We consider three types of multiple input problems in the context of property testing.Specifically, for a property 01n , a proximity parameter , and an integer m, we consider the following problems:

\begin{enumerate}\item Direct m-Sum Problem for and :Given a sequence of m inputs, output a sequence of m bitssuch that for each i[m] the i\xth bit satisfies the requirements froman -tester for regarding the ith input;that is, for each i, the ith output bit should be 1(w.p. 23 ) if the ith input is in ,and should be 0 (w.p. 23 )if the ith input is -far from .\item Direct m-Product Problem for and :Given a sequence of m inputs, output 1 (w.p. 23 )if all inputs are in , and output 0 (w.p. 23 )if at least one of the inputs is -far from .\item The m-Concatenation Problem for and :Here one is required to -test the m-product of ; that is,the property m=(x1xm):i[m]xi .\end{enumerate}We show that the query complexity of the first two problemsis (m) times the query complexity of -testing ,whereas (except in pathological cases) the query complexityof the third problem is almost of the same order of magnitudeas the query complexity of the problem of -testing .All upper bounds are shown via efficient reductions.

We also consider the nonadaptive and one-sided error versionsof these problems. The only significant deviation from thepicture in the general (adaptive and two-sided error) modelis that the one-sided error query complexity ofthe Direct Product Problem equals (m) timesthe (two-sided error) query complexity of -testing plus (1) timesthe one-sided error query complexity of -testing .