24 research outputs found
Sub-linear Upper Bounds on Fourier dimension of Boolean Functions in terms of Fourier sparsity
We prove that the Fourier dimension of any Boolean function with Fourier
sparsity is at most . Our proof method yields an
improved bound of assuming a conjecture of
Tsang~\etal~\cite{tsang}, that for every Boolean function of sparsity there
is an affine subspace of of co-dimension O(\poly\log s)
restricted to which the function is constant. This conjectured bound is tight
upto poly-logarithmic factors as the Fourier dimension and sparsity of the
address function are quadratically separated. We obtain these bounds by
observing that the Fourier dimension of a Boolean function is equivalent to its
non-adaptive parity decision tree complexity, and then bounding the latter
The zero-error randomized query complexity of the pointer function
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to
prove separation results among various measures of complexity such as
deterministic, randomized and quantum query complexities, exact and approximate
polynomial degrees, etc. In particular, the widest possible (quadratic)
separations between deterministic and zero-error randomized query complexity,
as well as between bounded-error and zero-error randomized query complexity,
have been obtained by considering {\em
variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function.
However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the
precise zero-error complexity of the original pointer function was not known.
We show a lower bound of on the zero-error
randomized query complexity of the pointer function on bits;
since an upper bound is already known
\cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a
factor of \polylog\, n
Towards Better Separation between Deterministic and Randomized Query Complexity
We show that there exists a Boolean function which observes the following
separations among deterministic query complexity , randomized zero
error query complexity and randomized one-sided error query
complexity : and
. This refutes the conjecture made by Saks
and Wigderson that for any Boolean function ,
. This also shows widest separation between
and for any Boolean function. The function was defined by
G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation
between deterministic decision tree complexity and unambiguous
non-deterministic decision tree complexity. Independently of us, Ambainis et al
proved that different variants of the function certify optimal (quadratic)
separation between and , and polynomial separation between
and . Viewed as separation results, our results are subsumed
by those of Ambainis et al. However, while the functions considerd in the work
of Ambainis et al are different variants of , we work with the original
function itself.Comment: Reference adde
On Parity Decision Trees for Fourier-Sparse Boolean Functions
We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows. Let f : ??? ? {-1, 1} be a Boolean function with Fourier support ? and Fourier sparsity k.
- We prove via the probabilistic method that there exists a parity decision tree of depth O(?k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices.
- We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural "folding property", then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(? k). More concretely, the folding property we consider is that for most distinct ?, ? in ?, there are at least a polynomial (in k) number of pairs (?, ?) of parities in ? such that ?+? = ?+?. We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016).
- Motivated by the above, we present some structural results about the Fourier spectra of Boolean functions. It can be shown by elementary techniques that for any Boolean function f and all (?, ?) in binom(?,2), there exists another pair (?, ?) in binom(?,2) such that ? + ? = ? + ?. One can view this as a "trivial" folding property that all Boolean functions satisfy. Prior to our work, it was conceivable that for all (?, ?) ? binom(?,2), there exists exactly one other pair (?, ?) ? binom(?,2) with ? + ? = ? + ?. We show, among other results, that there must exist several ? ? ??? such that there are at least three pairs of parities (??, ??) ? binom(?,2) with ??+?? = ?. This, in particular, rules out the possibility stated earlier
Decision Tree Complexity versus Block Sensitivity and Degree
Relations between the decision tree complexity and various other complexity
measures of Boolean functions is a thriving topic of research in computational
complexity. It is known that decision tree complexity is bounded above by the
cube of block sensitivity, and the cube of polynomial degree. However, the
widest separation between decision tree complexity and each of block
sensitivity and degree that is witnessed by known Boolean functions is
quadratic. In this work, we investigate the tightness of the existing cubic
upper bounds.
We improve the cubic upper bounds for many interesting classes of Boolean
functions. We show that for graph properties and for functions with a constant
number of alternations, both of the cubic upper bounds can be improved to
quadratic. We define a class of Boolean functions, which we call the zebra
functions, that comprises Boolean functions where each monotone path from 0^n
to 1^n has an equal number of alternations. This class contains the symmetric
and monotone functions as its subclasses. We show that for any zebra function,
decision tree complexity is at most the square of block sensitivity, and
certificate complexity is at most the square of degree.
Finally, we show using a lifting theorem of communication complexity by
G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper
bound on the decision tree complexity for all functions is in a sense
equivalent to the potentially easier task of proving a similar upper bound on
communication complexity for each bi-partition of the input variables, for all
functions. In particular, this implies that to bound the decision tree
complexity it suffices to bound smaller measures like parity decision tree
complexity, subcube decision tree complexity and decision tree rank, that are
defined in terms of models that can be efficiently simulated by communication
protocols
A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity
For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)})where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g
Linear Sketching over F_2
We initiate a systematic study of linear sketching over F_2. For a given Boolean function treated as f : F_2^n -> F_2 a randomized F_2-sketch is a distribution M over d x n matrices with elements over F_2 such that Mx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms.
Motivated by these applications we study a connection between F_2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F_2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F_2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F_2-sketching is optimal (up to constant factors) for uniformly distributed inputs.
Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F_2 can be constructed as F_2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC\u2714) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates
Lifting to Parity Decision Trees via Stifling
We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f ◦ g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(☉), of the unsatisfiability of closely related constant-width CNF formulas
Quadratically Tight Relations for Randomized Query Complexity
Let be a Boolean function. The certificate
complexity is a complexity measure that is quadratically tight for the
zero-error randomized query complexity : . In this paper we study a new complexity measure that we call
expectational certificate complexity , which is also a quadratically
tight bound on : . We prove that and show that there is a quadratic separation between
the two, thus gives a tighter upper bound for . The measure is
also related to the fractional certificate complexity as follows:
. This also connects to an open question by
Aaronson whether is a quadratically tight bound for , as
is in fact a relaxation of .
In the second part of the work, we upper bound the distributed query
complexity for product distributions by the square of
the query corruption bound () which improves upon a
result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for
communication complexity is open.Comment: 14 page