# Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity II: A Simple Example

@article{Friedman2006CohomologyIG, title={Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity II: A Simple Example}, author={J. Friedman}, journal={ArXiv}, year={2006}, volume={abs/cs/0512008} }

In a previous paper we have suggested a number of ideas to attack circuit size complexity with cohomology. As a simple example, we take circuits that can only compute the AND of two inputs, which essentially reduces to SET COVER. We show a very special case of the cohomological approach (one particular free category, using injective and superskyscraper sheaves) gives the linear programming bound coming from the relaxation of the standard integer programming reformulation of SET COVER.

## Topics from this paper

## 6 Citations

Linear Transformations in Boolean Complexity Theory

- Mathematics, Computer ScienceCiE
- 2007

This work attempts to understand a cohomological approach to lower bounds in Boolean circuits by studying a very restricted case by studying the kernel (or nullspace) of a fairly simple linear transformation and its transpose.

The Strengthened Hanna Neumann Conjecture I: A Combinatorial Proof ∗

- Mathematics
- 2010

We prove the Strengthened Hanna Neumann Conjecture, using a common \bre product in graphs" formulation of the conjecture. Our original approach used sheaf theory, although here we give a proof with…

Extreme Rays of AND-Measures in Circuit Complexity

- Mathematics
- 2008

This paper is motivated by the problem of proving lower bounds on the formula size of boolean functions, which leads to lower bounds on circuit depth. We know that formula size is bounded from below…

A Proof of the Strengthened Hanna Neumann Conjecture

- Mathematics
- 2009

We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of “typical” covering maps, and use graph Galois theory to…

A sheaf-theoretic description of Khovanov's knot homology

- Mathematics
- 2012

We give a description of Khovanov's knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by…

Sheaves on Graphs and Their Homological Invariants

- Mathematics
- 2011

We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves.
One sheaf invariant, its "maximum excess," has a number of remarkable…

## References

SHOWING 1-10 OF 35 REFERENCES

Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems

- Mathematics, Computer ScienceSIAM J. Comput.
- 2001

The notion of a partially stable point in a reductive-group representation is introduced, which generalizes the notion of stability in geometric invariant theory due to Mumford and reduces fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points.

Lower bounds for algebraic computation trees

- Computer Science, MathematicsSTOC
- 1983

All the apparently known lower bounds for linear decision trees are extended to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20].

Grothendieck ring of pretriangulated categories

- Mathematics
- 2004

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated…

Combinatorial duality and intersection product: A direct approach

- Mathematics
- 2003

The proof of the Combinatorial Hard Lefschetz Theorem for the “virtual” intersection cohomology of a not necessarily rational polytopal fan as presented by Karu completely establishes Stanley’s…

Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 /64

- Mathematics
- 1966

The derived category.- Application to preschemes.- Duality for projective morphisms.- Local cohomology.- Dualizing complexes and local duality.- Residual complexes.- The duality theorem.

Lower Bounds for Algebraic Decision Trees

- Mathematics, Computer ScienceJ. Algorithms
- 1982

A topological method is given for obtaining lower bounds for the height of algebraic decision trees and an Ω(n2) bound is obtained for trees with bounded-degree polynomial tests, thus extending the Dobkin-Lipton result for linear trees.

Topological hypercovers and 1-realizations

- Mathematics
- 2004

Abstract.We show that if U* is a hypercover of a topological space X then the natural map hocolim U* → X is a weak equivalence. This fact is used to construct topological realization functors for the…

Natural Proofs

- MathematicsCOLT 1997
- 1997

We introduce the notion ofnaturalproof. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in nonmonotone models fall within our definition of natural. We…

Reconstruction of a Variety from the Derived Category and Groups of Autoequivalences

- MathematicsCompositio Mathematica
- 2001

We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also…

The complexity of Boolean functions

- Computer Science, Mathematics
- 1987

This chapter discusses Circuits and other Non-Uniform Computation Methods vs. Turing Machines and other Uniform Computation Models, and the Design of Efficient Circuits for Some Fundamental Functions.