Circuits and local computation

This paper contains two parts. In Part I, we show that polynomial-size monotone threshold circuits of depth <italic>k</italic> form a proper hierarchy in parameter <italic>k</italic>. This implies in particular that monotone <italic>TC</italic><supscrpt>0</supscrpt> is properly contained in <italic>NC</italic><supscrpt>1</supscrpt>. In Part II, we introduce a new concept, called <italic>local function</italic>, which tries to characterize when a function can be efficiently computed using only localized processing elements. It serves as a unifying framework for viewing related and sometimes apparently unrelated results. In particular, it will be demonstrated that the recent results on lower bounds for monotone circuits by Razborov [Ra1] and Karchmer and Wigderson [KW], as well as a main theorem in Part I of this paper, can be regarded as proving certain functions to be nonlocal. We will also suggest an approach based on locality for attacking the conjecture that (nonmonotone) <italic>TC</italic><supscrpt>0</supscrpt> is properly contained in <italic>NC</italic><supscrpt>1</supscrpt>.