19,592 research outputs found
Making Classical Ground State Spin Computing Fault-Tolerant
We examine a model of classical deterministic computing in which the ground
state of the classical system is a spatial history of the computation. This
model is relevant to quantum dot cellular automata as well as to recent
universal adiabatic quantum computing constructions. In its most primitive
form, systems constructed in this model cannot compute in an error free manner
when working at non-zero temperature. However, by exploiting a mapping between
the partition function for this model and probabilistic classical circuits we
are able to show that it is possible to make this model effectively error free.
We achieve this by using techniques in fault-tolerant classical computing and
the result is that the system can compute effectively error free if the
temperature is below a critical temperature. We further link this model to
computational complexity and show that a certain problem concerning finite
temperature classical spin systems is complete for the complexity class
Merlin-Arthur. This provides an interesting connection between the physical
behavior of certain many-body spin systems and computational complexity.Comment: 24 pages, 1 figur
Non-causal computation
Computation models such as circuits describe sequences of computation steps
that are carried out one after the other. In other words, algorithm design is
traditionally subject to the restriction imposed by a fixed causal order. We
address a novel computing paradigm beyond quantum computing, replacing this
assumption by mere logical consistency: We study non-causal circuits, where a
fixed time structure within a gate is locally assumed whilst the global causal
structure between the gates is dropped. We present examples of logically
consistent non- causal circuits outperforming all causal ones; they imply that
suppressing loops entirely is more restrictive than just avoiding the
contradictions they can give rise to. That fact is already known for
correlations as well as for communication, and we here extend it to
computation.Comment: 6 pages, 4 figure
Lower Bounds for RAMs and Quantifier Elimination
We are considering RAMs , with wordlength , whose arithmetic
instructions are the arithmetic operations multiplication and addition modulo
, the unary function , the binary
functions (with ), ,
, and the boolean vector operations defined on
sequences of length . It also has the other RAM instructions. The size
of the memory is restricted only by the address space, that is, it is
words. The RAMs has a finite instruction set, each instruction is encoded by a
fixed natural number independently of . Therefore a program can run on
each machine , if is sufficiently large. We show that there
exists an and a program , such that it satisfies the following
two conditions.
(i) For all sufficiently large , if running on gets an
input consisting of two words and , then, in constant time, it gives a
output .
(ii) Suppose that is a program such that for each sufficiently large
, if , running on , gets a word of length as an
input, then it decides whether there exists a word of length such that
. Then, for infinitely many positive integers , there exists a
word of length , such that the running time of on at
input is at least
Empirical Encounters with Computational Irreducibility and Unpredictability
There are several forms of irreducibility in computing systems, ranging from
undecidability to intractability to nonlinearity. This paper is an exploration
of the conceptual issues that have arisen in the course of investigating
speed-up and slowdown phenomena in small Turing machines. We present the
results of a test that may spur experimental approaches to the notion of
computational irreducibility. The test involves a systematic attempt to outrun
the computation of a large number of small Turing machines (all 3 and 4 state,
2 symbol) by means of integer sequence prediction using a specialized function
finder program. This massive experiment prompts an investigation into rates of
convergence of decision procedures and the decidability of sets in addition to
a discussion of the (un)predictability of deterministic computing systems in
practice. We think this investigation constitutes a novel approach to the
discussion of an epistemological question in the context of a computer
simulation, and thus represents an interesting exploration at the boundary
between philosophical concerns and computational experiments.Comment: 18 pages, 4 figure
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