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Hajime Machida

The International Symposium on Multiple-Valued Logic (ISMVL) is an annual symposium organized with the support of IEEE. Its first meeting took place in 1971. The contributions to ISMVL range over various areas in multiple-valued logic; from practical side to theoretical side, that is, from electric and electronic engineering to mathematics and philosophy.

The 39th International Symposium on Multiple-Valued Logic (ISMVL-2009) was held in May, 2009, in Okinawa, Japan. More than 50 papers were presented there. During the symposium, Dan Simovici, an editor-in-chief of the Journal of Multiple-Valued Logic and Soft Computing, asked me to be the editor of a special issue of the journal which contains the extended and refined version of the selected papers on clone theory and its related areas presented at ISMVL-2009.

The term clone may not be familiar to some of the readers.A clone is a purely mathematical concept and clone theory sits at the heart of the theoretical side of multiple-valued logic. By definition, a clone is a set of multi-variable functions on a fixed set which contains all projections and is closed with respect to (functional) composition. Clone theory has a long history, initiated nearly a century ago, and prominent logicians and mathematicians as well as experts in engineering have contributed to give the theory solid and profound basis.

This special issue contains the selection of six papers presented at ISMVL- 2009. The authors were requested to submit their papers in extended and refined form. All papers in this issue went through the refereeing process. The papers study various aspects and problems in clone theory: (total) clones, partial clones and hyperclones. Most noteworthy is the paper by A. A. Bulatov and A. Hedayaty, which evolved from the invited lecture given by the first author at ISMVL-2009. In constraint satisfaction problem (CSP for short) clone theory is connected to computational complexity theory. This is a fascinating application of clone theory to other research field. A. A. Bulatov is one of the pioneers in CSP who has led the field for more than ten years. Their paper discusses an advanced topic of CSP, the counting CSP.

We are deeply grateful to the editors-in-chief, D. Simovici and I. Stojmenovic, for giving us this opportunity to publish our works. We hope that this issue will help the reader to get better understanding of clone theory and promote a further study of the theory in the future.

Hajime Machida
Tokyo, Japan

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