14,823 research outputs found
Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V
Using a result of Kari and Ollinger, we prove that the torsion problem for
elements of the Brin-Thompson group 2V is undecidable. As a result, we show
that there does not exist an algorithm to determine whether an element of the
rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite
order. A modification of the construction gives other undecidability results
about the dynamics of the action of elements of 2V on Cantor Space.
Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely
presented group with solvable word problem and unsolvable torsion problem. To
our knowledge, 2V furnishes the first concrete example of such a group, and
gives an example of a direct undecidability result in the extended family of R.
Thompson type groups.Comment: 16 pages, 3 figure
Perfect Computational Equivalence between Quantum Turing Machines and Finitely Generated Uniform Quantum Circuit Families
In order to establish the computational equivalence between quantum Turing
machines (QTMs) and quantum circuit families (QCFs) using Yao's quantum circuit
simulation of QTMs, we previously introduced the class of uniform QCFs based on
an infinite set of elementary gates, which has been shown to be computationally
equivalent to the polynomial-time QTMs (with appropriate restriction of
amplitudes) up to bounded error simulation. This result implies that the
complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its
counterpart for uniform QCFs. However, the complexity classes ZQP and EQP for
QTMs do not appear to equal their counterparts for uniform QCFs. In this paper,
we introduce a subclass of uniform QCFs, the finitely generated uniform QCFs,
based on finite number of elementary gates and show that the class of finitely
generated uniform QCFs is perfectly equivalent to the class of polynomial-time
QTMs; they can exactly simulate each other. This naturally implies that BQP as
well as ZQP and EQP equal the corresponding complexity classes of the finitely
generated uniform QCFs.Comment: 11page
Universality and programmability of quantum computers
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of
universal physical simulation procedure. Much of the writing about quantum
logic circuits and quantum Turing machines has shown how these machines can
simulate an arbitrary unitary transformation on a finite number of qubits. The
problem of universality has been addressed most famously in a paper by Deutsch,
and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum
logic circuit model, developed by Feynman and Deutsch, has been more prominent
in the research literature than Deutsch's quantum Turing machines. Quantum
Turing machines form a class closely related to deterministic and probabilistic
Turing machines and one might hope to find a universal machine in this class. A
universal machine is the basis of a notion of programmability. The extent to
which universality has in fact been established by the pioneers in the field is
examined and this key notion in theoretical computer science is scrutinised in
quantum computing by distinguishing various connotations and concomitant
results and problems.Comment: 17 pages, expands on arXiv:0705.3077v1 [quant-ph
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
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