1,914 research outputs found
Verifying whether One-Tape Non-Deterministic Turing Machines Run in Time
We discuss the following family of problems, parameterized by integers and : Does a given one-tape non-deterministic -state Turing
machine make at most steps on all computations on all inputs of length
, for all ?
Assuming a fixed tape and input alphabet, we show that these problems are
co-NP-complete and we provide good non-deterministic and co-non-deterministic
lower bounds. Specifically, these problems can not be solved in
non-deterministic time by multi-tape Turing machines. We also
show that the complements of these problems can be solved in
non-deterministic time and not in non-deterministic time by
multi-tape Turing machines.Comment: 12 pages + 5 pages appendi
A Concrete View of Rule 110 Computation
Rule 110 is a cellular automaton that performs repeated simultaneous updates
of an infinite row of binary values. The values are updated in the following
way: 0s are changed to 1s at all positions where the value to the right is a 1,
while 1s are changed to 0s at all positions where the values to the left and
right are both 1. Though trivial to define, the behavior exhibited by Rule 110
is surprisingly intricate, and in (Cook, 2004) we showed that it is capable of
emulating the activity of a Turing machine by encoding the Turing machine and
its tape into a repeating left pattern, a central pattern, and a repeating
right pattern, which Rule 110 then acts on. In this paper we provide an
explicit compiler for converting a Turing machine into a Rule 110 initial
state, and we present a general approach for proving that such constructions
will work as intended. The simulation was originally assumed to require
exponential time, but surprising results of Neary and Woods (2006) have shown
that in fact, only polynomial time is required. We use the methods of Neary and
Woods to exhibit a direct simulation of a Turing machine by a tag system in
polynomial time
The reduction of tape reversals for off-line one-tape Turing machines
AbstractFor off-line one-tape Turing machines the number of tape reversals required for various computations may be uniformly reduced by an arbitrary constant factor
Length and Area Functions on Groups and Quasi-Isometric Higman Embeddings
We survey recent results about asymptotic functions of groups, obtained by
the authors in collaboration with J.-C.Birget, V. Guba and E. Rips. We also
discuss methods used in the proofs of these results.Comment: 33 page
Some Thoughts on Hypercomputation
Hypercomputation is a relatively new branch of computer science that emerged
from the idea that the Church--Turing Thesis, which is supposed to describe
what is computable and what is noncomputable, cannot possible be true. Because
of its apparent validity, the Church--Turing Thesis has been used to
investigate the possible limits of intelligence of any imaginable life form,
and, consequently, the limits of information processing, since living beings
are, among others, information processors. However, in the light of
hypercomputation, which seems to be feasibly in our universe, one cannot impose
arbitrary limits to what intelligence can achieve unless there are specific
physical laws that prohibit the realization of something. In addition,
hypercomputation allows us to ponder about aspects of communication between
intelligent beings that have not been considered befor
Polynomially-bounded Dehn functions of groups
On the one hand, it is well known that the only subquadratic Dehn function of
finitely presented groups is the linear one. On the other hand there is a huge
class of Dehn functions with growth at least (essentially all
possible such Dehn functions) constructed in \cite{SBR} and based on the time
functions of Turing machines and S-machines. The class of Dehn functions
with remained more mysterious even though it
has attracted quite a bit of attention (see, for example, \cite{BB}). We fill
the gap obtaining Dehn functions of the form (and much more) for
all real computable in reasonable time, for example,
or , or is any algebraic number. As in \cite{SBR}, we use
S-machines but new tools and new way of proof are needed for the best possible
lower bound .Comment: 98 pages, 18 figures, replaced figures, correction
Hypercomputability of quantum adiabatic processes: Fact versus Prejudices
We give an overview of a quantum adiabatic algorithm for Hilbert's tenth
problem, including some discussions on its fundamental aspects and the emphasis
on the probabilistic correctness of its findings. For the purpose of
illustration, the numerical simulation results of some simple Diophantine
equations are presented. We also discuss some prejudicial misunderstandings as
well as some plausible difficulties faced by the algorithm in its physical
implementation.Comment: 25 pages, 4 figures. Invited paper for a special issue of the Journal
of Applied Mathematics and Computatio
Finite Computational Structures and Implementations
What is computable with limited resources? How can we verify the correctness
of computations? How to measure computational power with precision? Despite the
immense scientific and engineering progress in computing, we still have only
partial answers to these questions. In order to make these problems more
precise, we describe an abstract algebraic definition of classical computation,
generalizing traditional models to semigroups. The mathematical abstraction
also allows the investigation of different computing paradigms (e.g. cellular
automata, reversible computing) in the same framework. Here we summarize the
main questions and recent results of the research of finite computation.Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version
published by IEEE Computer Societ
Periodicity in tilings
Tilings and tiling systems are an abstract concept that arise both as a
computational model and as a dynamical system. In this paper, we characterize
the sets of periods that a tiling system can produce. We prove that up to a
slight recoding, they correspond exactly to languages in the complexity classes
\nspace{n} and \cne
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