Granular Computing and hypercomputation
The development of the paradigm of Granular Computing as a framework
for processing of information abstractions has raised an interesting
question whether the computational power of the multi-resolution information
processing is higher than that of a Universal Turing Machine (UTM).
There are good reasons to believe that this is so. The UTM is essentially
a state machine that acts on a recursively enumerable input. This is
why it is unable to evaluate functions that are either partial or
non-recursive. These limitations of UTM are well known and Turing
has proposed the augmentation of UTMs by oracles OTM (which are essentially
look-up tables). Although, theoretically such an augmented Turing machine
can overcome the problems with evaluation of halting functions but it
does so at the cost of requiring infinite resources to implement oracles.
However, as Turing has shown, even using the OTM the famous Hilbert's
Entscheidungsproblem is not computable (the statement, T is in the
set of theorems provable from axioms A, can be solved by computation,
but the problem of showing that T is not provable cannot). Similarly,
the set of all truth of elementary number theory is not computable
(it is not recursively enumerable - which is what Goedel's
Incompleteness Theorem says). Nevertheless, both of these problems can
be solved by human thought processes that are not confined exclusively
to computing. For example one can start by asserting the negative answer
to the Entscheidungsproblem as a first approximation and then proceed
with checking whether the theorem T is in a set of theorems that are
provable from the axioms A (this is an OTM computational process).
If such a theorem can be found the initial answer is changed from negative
to positive (this change of mind contradicts the definition of computations
but it captures what human reasoning is like). Such a two-trial approach to
achieving hypercomputational powers has been suggested by Kugel, 1986, and
subsequently developed by other researchers. What is promising with
Granular Computing in this context is that it provides a formal basis
for such mind changing that reflects human analysis of complex problems
at various levels of abstraction (granularity).
Publications
- Bargiela, A., Pedrycz, W., Granular computing: an introduction, Springer, 2003
- Bargiela, A., Pedrycz, W., (Eds.) Human-centric information processing through granular modelling, Studies in Computational Intelligence 182, Springer Berlin Heidelberg, 2009, (doi: 10.1007/978-3-540-92916-1)
- Bargiela A., Pedrycz W., Hirota K., Data granulation through optimization of similarity measure, Archives of Control Sciences, 12, 4, 469-491, 2002
- Pedrycz, W., Bargiela, A., Granular clustering: a granular signature of data, IEEE Trans. on Systems Man and Cybernetics, SMC-B, 32, 2, April 2002, 212-224, (doi: 10.1109/3477.990878 )
- Bargiela, A., Pedrycz, W., Recursive information granulation: Aggregation and interpretation issues, IEEE Trans. on Systems Man and Cybernetics SMC-B, 33, 1, 17, 2003, 96-112. (10.1109/TSMCB.2003.808190)
- Bargiela A., Pedrycz W., Hirota K., Granular prototyping in fuzzy clustering, IEEE Transactions on Fuzzy Systems, 12, 5, 2004, 697-709 (doi: 10.1109/TFUZZ.2004.834808)
- Bargiela, A., Pedrycz, W., Granular mappings, IEEE Transactions on Systems Man and Cybernetics SMC-A, vol. 35, 2, March 2005, 288-301 (doi: 10.1109/TSMCA.2005.843381)
- Bargiela, A., Pedrycz, W., A model of granular data: a design problem with the Tchebyschev FCM, Soft Computing, 9, 3, March 2005, 155-163 (doi: 10.1007/s00500-003-0339-2)
- Bargiela A., Homenda W., Information structuring in natural language communication Ð syntactical approach, Journal of Intelligent and Fuzzy Systems, 17(6), 2006, 575-582.
- Bargiela, A., Pedrycz, W., The roots of granular computing, Proceedings of 2006 IEEE International Conference on Granular Computing, 806-809
- Bargiela, A., Pedrycz, W., Toward a theory of Granular Computing for human-centred information processing, IEEE Trans. on Fuzzy Systems, vol. 16, 2, 2008, 320-330. (doi: 10.1109/TFUZZ.2007.905912)
- Pedrycz W., Bargiela A., Fuzzy clustering with semantically distinct families of variables: descriptive and predictive aspects, Pattern Recognition Letters, 31(13), 1952-1958, 2010. (doi: 10.1016/j.patrec.2010.06.016)
- Pedrycz, W., Bargiela, A., An Optimisation of allocation of information gramularity in the interpretation of data structures, IEEE Transactions on Systems Man and Cybernetics Part B , 42(3), 2012, 582-590 (doi: 10.1109/TSMCB.2011.2170067)
- Bargiela A., Pedrycz W., Optimised information abstraction in granular Min/Max clustering, in S. Ramanna, R.J. Howlett, L. Jain (ed.), Emerging Paradigms in Machine Learning, Springer, June 2012. ISBN 978-3-642-28698-8
- Bargiela A., Pedrycz W., Supervised and unsupervised information granulation: A study in hyperbox design, Chapter 3 in J.T.Yao (ed.), Novel developments in Granular Computing: Applications for Advanced Human Reasoning and Soft Computation, IGI Global, 2010.
- Bargiela, A., Pedrycz W., Self-organising maps for interactive information granulation, in Neural Network Applications in IT, In : D. Wang, N.K.Lee (eds.), University of Malasia Press, 1-19, 2004
- Bargiela, A. Pedrycz, W., Combined physical and mathematical model of Granular Computing, Proc. Int. Conf. On Soft Computing and Intelligent Systems SCIS2006, Tokyo, Japan, Sept. 2006
- Pedrycz W, Bargiela A, Fuzzy fractal dimensions and fuzzy modeling, Information Sciences , 153, 2003, 199-216, doi:10.1016/S0020-0255(03)00075-6
- Cichocki A., Bargiela A., Neural Networks for Solving Linear Inequality Systems, Parallel Computing , Vol.22, No.11, Jan. 1997, pp.1455-1475, doi: 10.1016/S0167-8191(96)00065-8
- Bargiela A., Irving M., Sterling M., Observability determination in power system state estimation, IEEE. Transactions on Power Systems , PWR, 1, 2, May 1986, pp. 108-112, doi: 10.1109/TPWRS.1986.4334914