scholarly journals Nonexistence of minimal pairs for generic computability

2013 ◽  
Vol 78 (2) ◽  
pp. 511-522 ◽  
Author(s):  
Gregory Igusa
Keyword(s):  
Group Theory ◽  
Complexity Theory ◽  
Primary Result ◽  
Theoretic Analysis ◽  
Minimal Pairs ◽  
Generic Computation

AbstractA generic computation of a subset A of ℕ consists of a computation that correctly computes most of the bits of A, and never incorrectly computes any bits of A, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.

Preliminaries

2020 ◽  
pp. 5-56
Author(s):  
Andreas Bolfing
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Group Theory ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Ring Theory ◽  
Cryptographic Primitives ◽  
Mathematical Concepts ◽  
Cyclic Groups ◽  
Efficiency Of Algorithms

Blockchains are heavily based on mathematical concepts, in particular on algebraic structures. This chapter starts with an introduction to the main aspects in number theory, such as the divisibility of integers, prime numbers and Euler’s totient function. Based on these basics, it follows a very detailed introduction to modern algebra, including group theory, ring theory and field theory. The algebraic main results are then applied to describe the structure of cyclic groups and finite fields, which are needed to construct cryptographic primitives. The chapter closes with an introduction to complexity theory, examining the efficiency of algorithms.

scholarly journals THE GENERIC DEGREES OF DENSITY-1 SETS, AND A CHARACTERIZATION OF THE HYPERARITHMETIC REALS

2015 ◽  
Vol 80 (4) ◽  
pp. 1290-1314 ◽  
Author(s):  
GREGORY IGUSA
Keyword(s):  
Complexity Theory ◽  
Point Of View ◽  
Turing Degrees ◽  
Worst Case ◽  
Know How ◽  
Theoretic Point ◽  
Image Position ◽  
Generic Computation

AbstractA generic computation of a subsetAof ℕ is a computation which correctly computes most of the bits ofA, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory, where it has been noticed that frequently, it is more important to know how difficult a type of problem is in the general case than how difficult it is in the worst case. When we study this concept from a recursion theoretic point of view, to create a transitive relationship, we are forced to consider oracles that sometimes fail to give answers when asked questions. Unfortunately, this makes working in the generic degrees quite difficult. Indeed, we show that generic reduction is$\Pi _1^1$―complete. To help avoid this difficulty, we work with the generic degrees of density-1 reals. We demonstrate how an understanding of these degrees leads to a greater understanding of the overall structure of the generic degrees, and we also use these density-1 sets to provide a new a characterization of the hyperartithmetical Turing degrees.

Final Consonant Deletion in African American Children Speaking Black English

1993 ◽  
Vol 24 (3) ◽  
pp. 161-166 ◽  
Author(s):  
Michael J. Moran
Keyword(s):  
African American ◽  
Black English ◽  
African American Children ◽  
Vowel Length ◽  
Minimal Pairs ◽  
Speech Evaluation ◽  
American Children

The purpose of this study was to determine whether African American children who delete final consonants mark the presence of those consonants in a manner that might be overlooked in a typical speech evaluation. Using elicited sentences from 10 African American children from 4 to 9 years of age, two studies were conducted. First, vowel length was determined for minimal pairs in which final consonants were deleted. Second, listeners who identified final consonant deletions in the speech of the children were provided training in narrow transcription and reviewed the elicited sentences a second time. Results indicated that the children produced longer vowels preceding "deleted" voiced final consonants, and listeners perceived fewer deletions following training in narrow transcription. The results suggest that these children had knowledge of the final consonants perceived to be deleted. Implications for assessment and intervention are discussed.

Group Theory

2009 ◽  
Author(s):  
Pierre Ramond
Keyword(s):  
Group Theory
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