In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 also admit arithmetic circuits of depth four and size 2. This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense

We study the Minimum Circuit Size Problem (MCSP): given the truth-table of a Boolean function f and a number k, does there exist a Boolean circuit of size at most k computing f? This is a fundamental NP problem that is not known to be NP-complete. Previous work has studied consequences of the NP-completeness of MCSP. We extend this work and consider whether MCSP may be complete for NP under mor...

We consider the size of monotone circuits for quadratic boolean functions, that is, disjunctions of length-2 monomials. Our motivation is that a good (linear in the number of variables) lower bound on the monotone circuit size for a certain type of quadratic function would imply a good (even exponential) lower bound on the general non-monotone circuit size. To get more insight into the structur...

This paper provides a refinement of the minimum DNA computational model by [OR98b], which assumes the smallest set of permissible biochemical operations. The power of the refined model is characterized in terms of standard Boolean circuit complexity classes.

From 08.11.06 to 10.11.06, the Dagstuhl Seminar 06451 Circuits, Logic, and Games was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ide...

We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field.

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Ra...

We provide a zero-knowledge argument for arithmetic circuit satisfiability with a communication complexity that grows logarithmically in the size of the circuit. The round complexity is also logarithmic and for an arithmetic circuit with fan-in 2 gates the computation of the prover and verifier is linear in the size of the circuit. The soundness of our argument relies solely on the well-establi...

We suggest practical sub-linear size zero-knowledge arguments for statements involving linear algebra. Given commitments to matrices over a finite field, we give a sub-linear size zero-knowledge argument that one committed matrix is the product of two other committed matrices. We also offer a sub-linear size zero-knowledge argument for a committed matrix being equal to the Hadamard product of t...

This paper summarizes a series of three lectures the first author was invited to present at the NZMRI summer 2000 workshop, held in Kaikoura, New Zealand. Lecture 1 presents the goals of computational complexity theory. We discuss (a) what complexity provably can never deliver, (b) what it hopes to deliver but thus far has not, and finally (c) where it has been extremely successful in providing...