Some Meet-in-the-Middle Circuit Lower Bounds Kristoffer Arnsfelt Hansen and Peter Bro Miltersen Department of Computer Science, University of Aarhus, Denmark {arnsfelt,bromille}@daimi.au.dk June 7, 2004 Abstract We observe that a combination of known top-down and bottom-up lower bound techniques of circuit complexity may yield new circuit lower bounds. An important example is this: Razborov and Wigderson showed that a certain function f in ACC 0 cannot be computed by polynomial size circuits consisting of two layers of MAJORITY gates at the top and a layer of AND gates at the bottom. We observe that a simple combination of their result with the H˚astad switching lemma yields the following seem- ingly much stronger result: The same function f cannot be computed by polynomial size circuits consisting of two layers of MAJORITY gates at the top and an arbitrary AC 0 circuit feeding the MAJORITY gates. 1 Introduction During the 1980’s and 1990’s significant progress was made understanding the power of bounded depth computation, i.e. computation performed by unbounded fan-in circuits of constant depth and polynomial size containing layers of gates of various kinds. Such research was motivated by its fundamental nature. Indeed, it can be argued that understanding such restricted computation must be a prerequisite to solving problems such as P vs. NP. Two methodologies for proving lower bounds on the power of constant depth computation were developed in the 1980’s: 1. top-down methods, exemplified by lower bounds obtained using two- or multi-party communication complexity [11,12] and 2. bottom-up methods, exemplified by lower bounds obtained using the H˚astad switching lemma [4, 6–8, 10, 13, 14]. Roughly speaking, a top-down argument proves that a problem cannot be solved by a certain kind of circuit by assuming to the contrary that it can, looking at the output gate of the circuit and working its way downwards to an input wire where a contradiction is reached. In contrast, a bottom-up argument starts at the input and successively eliminates gates at the bottom of the circuit, until the circuit is gone and a contradiction is reached. Progress on the understanding of bounded depth computation in the last decade has been less encouraging than the rate of progress in the 1980s. One 1