ORIGINAL ARTICLE IF-preorder, IF-topology and IF-automata S. P. Tiwari • Anupam K. Singh Received: 23 February 2013 / Accepted: 9 August 2013 Ó Springer-Verlag Berlin Heidelberg 2013 Abstract The concept of topologies are widely used in machine learning, and fuzzy automata are used as models of machine learning systems. An advantage of employing fuzzy automaton as a model of machine learning system is its simplicity in design and computation. Also, some standard topological concepts and ideas are used in fuzzy automata and IF-automata to obtain certain results therein. In view of above, it seems that IF-topologies and IF- automata may play vital role in the study of machine learning; the purpose of this work is to introduce and study IF-topologies and to use such studies in the theory of IF- automata. Keywords IF-set  IF-preordered set  Upper set  Lower set  IF-topology  IF-automaton 1 Introduction It is well known that the concept of topology exists not only in mathematics but also in many real life applications. In computer science, topologies are widely used in machine learning and cybernetics (cf., [1, 2, 7, 10, 24, 25]), and in the study of automata as well as fuzzy automata theory (cf., [29–31]). A generalization of topology in fuzzy environ- ment, namely fuzzy topology has also been successfully used in fuzzy automata theory (cf., [14, 32, 34]). The topological and fuzzy topological methods are beneficial in the study of the crisp as well as fuzzy automata atleast by two reasons (1) by way of the resulting economy in the arguments used to prove several results and (2) the addi- tional insights provided by the established discipline of topology (cf., [14, 27–33]). It is thus imperative to carry out an analogous study for other types of fuzzy automata. Atanassov [3] introduced a useful generalization of fuzzy sets in the form of the notion of an intuitionistic fuzzy set, which has since been pursued vigorously (At this point, we mention that in [18], it has been argued, rather convincingly, that the use of the term intuitionistic, for the concept introduced by Atanassov [3], is inappropriate. Accordingly, we use in this paper the prefix IF- in place of intuitionistic fuzzy; thus for example, an intuitionistic fuzzy set is renamed here as an IF-set. This terminology has already been used in [33, 35]). The generalization of fuzzy sets to IF-sets may have perhaps led Jun [21] to analogously generalize fuzzy automata by introducing and studying the concept of an IF-automaton. In application point of view, fuzzy automata have been shown to be useful in numerous engineering applications such as pat- tern recognition, clinical monitoring, and also used as models of machine learning systems (cf., [26, 36]). Spe- cifically, in [36], a nonsupervised learning scheme in automatic control and pattern recognition is proposed and it has been pointed out that advantage of employing fuzzy automaton as a model of machine learning system is its simplicity in design and computation. Also, on the other hand, the usefulness of IF-automaton has been shown in the study of social sciences (cf., [8, 9]). In IF-set theory, t-norms and conorms are important notions [16]. Using IF t-norms the composition of fuzzy relations has been extended to the IF-case (cf., [13, 15, 17]). The usefulness of these compositions have already been shown in approximate reasoning, e.g., for medical S. P. Tiwari (&)  A. K. Singh Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India e-mail: sptiwarimaths@gmail.com A. K. Singh e-mail: anupam09.bhu@gmail.com 123 Int. J. Mach. Learn. & Cyber. DOI 10.1007/s13042-013-0191-3