Revision #1 Authors: Arnab Bhattacharyya, Elena Grigorescu, Jakob NordstrÃ¶m, Ning Xie

Accepted on: 27th August 2010 19:39

Downloads: 1582

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Properties of Boolean functions on the hypercube that are invariant with respect to linear transformations of the domain are among some of the most well-studied properties in the context of property testing. In this paper, we study the fundamental class of linear-invariant properties called matroid freeness properties. These properties have been conjectured to essentially coincide with all testable linear-invariant properties, and a recent sequence of works has established testability for increasingly larger subclasses of matroid freeness properties. One question that has been left open, however, is whether the infinitely many syntactically different matroid freeness properties recently shown to be testable in fact correspond to new, semantically distinct properties. This is a crucial issue since it has also been shown previously that there exist subclasses of matroid freeness properties for which an infinite set of syntactically different representations collapse into one of a small, finite set of properties, all previously known to be testable.

An important question is therefore to understand the semantics of matroid freeness properties, and in particular when two syntactically different properties are truly distinct. We shed light on this problem by developing a method for determining the relation between two matroid freeness properties P and Q. Furthermore, we show that there is a natural subclass of matroid freeness properties such that for any two properties P and Q from this subclass, a strong dichotomy must hold: either P is contained in Q or the two properties are "well separated" from one another. As an application of this method, we exhibit new, infinite hierarchies of testable matroid freeness properties such that at each level of the hierarchy, there are explicit functions that are far in Hamming distance from all functions lying in the lower levels of the hierarchy. Our key technical tool is an apparently new notion of maps between linear matroids, which we call labeled matroid homomorphisms, that might be of independent interest.

Mostly fixing of typos and corrections of some minor bugs.

TR10-136 Authors: Arnab Bhattacharyya, Elena Grigorescu, Jakob NordstrÃ¶m, Ning Xie

Publication: 26th August 2010 13:07

Downloads: 1827

Keywords:

Properties of Boolean functions on the hypercube that are invariant

with respect to linear transformations of the domain are among some of

the most well-studied properties in the context of property testing.

In this paper, we study a particular natural class of linear-invariant

properties, called matroid freeness properties. These properties have

all been conjectured to be testable, and a recent sequence of works

have established testability for increasingly larger subclasses of

matroid freeness properties. One question that has been left open,

however, is whether the infinitely many syntactically different

matroid freeness properties recently shown to be testable in fact

correspond to new, semantically distinct properties. This is a crucial

issue since it has also been shown previously that there exist

subclasses of matroid freeness properties for which an infinite set of

syntactically different representations collapse into one of a small,

finite set of properties, all previously known to be testable.

An important question is therefore to understand the semantics of

matroid freeness properties, and in particular when two syntactically

different properties are truly distinct. We shed light on this problem

by developing a method for determining the relation between two

matroid freeness properties P and Q. Furthermore, we show that there

is a natural subclass of matroid freeness properties such that for any

two properties P and Q from this subclass, a strong dichotomy must

hold: either P is contained in Q or the two properties are "well

separated" from one another. As an application of this method, we

exhibit new, infinite hierarchies of testable matroid freeness

properties such that at each level of the hierarchy, there are

explicit functions that are far in Hamming distance from all functions

lying in the lower levels of the hierarchy. Our key technical tool is

an apparently new notion of maps between linear matroids, which we

call labeled matroid homomorphisms, that might be of independent

interest.