Straight-line program length as a parameter for complexity measures

A definition is proposed for a size measure to be used as a parameter for algorithm analysis in any algebra. The parameter is simply the straight-line program length in the associated free algebra. This parameter generalizes the usual measures in basic arithmetic and string algebras, as well as some apparently different measures used for data structure algorithms. Another use is illustrated with an introduction to complexity-bounded group theory. I. INT~00ucT10N This paper continues the work in [9-l 21 directed toward the development of a unified, relative framework for complexity theory. Those papers establish a natural model for situations in which each data element is regarded as “atomic”; for example, it can be copied in one computation step. The attractiveness of the model is demonstrated by its use in stating and proving a variety of technical results, prin-cipally involving data types whose elements are bit strings or natural numbers. It would be useful to extend those ideas to.data types whose elements are not usually regarded as atomic (such as matrices, graphs, or, storage-retrieval structures of various kinds). This paper treats a problem that permeates the other work: Ho