Computations over finite monoids and their test complexity

Abstract

We consider the test pattern generation problem for circuits, which compute expressions over some algebraic structure. The relation between algebraic properties of this structure and test complexity (i.e. the best possible test size) is analyzed. Here, this relation is looked at in detail for the family of all finite monoids. The test complexity of a monoid with respect to a problem is measured by the number of tests needed to check the best testable circuit (in a certain computational model) solving the problem. Two important computations over finite monoids namely expression evaluation and parallel prefix computation are considered. The relation between algebraic properties of a monoid and its test complexity (with respect to these problems) is studied. In both cases it can be shown that the set of all finite monoids partitions into exactly three classes with constant, logarithmic and linear test complexity, respectively. These classes are characterized using algebraic properties. For example groups are exactly the monoids with constant test complexity. For each class we provide circuits with optimal test sets and efficient methods, which decide the membership problem for a given finite monoid M, i.e. determine the test complexity for M.