scholarly journals Low-complexity computations for nilpotent subgroup problems

2019 ◽  
Vol 29 (04) ◽  
pp. 639-661
Author(s):  
Jeremy Macdonald ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Nilpotent Groups ◽  
Low Complexity ◽  
Nilpotency Class ◽  
Nilpotent Subgroup ◽  
Torsion Subgroup ◽  
Straight Line ◽  
Algorithmic Problems ◽  
Straight Line Programs

We solve the following algorithmic problems using [Formula: see text] circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal’cev coordinates, the algorithms run in quartic time.

scholarly journals POWER CIRCUITS, EXPONENTIAL ALGEBRA, AND TIME COMPLEXITY

2012 ◽  
Vol 22 (06) ◽  
pp. 1250047 ◽  
Author(s):  
ALEXEI G. MIASNIKOV ◽  
ALEXANDER USHAKOV ◽  
DONG WOOK WON
Keyword(s):  
Polynomial Time ◽  
Binary Operations ◽  
Large Numbers ◽  
Straight Line ◽  
Algorithmic Problems ◽  
Power Circuits ◽  
Combinatorial Group ◽  
Computational Properties ◽  
Exponential Space ◽  
Straight Line Programs

Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x ⋅ 2y, z = x ⋅ 2-y (the former two are partial) and predicates < and =. Notice that in this case very large numbers, which are obtained as n towers of exponentiation in the base 2 can be realized as n applications of the operation x ⋅ 2y, so working with such numbers given in the usual binary expansions requires super exponential space. We define a new compressed representation for integers by power circuits (a particular type of straight-line programs) which is unique and easily computable, and show that the operations above can be performed in polynomial time if the numbers are presented by power circuits. We mention several applications of this technique to algorithmic problems, in particular, we prove that the quantifier-free theories of various exponential algebras are decidable in polynomial time.

Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

scholarly journals A NEW LINEAR GENETIC PROGRAMMING APPROACH BASED ON STRAIGHT LINE PROGRAMS: SOME THEORETICAL AND EXPERIMENTAL ASPECTS

2009 ◽  
Vol 18 (05) ◽  
pp. 757-781 ◽  
Author(s):  
CÉSAR L. ALONSO ◽  
JOSÉ LUIS MONTAÑA ◽  
JORGE PUENTE ◽  
CRUZ ENRIQUE BORGES
Keyword(s):  
Data Structure ◽  
Genetic Programming ◽  
Computer Programs ◽  
Symbolic Regression ◽  
Programming Approach ◽  
Linear Genetic Programming ◽  
Straight Line ◽  
Structured Representations ◽  
Regression Problems ◽  
Straight Line Programs

Tree encodings of programs are well known for their representative power and are used very often in Genetic Programming. In this paper we experiment with a new data structure, named straight line program (slp), to represent computer programs. The main features of this structure are described, new recombination operators for GP related to slp's are introduced and a study of the Vapnik-Chervonenkis dimension of families of slp's is done. Experiments have been performed on symbolic regression problems. Results are encouraging and suggest that the GP approach based on slp's consistently outperforms conventional GP based on tree structured representations.

On the average-case complexity of underdetermined functions

2018 ◽  
Vol 28 (4) ◽  
pp. 201-221
Author(s):  
Aleksandr V. Chashkin
Keyword(s):  
Boolean Functions ◽  
Correct Order ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity ◽  
Straight Line ◽  
Straight Line Programs

Abstract The average-case complexity of computation of underdetermined functions by straight-line programs with conditional stop over the basis of all at most two-place Boolean functions is considered. Correct order estimates of the average-case complexity of functions with maximum average-case complexity among all underdetermined functions are derived depending on the degree of their determinacy, the size of their domain, and the size of their support.

Sign in / Sign up

Export Citation Format

Share Document