**by Venkatesan Guruswami (Simons Institute)**

This semester at the Simons Institute, the Meta-Complexity program is buzzing along with intense activity in the form of multiple reading groups and a weekly seminar, on top of the usual three workshops and boot camp. A large number of complexity theorists have converged at the Institute, including several students, postdocs, and junior researchers. And all this complexity action is intermixed with some good and varied fun in the form of many self-organized social activities.

Theory readers know what complexity is, but what is the “meta” for? An online dictionary definition of “meta” is a term describing something that “consciously references or comments upon its own subject or features” — e.g., “A movie about making a movie is just so meta—especially when the actors criticize the acting.”

So meta-complexity is the study of complexity of problems that themselves pertain to complexity. A prototypical example is circuit minimization, where the goal is to find the smallest circuit for a given specification. This is modeled as the minimum circuit size problem (MCSP), which is one of the lead actors in meta-complexity: Given the truth table of an *n*-bit Boolean function *f*, and a size parameter *s*, does *f* have a Boolean circuit of size at most *s*? (Note that the input size is 2* ^{n}*.)

The MCSP problem was explicitly defined in 2000 by Kabanets and Cai, inspired by Razborov and Rudich’s seminal work on natural proofs from 1994. (Upon some reflection, one can realize that a natural property against some class of circuits is really an average-case, one-sided error algorithm for MCSP on circuits of that class.) Being a “natural” problem, MCSP was in fact considered implicitly even earlier, including in early works on complexity in the former Soviet Union. Yablonski claimed in 1959 that MCSP is not in *P* (to use modern terminology — the class *P* wasn’t even defined then). MCSP is easily seen to be in *NP* — indeed, one can guess a circuit and then check if it computes the function correctly on all inputs — so we of course can’t yet prove that it lies outside *P*.

Leading up to his seminal work on *NP*-completeness, Levin was in fact trying to show *NP*-hardness of MCSP, but didn’t succeed. One of the six problems that Levin showed *NP*-hardness of was DNF-MCSP, where one tries to find a DNF formula of minimum size to compute a given truth table. (Technically, Levin proved only *NP*-hardness of the partial-function version of the question, where we don’t care about the function value on some inputs.) In fact, this was Problem 2 on Levin’s list, between Problem 1, which was set cover, and Problem 3, which was Boolean formula satisfiability.

The complexity of MCSP remains open — we do not know if it is in *P* or *NP*-complete. Resolving this is a natural challenge, but on the surface it might seem like a rather specific curiosity. One reason to care about MCSP is that any *NP*-hardness proof faces some fundamental challenges — in particular, the function produced on No instances must have large circuits, giving an explicit function in EXP that has large circuits, which seems beyond the reach of current techniques. One might try to circumvent this via randomized or exponential-time reductions, and this has indeed fueled some very interesting recent results, as well as intriguing connections to learning theory, average-case complexity, cryptography, and pseudorandomness.

An “extraneous” reason to care about MCSP and meta-complexity is that the underlying techniques have led to some of the best insights on certain foundational questions that have nothing to do with meta-complexity per se. Meta-complexity has been energized by some significant recent progress on both of these (internal and extraneous) fronts. These advances and momentum are fueling a lot of activity and optimism in the Simons Institute program this semester.

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