Complexity of Boolean Functions

Summary

One of the most fundamental problems in computer science is to estimate the complexity of Boolean functions with respect to different models and complexity measures. It is frustrating that several central problems have remained open for a long time, such as proving (1) nonlinear size lower bounds for circuits of logarithmic depth, (2) nonpolynomial size lower bounds for formulas, or (3) nonpolynomial size lower bounds for threshold circuits of depth three. Nevertheless, there has been a lot of progress on some of the classical research problems. Also, new methods such as communication complexity are now available, and new applications (such as hardware verification) pose new problems which can be answered by those people active in this area.

The organizers (David Mix Barrington, Noam Nisan, Rüdiger Reischuk, and Ingo Wegener) are happy that 40 researchers came to the Dagstuhl seminar, only 14 of them from Germany (including three guests from India and Lithuania) with the others from the USA (10), Israel (5), Czech Republic (3), Austria, Canada, Denmark, Hungary, the Netherlands, Russia, Spain, and Sweden.

The 31 talks captured many of the aspects of Boolean function complexity: lower bounds for different types of circuits and branching programs, the average delay of circuits, the power of restrictions, communication complexity, applications to neural nets, and structural results on circuit-based complexity classes. It was discussed whether some lower bound proofs, including proofs that are not ”natural” in the sense of Razborov and Rudich, are even possible. Furthermore, some talks considered related areas such as the PCP theorem, Yao’s XOR lemma, visual cryptography, PRAM complexity, and hashing.

A lively problem session was organized, where 13 open problems were presented. There was also an open discussion on the future of this research topic. Needless to say, the participants took advantage of the Dagstuhl facilities and the excellent atmosphere to hold many informal discussions as well.