Minimal realization of sequential machines : the method of maximal adjacencies

Reducing the amount of hardware needed for implementing a sequential machine is very important. The problem of, in the strict sense, minimal realization of sequential machines is difficult and it is entirely unsolved (except complete enumeration). Typically, this problem can be replaced with a number of subproblems such as: state minimization, state assignment, choice of flip-flops and minimization of Boolean functions representing the next-state and output functions of a sequential machine. In this work, the greatest attention has been paid to state assignment and state minimization; however, the method, as a whole, covers all the subjects listed. Our aim was to find a practical method of state assignment for medium and large sequential machines. Two traditionally independent steps: state minimization and state assignment are replaced here with a single process of concurrent state minimization and assignment. Here, minimization (or partial minimization) of internal states is obtained as a byproduct of the state assignment and it results from assigning the same code to two or more internal states. The problem of, in the strict sense, optimal state assignment is unsolved, but some approximate approaches have been proposed. The best known of them are: the partition theory [7][8][12][13][16][22][24][25], the column based approach [4], the graph embedding approach [1] [2] [15] and related to it multivalued, multi-output, non-univocal function minimization methods [3][20][21]. Two first approaches suffer from many shortcomings. The method presented here is related to the third group of methods. Some of the observations, on which our method is based, are like those used by Armstrong [1) [2). However, many important differences exist between the method presented here and Armstrong's method. The method of maximal adjacencies uses much more information about the factors, that can have influence on the quality of the resultant assignments than the method of Armstrong [1][2] and, also, than all the other related methods [3] [15] [20] [21]. Therefore, in many cases, it can produce better assignments than the methods of the third group. For the same reason, it can give better assignments than the column based approach. The method presented here uses adjacency conditions, that are ordered according to the number of adjacencies reached when a given condition is satisfied by the assignment. The number of adjacencies reflects the condition's quality. Since the conditions are considered and combined, starting with the best, the first assignments constructed will be always nearly optimal and, almost always, the best of the nearly optimal solutions will be one of the first to be obtained. In the methods using minimization of multi-valued, multioutput non-univocal functions for creating the conditions used further in order to construct the near optimal assignments, such a measure of quality of conditions and appropriate ordering relation on the set of conditions did not exist. Thus, the method of maximal adjacencies seems to be most effective, i. e. it produces good results more quickly. The capacity of this method is very important, especially, for large machines with "difficult" algebraic structures, for which the construction of assignments is time and memory consuming. The method does not assume minimum numbers of states and memory elements. Furthermore, some of the best types of flip-flops can be adopted in order to realize each excitation function and the complexity of the realization of the output function is taken into account. The method of maximal adjacencies contains none of the shortcomings of the first two state assignment approaches.