Standard Linear ◽  
General Abstract ◽  
Nonstandard Approach ◽  
Abstract Probability

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.

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