Fooling predators

Fooling-sets and rank

ABSTRACT. An n×n matrix M is called a fooling-set matrix of size n if its diagonal entries are nonzero and Mk,lMl,k = 0 for every k 6= l. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n ≤ (rkM)2 , regardless of over which field the rank is computed, and asked whether the exponent on rkM can be improved. We settle this question. In characteristic zero, we construct an infinite fami...

Neoatherosclerosis: fooling mother nature.

Rank and fooling set size

Say that A is a Hadamard factorization of the identity In of size n if A ◦AT = In, where ◦ denotes the Hadamard or entrywise product, and AT denotes the transpose of A. As n = rk(In) = rk(A ◦ AT ) ≤ rk(A)2, it is clear that the rank of any Hadamard factorization of the identity must be at least √ n. Dietzfelbinger et al. [DHS96] raised the question if this bound can be achieved, and showed a bo...

Fooling One-Sided Quantum Protocols

We use the venerable “fooling set” method to prove new lower bounds on the quantum communication complexity of various functions. Let f : X × Y → {0, 1} be a Boolean function, fool1(f) its maximal fooling set size among 1-inputs, Q1(f) its one-sided-error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior enta...

Transplantation tolerance: fooling mother nature.

Foreignness is threatening. The immune system, the primary arbiter of self and nonself, will reject an organ transplant unless constrained by life-long administration of multiple immune suppressive drugs. Immunosuppressive therapy, while facilitating graft success, heightens susceptibility to infection and the development of malignancy. The challenge then is to educate the immune system to tole...