# Optimal State Assignment for Finite State Machines

## Abstract

Computer-Aided synthesis of sequential functions of VLSI systems, such as microprocessor control units, must include design optimization procedures to yield area-effective circuits. We model sequential functions as deterministic synchronous Finite State Machines (FSM's), and we consider a regular and structured implementation by means of Programmable Logic Arrays (PLA's) and feedback registers. State assignment, i.e., binary encoding of the internal states of the finite state machine, affects substantially the silicon area taken by such an implementation. Several state assignment techniques have been proposed in the past. However, to the best of our knowledge, no Computer-Aided Design tool is in use today for an efficient encoding of control logic. We propose an algorithm for optimal state assignment. Optimal state assignment is based on an innovative strategy: Logic minimization of the combinational component of the finite state machine is applied before state encoding. Logic minimization is performed on a symbolic (code independent) description of the finite state machine. The minimal symbolic representation defines the constraints of a new encoding problem, whose solutions are the state assignments that allow the implementation of the PLA with at most as many product-terms as the cardinality of the minimal symbolic representation. In this class, an optimal encoding is one of minimal length satisfying these constraints. A heuristic algorithm constructs a solution to the constrained encoding problem. The algorithm has been coded in a computer program, KISS, and tested on several examples of finite state machines. Experimental results have shown that the method is an effective tool for designing finite state machines. © 1985 IEEE