Embeddings of the 1-3-1 lattice

2020 ◽  
pp. 149-187
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Group Theory ◽  
Computability Theory ◽  
Solutions To Equations ◽  
Degrees Of Unsolvability ◽  
Computably Enumerable

This chapter studies embeddings of the 1–3–1 lattice, proving Theorem 1.6. One of the central and longstanding areas of classical computability theory concerns the structure of the degrees of unsolvability, and particularly the computably enumerable degrees. In the same way that studying symmetries in nature and solutions to equations leads to group theory, studies of the computational content of mathematics lead naturally to the structure of sets of integers under reducibilities. Understanding these structures should lead to insights into relative computability. Notable in these studies is the question of embeddability into the c.e. degrees. There are nondistributive lattices that can be embedded. Ultimately, the embedding of the 1–3–1 lattice was an amazing result, and introduced the “continuous tracing” technique into computability theory.

scholarly journals Generalized cohesiveness

1999 ◽  
Vol 64 (2) ◽  
pp. 489-516 ◽  
Author(s):  
Tamara Hummel ◽  
Carl G. Jockusch
Keyword(s):  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Set ◽  
Degrees Of Unsolvability ◽  
Computably Enumerable

AbstractWe study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

2018 ◽  
Vol 24 (1) ◽  
pp. 53-89 ◽  
Author(s):  
ROD DOWNEY ◽  
NOAM GREENBERG
Keyword(s):  
Computability Theory ◽  
Ordinal Notations ◽  
Image Position ◽  
Computably Enumerable

AbstractWe introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

scholarly journals Definable Encodings in the Computably Enumerable Sets

2000 ◽  
Vol 6 (2) ◽  
pp. 185-196 ◽  
Author(s):  
Peter A. Cholak ◽  
Leo A. Harrington
Keyword(s):  
Computability Theory ◽  
Finite Sets ◽  
Warm Up ◽  
Computably Enumerable Sets ◽  
Disjoint Sets ◽  
New Form ◽  
Historical Remarks ◽  
Equivalent Class ◽  
The Ideal ◽  
Computably Enumerable

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an infinite non-computable subset). We use A* to denote the equivalent class of A under the ideal of finite sets.

scholarly journals Hierarchy of Computably Enumerable Degrees II

10.53733/133 ◽  
2021 ◽  
Vol 52 ◽  
pp. 175-231
Author(s):  
Rod Downey ◽  
Noam Greenberg ◽  
Ellen Hammatt
Keyword(s):  
Computability Theory ◽  
Turing Degrees ◽  
Computably Enumerable

A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.

scholarly journals Randomness and Computability

2021 ◽  
Author(s):  
◽  
Adam Richard Day
Keyword(s):  
Computability Theory ◽  
Binary Sequences ◽  
Cantor Space ◽  
Computably Enumerable Sets ◽  
Schnorr Randomness ◽  
Separation Result ◽  
The Difference ◽  
Deep Relationship ◽  
Scott Set ◽  
Computably Enumerable

<p>This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets.</p>

scholarly journals Randomness and Computability

2021 ◽  
Author(s):  
◽  
Adam Richard Day
Keyword(s):  
Computability Theory ◽  
Binary Sequences ◽  
Cantor Space ◽  
Computably Enumerable Sets ◽  
Schnorr Randomness ◽  
Separation Result ◽  
The Difference ◽  
Deep Relationship ◽  
Scott Set ◽  
Computably Enumerable

<p>This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets.</p>

A Hierarchy of Turing Degrees

2020 ◽  
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Mathematical Logic ◽  
Group Theory ◽  
Graduate Students ◽  
Computability Theory ◽  
Structure Theory ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Closed Sets ◽  
Construction Techniques ◽  
High Level

Computability theory is a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field has developed growing connections in diverse areas of mathematics, with applications in topology, group theory, and other subfields. This book introduces a new hierarchy that allows them to classify the combinatorics of constructions from many areas of computability theory, including algorithmic randomness, Turing degrees, effectively closed sets, and effective structure theory. This unifying hierarchy gives rise to new natural definability results for Turing degree classes, demonstrating how dynamic constructions become reflected in definability. The book presents numerous construction techniques involving high-level nonuniform arguments, and their self-contained work is appropriate for graduate students and researchers. Blending traditional and modern research results in computability theory, the book establishes novel directions in the field.

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