131 research outputs found
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
Subspace-Invariant AC^0 Formulas
The n-variable PARITY function is computable (by a well-known recursive construction) by AC^0 formulas of depth d+1 and leaf size n2^{dn^{1/d}}. These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0,1}^n, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2^{d(n^{1/d}-1)} lower bound on the size of syntactically P-invariant depth d+1 formulas for PARITY. Quantitatively, this beats the best 2^{Omega(d(n^{1/d}-1))} lower bound in the non-invariant setting
An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity
Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k
Shrinkage of Decision Lists and DNF Formulas
We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the p-random restriction R_p for all values of p ? [0,1]. For a function f with domain {0,1}?, let DL(f) denote the minimum size of a decision list that computes f. We show that E[DL(f ? R_p)] ? DL(f)^log_{2/(1-p)}((1+p)/(1-p)). For example, this bound is ?{DL(f)} when p = ?5-2 ? 0.24. For Boolean functions f, we obtain the same shrinkage bound with respect to DNF formula size plus 1 (i.e., replacing DL(?) with DNF(?)+1 on both sides of the inequality)
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
Symmetric Formulas for Products of Permutations
We study the formula complexity of the word problem : given -by- permutation matrices
, compute the -entry of the matrix product . An important feature of this function is that it is invariant under
action of given by This symmetry is also exhibited in the smallest known unbounded fan-in
-formulas for
, which have size .
In this paper we prove a matching lower bound for
-invariant formulas computing . This result
is motivated by the fact that a similar lower bound for unrestricted
(non-invariant) formulas would separate complexity classes and
.
Our more general main theorem gives a nearly tight lower
bound on the -invariant depth-
-formula size of
for any finite simple group whose minimum permutation
representation has degree~. We also give nearly tight lower bounds on the
-invariant depth-
-formula size in the case where
is an abelian group.Comment: ITCS 202
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