SOME RESULTS CONCERNING THE IDENTIFIABILITY OF PARALLEL AND SERIAL PROCESSES

Abstract

A mathematical characterization of serial, parallel and hybrid processes is given, and this characterization is related to several current experimental paradigms. Non‐identifiability (mimicking) between two systems (i.e. models of systems) is defined as equivalence of probability distributions on element completion times for the two systems, where n elements are available for processing by each. Results are then presented for a class of systems with exponential processing times, and it is seen that several interesting cases of parallel and serial systems are equivalent to systems of the opposite type. Evidence that will allow accurate discrimination between parallel and serial processing for this and other classes of systems either requires more complete and precise information about the actual probability distributions of the systems or more specialized sets of converging operations than is usually obtained in psychological experimentation. For example, it is noted that at the level of first moments (means), even a parallel independent system can predict results usually associated with a serial system (an overall increasing linear mean reaction time curve as a function of the number of elements to be processed). Next, a functional equation is developed that must hold in order for mimicking to occur between parallel and serial systems within the same general family of probability distributions, and three special cases are examined. A parallel system with gamma‐distributed processing times for element completion is then investigated, and it is shown that a strictly serial system cannot mimic it, but an interesting hybrid system can. This is followed by discussion of two kinds of partial identifiability, mimicking at the level of means and possible predicted differences at higher levels, and mimicking by approximation. Some qualitative considerations that may enter into conclusions as to parallelity or seriality of processing are then introduced. Last, it is suggested that in a broad sense questions related to parallel and serial systems concern fundamental aspects of information‐processing structure and distribution of processing energy and hence merit further mathematical investigation.