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.

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 Controlling Shipping Along the North European Trade Axis

1971 ◽  
Vol 24 (4) ◽  
pp. 553-556
Author(s):  
D. J. Lindsay
Keyword(s):  
Weather Conditions ◽  
North Coast ◽  
Baltic Basin ◽  
Long Distance ◽  
The North ◽  
Large Numbers ◽  
Straight Line ◽  
European Trade ◽  
Trade Route ◽  
The Baltic

By the North European Trade Axis is meant the trade route from Ushant and Land's End, up the English Channel, through the Dover Strait fanning out to serve eastern England, the north coast of continental Europe and leading to the Baltic Basin. Recent events in this area have left a feeling that some form of tightening of control is not only desirable, but is rapidly becoming imperative. There is a basic conflict between the two forms of shipping using the area: the local users who use the area more or less constantly, and the long-distance traders, usually much larger, which arrive in the area for a brief stay after a prolonged period at sea, which has usually been in good weather conditions. Frequently these latter ships have a very poor notion of the hornet's nest into which they are steaming when they arrive. The net result is all too often the same: the local users, with familiarity breeding contempt, wander about as they see fit, with scant regard for routing or the regulations; all too often the big ships arrive from sea with navigating staffs who are too confused, sometimes too ignorant—and sometimes too terrified—to do much more than blunder forward in a straight line hoping for the best. Quite obviously this is not a total picture, and there are large numbers of ships which navigate perfectly competently, but the minority of those which do not seem to be rising rapidly, and show every sign of continuing to increase.

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.

scholarly journals How the Brain might use Division

2020 ◽  
Vol 8 ◽  
pp. 126-137
Author(s):  
Kieran Greer
Keyword(s):  
Artificial Intelligence ◽  
Binary System ◽  
Problem Size ◽  
Symbol Manipulation ◽  
Mathematical Functions ◽  
Binary Operations ◽  
Neural Architecture ◽  
Large Numbers ◽  
The Brain

One of the most fundamental questions in Biology or Artificial Intelligence is how the human brainperforms mathematical functions. How does a neural architecture that may organise itself mostly throughstatistics, know what to do? One possibility is to extract the problem to something more abstract. This becomesclear when thinking about how the brain handles large numbers, for example to the power of something, whensimply summing to an answer is not feasible. In this paper, the author suggests that the maths question can beanswered more easily if the problem is changed into one of symbol manipulation and not just number counting.If symbols can be compared and manipulated, maybe without understanding completely what they are, then themathematical operations become relative and some of them might even be rote learned. The proposed systemmay also be suggested as an alternative to the traditional computer binary system. Any of the actual maths stillbreaks down into binary operations, while a more symbolic level above that can manipulate the numbers andreduce the problem size, thus making the binary operations simpler. An interesting result of looking at this is thepossibility of a new fractal equation resulting from division, that can be used as a measure of good fit and wouldhelp the brain decide how to solve something through self-replacement and a comparison with this good fit.

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