 
Summary: Cracks in the Defenses: Scouting Out Approaches on
Circuit Lower Bounds
Eric Allender
Department of Computer Science, Rutgers University, Piscataway, NJ 08855,
allender@cs.rutgers.edu
Abstract. Razborov and Rudich identified an imposing barrier that stands in the
way of progress toward the goal of proving superpolynomial lower bounds on
circuit size. Their work on "natural proofs" applies to a large class of arguments
that have been used in complexity theory, and shows that no such argument can
prove that a problem requires circuits of superpolynomial size, even for some
very restricted classes of circuits (under reasonable cryptographic assumptions).
This barrier is so daunting, that some researchers have decided to focus their
attentions elsewhere. Yet the goal of proving circuit lower bounds is of such im
portance, that some in the community have proposed concrete strategies for sur
mounting the obstacle. This lecture will discuss some of these strategies, and will
dwell at length on a recent approach proposed by Michal Kouck´y and the author.
1 Introduction and Ancient History
More than a decade ago, the author wrote a survey of results in circuit complexity [7].
That survey is still depressingly uptodate. Despite some interesting recent progress on
circuit lower bounds (see, for example [16, 22,40, 21,17]), it is fairly accurate to say
