Real-Number Models of Computation
and HP-Memristors: A Postscript
J.L. Nazareth
September, 2014
This is a brief postscript to the research monograph [1] and its motivating,
companion essay [2], both being derived from an earlier article [3]. A
continuous-discrete random-access machine and stored program (CDRAM/RASP) “real-number” model of computation was proposed as a
theoretical foundation for numerical algorithmic science & engineering, the
discipline within computer science whose focus is the algorithmic solution
of finite-dimensional, continuous and discrete numerical problems. Within
this theoretical model, a real number is defined by a mantissa and an
exponent, the former being represented by an analog “magnitude”, or Abit, and the latter by a finite, digital sequence of unary (or binary) bits.
Basic arithmetic operations between A-bits are defined geometrically (on
magnitudes). Is this abstract CD-RAM/RASP model realistic, i.e., does it
conceptualize computing machinery that could actually be built from
available analog (and digital) hardware?
Analog computing predates digital computing and has taken many forms,
and the literature on analog memory devices, in particular, is vast. A 1963
survey article of Nagy [4] reaches the following conclusion (italics ours):
Even the considerable improvements in components almost within reach are not likely
to close the existing gap between proposed theoretical models and their hardware
realization. As much as ever it will remain up to the individual designer to maximize the
yield of machines severely handicapped by the lack of a really cheap, reliable and fast
analog memory component.
Analog memory remained essentially at this standstill for several decades,
until the recent breakthrough in 2008 by a team at Hewlett-Packard led by
R. Stanley Williams and announced in the journal Nature by Strukov et al.
[5]. As described by the authors, their creation of a nanoscale, platinumtitanium-oxide resistor with memory---“HP-memristor” hereinafter-----was
inspired by a prescient theoretical formulation in 1971 by Chua [6] at U.C.,
Berkeley, and an informal and very readable account of the background
and potential for their revolutionary discovery can be found in Williams [7].
HP-memristors can provide an analog memory that, for the first time, has
the potential of overcoming the severe handicaps mentioned above. They
can store a “continuum” of numerical values in the range [0,1], and circuits
based on them for performing basic arithmetic operations on numbers
stored in analog form (as magnitudes) have recently been proposed in [8]
and [9] (and surveyed in [10] and Section II of [11]). Thus, in much the
same way that the standard RAM/RASP model of computation can be
realized by computers based on standard CMOS technology, the
aforementioned CD-RAM/RASP model can potentially be realized by HPmemristors (and CMOS) technology, i.e., it is indeed a highly-abstract
conceptualization of computing machines that are now within practical
reach.
The implications of memristors and, more generally, memristive devices
are far-reaching and the associated literature is already vast; see, for
example, Yang et al. [12] and its extensive bibliography. The subject is
also surrounded by considerable controversy. Here, our focus has been
only on the original HP-memristors and one particular use to which they
could be put, namely, to make practical our CD-RAM/RASP real-number
model of computation, which to date has been only theoretical and
abstract.
References
[1] Nazareth, J.L. (2012), Numerical Algorithmic Science and Engineering:
Foundations and Organization, PSIPress, Portland, Oregon, USA.
As a result of the closure of the publishing house PSIPress and its website in 2014, following the sudden
and unexpected death in late-2012 of its founder, Dr. Richard Crandall, the above monograph has now
been made available, free-of-charge, at the author’s website: www.math.wsu.edu/faculty/nazareth :
follow the link `biographical summary’ and then the link to the pdf version of the monograph provided
near the end of the webpage. This pdf version can be redistributed completely freely.
[2] Nazareth, J.L. (2013), “Numerical algorist vis-à-vis numerical analyst”,
Journal of the Cambridge Computer Lab Ring, Issue XXXIII, pp. 8-10.
www.cl.cam.ac.uk/downloads/ring/ring-2013-05.pdf
[3] Nazareth, J.L. (2008), “Symbol-based vis-à-vis magnitude-based
computing: on the foundations and organization of algorithmic science and
engineering,” unpublished paper presented at the Santa Fe Institute, New
Mexico, USA (Seminar: April, 2008).
[4] Nagy, G. (1963), “A survey of analog memory devices,” IEEE
Transactions on Electronic Computers 12 (4), pp. 388-393.
[5] Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S. (2008), “The
missing memristor found,” Nature 435, pp. 80-83.
[6] Chua, L.O. (1971), “Memristor---the missing circuit element,” IEEE
Transactions on Circuit Theory 18 (5) pp. 507-519.
[7] Williams, R. Stanley (2008), “How we found the missing memristor,”
http://spectrum.ieee.org/semiconductors/processors/how-we-found-the-missing-memristor
[8] Laiho, M., Lehtonen, E. (2010), “Arithmetic operations within
memristor-based analog memory,” Proceedings of the 12th International
Workshop on Cellular Nanoscale Networks and their Applications (CNNA
2010), pp. 1-4, Berkeley, California, USA.
[9] Merrikh-Bayat, F., Shouraki, S. (2011), “Memristor-based circuits for
performing basic arithmetic operations,” Procedia Computer Science 3, pp.
128-132.
[10] Kirar, V.P.S. (2012), “Memristor: the missing circuit element and its
application,” World Academy of Science, Engineering and Technology, 6
2012-12-22, pp. 450-452.
[11] Bickerstaff, K., Swartzlander, E.E. (2010), “Memristor-based
arithmetic,” Conference Record of the 44th Asilomar Conference on Signals,
Systems and Computers, pp. 1173-1177, Pacific Grove, California, USA.
[12] Yang, J.J., Strukov, D.B., Stewart, D.R. (2013), “Memristive devices
for computing,” Nature Nanotechnology 8, pp. 13-24.