Minimization of functions with multiple-valued outputs: theory and applications

A theoretical framework for the minimization of logic functions with symbolic or multiple-valued outputs is presented. By use of this framework, efficient, exact algorithms for the problems of output encoding and finite-state-machine (FSM) state assignment are developed. All previous automatic approaches to these encoding problems have involved the use of heuristic techniques. A notion of generalized prime implicants is presented for functions with multiple-valued outputs, and a novel minimization procedure of prime implicant generation and covering for solving the output encoding problem is proposed. An optimum solution to this covering problem is also an optimum solution to the encoding problem. A single logic minimization step thus replaces on the order of n-factorial minimizations required by straightforward exhaustive search. It has been shown previously that the input encoding problem can be exactly solved using Boolean minimization over functions with multiple-valued inputs. An extension of the presented algorithm that handles functions with multiple-valued inputs and outputs can be used to solve the state assignment problem exactly. Experimental results are presented for a set of examples