arXiv subjects

Prohrak Kruengthomya

Publications and source records attributed to Prohrak Kruengthomya.

5 recordsLinked to original sources

arXiv2024

Extending the Synchronous Fellow Traveler Property

We introduce an extension of the fellow traveler property which allows fellow travelers to be at distance bounded from above by a function $f(n)$ growing slower than any linear function. We study normal forms satisfying this extended fellow traveler property and certain geometric constraints that naturally generalize two fundamental properties of an automatic normal form - the regularity of its language and the bounded length difference property. We show examples of such normal forms and prove some non-existence theorems.

math.GR↗

arXiv2024

Finite Automata Encoding Piecewise Polynomials

Finite automata are used to encode geometric figures, functions and can be used for image compression and processing. The original approach is to represent each point of a figure in $\mathbb{R}^n$ as a convolution of its $n$ coordinates written in some base. Then a figure is said to be encoded as a finite automaton if the set of convolutions corresponding to the points in this figure is accepted by a finite automaton. The only differentiable functions which can be encoded as a finite automaton in this way are linear. In this paper we propose a representation which enables to encode piecewise polynomial functions with arbitrary degrees of smoothness that substantially extends a family of functions which can be encoded as finite automata. Such representation naturally comes from the framework of hierarchical tensor product B-splines, which are piecewise polynomials widely utilized in numerical computational geometry. We show that finite automata provide a suitable tool for solving computational problems arising in this framework when the support of a function is unbounded.

cs.CG↗

arXiv2024

Cayley Linear-Time Computable Groups

This paper looks at the class of groups admitting normal forms for which the right multiplication by a group element is computed in linear time on a multi-tape Turing machine. We show that the groups $\mathbb{Z}_2 \wr \mathbb{Z}^2$, $\mathbb{Z}_2 \wr \mathbb{F}_2$ and Thompson's group $F$ have normal forms for which the right multiplication by a group element is computed in linear time on a $2$-tape Turing machine. This refines the results previously established by Elder and the authors that these groups are Cayley polynomial-time computable.

math.GR↗

arXiv2021

Cayley Polynomial-Time Computable Groups

We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the class $\mathcal C$ and multiplication by generators computable in linear time on a certain restricted Turing machine model (position-faithful one-tape). We show that many of the algorithmic properties of automatic groups are preserved (quadratic time word problem), prove various closure properties, and show that the class is quite large; for example it includes all virtually polycyclic groups. We then generalise to groups which have normal form language in the class $\mathcal C$ and multiplication by generators computable in polynomial time on a (standard) Turing machine. Of particular interest is when $\mathcal C= \mathrm{REG}$ (the class of regular languages). We prove that $\mathrm{REG}$-Cayley polynomial-time computable groups includes all finitely generated nilpotent groups, the wreath product $\mathbb Z_2 \wr \mathbb Z^2$, and Thompson's group $F$.

math.GR↗

arXiv2020

Nonstandard Cayley automatic representations of fundamental groups

We construct a new family of Cayley automatic representations of semidirect products $\mathbb{Z}^n \rtimes_A \mathbb{Z}$ for which none of the projections of the normal subgroup $\mathbb{Z}^n$ onto each of its cyclic components is finite automaton recognizable. For $n=2$ we describe a family of matrices from $\mathrm{GL}(2,\mathbb{Z})$ corresponding to these representations. We are motivated by a problem of characterization of all possible Cayley automatic representations of these groups.

math.GR↗