9,552 research outputs found
On Symmetric Circuits and Fixed-Point Logics
We study properties of relational structures such as graphs that are decided
by families of Boolean circuits. Circuits that decide such properties are
necessarily invariant to permutations of the elements of the input structures.
We focus on families of circuits that are symmetric, i.e., circuits whose
invariance is witnessed by automorphisms of the circuit induced by the
permutation of the input structure. We show that the expressive power of such
families is closely tied to definability in logic. In particular, we show that
the queries defined on structures by uniform families of symmetric Boolean
circuits with majority gates are exactly those definable in fixed-point logic
with counting. This shows that inexpressibility results in the latter logic
lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Intersecting Families of Permutations
A set of permutations is said to be {\em k-intersecting} if
any two permutations in agree on at least points. We show that for any
, if is sufficiently large depending on , then the
largest -intersecting subsets of are cosets of stabilizers of
points, proving a conjecture of Deza and Frankl. We also prove a similar result
concerning -cross-intersecting subsets. Our proofs are based on eigenvalue
techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that
the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is
Theorem 27 for k > 1. An alternative proof of the equality part of the
Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and
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