On the Max-Product Fuzzy Markov Chains: Fundamentals and Applications

Fundamentals of the fuzzy Markov chains have become my long term research topics. The first fuzzy Markov chain model was proposed by Sanchez in 1978. However, unlike its counterpart known the Markov chains which have been studied for quite a long time, many important fundamentals of the fuzzy Markov chains such as ergodicity, long term stability, etc. remain open (or only limited results obtained.) The long-term stability issues have closed relationship with the powers of the associated fuzzy matrix embedded in the fuzzy Markov chain. In the literature, around 40 papers have been published in this regard, and most of them discuss the convergence aspects of the powers of that fuzzy matrix (a few papers are in the oscillating behavior of the fuzzy Markov chain.) On the other hand, very few papers are on the ergodicity issue. Avrachenkov and Sanchez proposed the notion of ergodicity for the max-min fuzzy Markov chains. Unfortunately, they did not provide useful sufficient conditions to ensure the ergodicity. It turns out the finding sufficient conditions for the ergodicity of the max-min fuzzy Markov chains remain an open question. Indeed, finding sufficient conditions for the ergodicity of the max-min fuzzy Markov chains is a challenging problem. Our research experience shows that the difficulty lies on the max-min operation used. Therefore, we approach this issue by considering the max-generalized mean operation. This gave us great result: it can be shown that any max-generalized mean fuzzy Markov chain has the ergodic property. Hence, when study the fundamentals of a fuzzy Markov chain, we need to tell what kind of operation is being used in the system. In the literature, most of commonly seen operations (such as the max-min and max-product) are the max-t norms. And the max-generalized mean is not a max-t norm. Furthermore, from the literature, around 40 max-t norms have been documented. And the continuous t-norms can be further divided into three subgroups: one is the min operation, the second group is called the Archimedean t-norm, and the third group consists of complicated combination of the Archimedean t-norms. Note that every operation in the second group (the Archimedean t-norms) is either isomorphic to the product operation or to the nilpotent operation. That may be explained why the max-min and max-product, two operations, are so important when we studied the max t-norms. Our research proposal is a three year project. The main work in the first year is to find useful sufficient conditions to ensure the ergodicity of the max-min fuzzy Markov chains and the max-product fuzzy Markov chains, respectively. With strong confidence, the results (once obtained) should be able to get published in leading journals. The second year work is to apply our first year，s results to compute the FuzzRank, an index to establish order on the web search. The FuzzRank is the ' fuzzy version，，of the famous Google PageRank. We need to investigate on the web search which operation(s) (max-min or max-product, max-generalized mean, etc) are more appropriate? We also want to see what kinds of fuzzy Markov models are more robust when subjected to the unexpected perturbation (frequently seen on the web!) The third year’s work is to extend the ergodic results of the max-product fuzzy Markov chains to a more general setting, namely, the max-Archimedean t-norm fuzzy Markov chains. It is known that the Archimedean t-norm contains about 10 instances (including the product, Bolt operation, etc.). Hence, if successful, we don’t need to examine the fuzzy Markov chain with one operation by one operation. This will be a big contribution to the study of fuzzy Markov chains.

Project ID:PB10401-0087
External Project ID:NSC102-2221-E182-040-MY3